1. Introduction

Theory of multisets is an important generalization of classical set theory which has emerged by violating a basic property of classical sets that an element can belong to a set only once. The term multiset (mset in short) as Knuth [1] notes, was first suggested by De Bruijn [2] in a private communication to him. The concept of fuzzy sets proposed by Zadeh [3] is a mathematical tool for representing vague concepts. The idea of fuzzy multisets was conceived by Yager [4] as the generalization of fuzzy sets in multisets framework.

The concept of fuzzy multigroups was introduced by Shinoj \emph{et al.,} [5] as an application of fuzzy multisets to group theory, and some properties of fuzzy multigroups were presented. Ejegwa introduced the concept of fuzzy multigroupoids and presented the idea of fuzzy submultigroups with a number of results and more properties of abelian fuzzy multigroups were explicated [6, 7, 8]. Also Ejegwa introduced direct product in fuzzy multigroup setting as an extension of direct product of fuzzy subgroups [9]. In mathematics, a t-norm (also \( T \)-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A \( t\)-norm generalizes intersection in a lattice and conjunction in logic. The name triangular norm refers to the fact that in the framework of probabilistic metric spaces \( t\)-norms are used to generalize triangle inequality of ordinary metric spaces. The author by using norms, investigated some properties of fuzzy algebraic structures [10, 11, 12, 13, 14, 15]. The author [15] defined fuzzy multigroups under t-norms and some properties of them are explored and some related results are obtained.

In this paper, we introduce the concept of direct product of fuzzy multigroups under \(t\)-norms \((TFSM(G))\) and investigate some properties and results about them. We prove that direct products of \(TFSM(G)\) are also \(TFSM(G).\) Next we investigate and obtain some new results of strong upper alpha-cut, weak upper alpha-cut, strong lower alpha-cut and weak lower alpha-cut of direct product of fuzzy Multigroups under \(t\)-norms. Later we prove that if \( A,C \in TFMS(G)\) and \( B,D \in TFMS(H)\) such that \(A\) is conjugate to \(B\) and \(C\) is conjugate to \( D,\) then \(A\times C \) is conjugate to \( B \times D.\) Also \( A\) and \( B\) are commutative if and only if \( A \times B \) is a commutative. Finally, we define group homomorphisms on direct propduct of fuzzy multigroups under t-norms and we prove that image and pre image of direct propduct of fuzzy multigroups under t-norms is also fuzzy multigroups under t-norms.

2. Preliminaries

This section contains some basic definitions and preliminary results which will be needed in the sequel. For details we refer to [15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

Definition 1. Let \(G\) be an arbitrary group with a multiplicative binary operation and identity \(e\). A fuzzy subset of \(G\), we mean a function from \(G\) into \( [0,1]\). The set of all fuzzy subsets of \(G\) is called the \( [0,1]\)-power set of \(G\) and is denoted \( [0,1]^G.\)

Definition 2. Let \( X\) be a set. A fuzzy multiset \( A\) of \( X\) is characterized by a count membership function \[CM_{A} : X \to [0, 1]\] of which the value is a multiset of the unit interval \(I = [0, 1].\) That is, \[CM_{A}(x) = \lbrace \mu^{1}, \mu^{2}, ... , \mu^{n}, ... \rbrace \forall x \in X,\] where \( \mu^{1}, \mu^{2}, ... , \mu^{n}, ... \in [0,1]\) such that \[(\mu^{1} \geq \mu^{2} \geq ... \geq \mu^{n} \geq ... ).\] Whenever the fuzzy multiset is finite, we write \[CM_{A}(x) = \lbrace \mu^{1}, \mu^{2}, ... , \mu^{n} \rbrace,\] where \( \mu^{1}, \mu^{2}, ... , \mu^{n} \in [0,1]\) such that \[(\mu^{1} \geq \mu^{2} \geq ... \geq \mu^{n} ),\] or simply \[CM_{A}(x) = \lbrace \mu^{i} \rbrace,\] for \( \mu^{i} \in [0,1]\) and \(i = 1, 2, ... ,n.\) Now, a fuzzy multiset \( A\) is given as \[ A= \left\lbrace \dfrac{CM_{A}(x)}{x} : x \in X \right\rbrace \;\;\text{or}\;\; A= \left\lbrace (CM_{A}(x),x) : x \in X \right\rbrace. \] The set of all fuzzy multisets is depicted by \(FMS(X).\)

Example 1. Consider the set \(X = \lbrace a, b, c \rbrace \). Then for \( CM_{A}(a) = \lbrace 1, 0.5, 0.4 \rbrace \), \( CM_{A}(b) = \lbrace 0.9, 0.6 \rbrace \) and \( CM_{A}(c) = \lbrace 0 \rbrace \) we get that \( A\) is a fuzzy multiset of \( X\) written as \(A= \left\lbrace \dfrac{1, 0.5, 0.4}{a}, \dfrac{0.9, 0.6}{b} \right\rbrace.\)

Definition 3. Let \(A, B \in FMS(X).\) Then \( A\) is called a fuzzy submultiset of \( B\) written as \(A \subseteq B\) if \(CM_{A}(x) \leq CM_{B}(x)\) for all \(x \in X.\) Also, if \(A \subseteq B\) and \(A \neq B,\) then \( A\) is called a proper fuzzy submultiset of \( B\) and denoted as \(A \subset B.\)

Definition 4. Let \(A \in FMS(X)\) and \( \alpha \in [0,1]. \) Then we define the following notions:

• \( A_{\star}=\lbrace x \in X \hspace{0.1cm}|\hspace{0.1cm} CM_{A}(x)>0 \rbrace.\)
• \( A^{\star}=\lbrace x \in X \hspace{0.1cm}|\hspace{0.1cm} CM_{A}(x)=CM_{A}(e_{X}) \rbrace \) where \( e_{X}\) is the identity element of \(X. \)
• \( A_{[\alpha]}=\lbrace x \in X \hspace{0.1cm}|\hspace{0.1cm} CM_{A}(x) \geq \alpha \rbrace \) is called strong upper alpha-cut of \(A.\)
• \( A_{(\alpha)}=\lbrace x \in X \hspace{0.1cm}|\hspace{0.1cm} CM_{A}(x) > \alpha \rbrace \) is called weak upper alpha-cut of \(A.\)
• \( A^{[\alpha]}=\lbrace x \in X \hspace{0.1cm}|\hspace{0.1cm} CM_{A}(x) \leq \alpha \rbrace \) is called strong lower alpha-cut of \(A.\)
• \( A^{(\alpha)}=\lbrace x \in X \hspace{0.1cm}|\hspace{0.1cm} CM_{A}(x) < \alpha \rbrace \) is called weak lower alpha-cut of \(A.\)

Definition 5. Let \(A, B \in FMG(X).\) We say that \( A\) is conjugate to \( B\) if for all \( x,y \in X \) we have \( CM_{A}(x)=CM_{B}\left(yxy^{-1}\right).\)

Definition 6. Let \(A\in FMG(X).\) We say that \( A\) is commutative if \( CM_{A}(xy)=CM_{A}(yx) \) for all \( x,y \in X.\)

Definition 7. A \(t\)-norm \(T\) is a function \(T : [0,1]\times [0,1] \to [0,1]\) having the following four properties:
(T1) \(T(x,1)=x\) (neutralelement),
(T2) \(T(x,y)\leq T(x,z)\) if \(y\leq z\) (monotonicity),
(T3) \(T(x,y)= T(y,x)\) (commutativity),
(T4) \(T(x,T(y,z))=T(T(x,y),z)\) (associativity),\\ for all \(x,y,z \in[0,1].\)
We say that \(T\) be idempotent if \(T(x,x)=x\) for all \(x\in [0,1].\)

It is clear that if \(x_{1}\geq x_{2}\) and \(y_{1}\geq y_{2}\), then \(T(x_{1},y_{1}) \geq T(x_{2},y_{2}).\)

Example 2. (1) Standard intersection \(t\)-norm \(T_m(x,y) = \min \{ x,y \}.\)
(2) Bounded sum \(t\)-norm \(T_b(x,y) =\max\{0, x+y- 1 \}.\)
(3) Algebraic product \(t\)-norm \(T_p(x, y) = xy. \)
(4) Drastic \(T\)-norm \(T_{D}(x,y) = \left\{ \begin{array}{rl} y &\text{if } x=1\\ x &\text{if } y=1\\ 0 &\text{otherwise. } \\ \end{array} \right. \)
(5) Nilpotent minimum \(t\)-norm \(T_{nM}(x,y) = \left\{ \begin{array}{rl} \min \lbrace x , y \rbrace &\text{if } x+y >1\\ 0 &\text{otherwise. } \\ \end{array} \right. \)
(6) Hamacher product \(t\)-norm \(T_{H_{0}}(x,y) = \left\{ \begin{array}{rl} 0 &\text{if } x=y =0\\ \frac{xy}{x+y-xy} &\text{otherwise. } \\ \end{array} \right. \)

The drastic \(t\)-norm is the pointwise smallest \(t\)-norm and the minimum is the pointwise largest \(t\)-norm: \(T_{D}(x,y) \leq T(x,y) \leq T_{\min} (x ,y)\) for all \( x,y \in [0,1].\)

Lemma 1. Let \(T\) be a \(t\)-norm. Then \[T(T(x,y),T(w,z))= T(T(x,w),T(y,z)),\] for all \(x,y,w,z\in [0,1].\)

Definition 8. Let \(A \in FMS(G).\) Then \( A \) is said to be a fuzzy multigroup of \( G\) under \(t\)-norm \(T\) if it satisfies the following two conditions:
(1) \(CM_{A}(xy) \geq T(CM_{A}(x),CM_{A}(y)),\)
(2) \(CM_{A}(x^{-1}) \geq CM_{A}(x),\) for all \(x,y\in G.\)
The set of all fuzzy multisets of \( G\) under \(t\)-norm \(T\) is depicted by \(TFMS(G).\)

Theorem 1. Let \( A \in TFMS(G).\) If \(T\) be idempotent, then for all \(x \in G \), we have and \(n\geq 1,\)
(1) \(CM_{A}(e)\geq CM_{A}(x)\);
(2) \(CM_{A}(x^n)\geq CM_{A}(x);\)
(3) \(CM_{A}(x)=CM_{A}(x^{-1}).\)

3. Direct product of fuzzy multigroups under \(t\)-norms

Definition 9. Let \( A \in TFMS(G)\) and \( B \in TFMS(H).\) The direct product of \(A\) and \(B\), denoted by \(A \times B\), is characterized by a count membership function \[CM_{A \times B} : G \times H \to [0, 1]\] such that \[CM_{A \times B}(x,y)=T(CM_{A}(x) ,CM_{B}(y))\] for all \( x \in G \) and \( y \in H. \)

Example 3. Let \(G = \left\lbrace 1, x \right\rbrace\) be a group, where \(x^{2} = 1\) and \(H =\lbrace e, a, b, c \rbrace\) be a Klein 4-group, where \(a^{2} = b^{2} = c^{2}= e.\) Suppose \[A= \left\lbrace \dfrac{0.9,0.8}{1}, \dfrac{0.7, 0.6}{x} \right\rbrace \] and \[B= \left\lbrace \dfrac{1, 0.85}{e}, \dfrac{0.35, 0.25}{a}, \dfrac{0.10, 0.50}{b}, \dfrac{0.8, 0.6}{c} \right\rbrace\] be fuzzy multigroups of \( G\) and \( H.\) Let \[G \times H=\left\lbrace (1,e),(1,a),(1,b),(1,c),(x,e),(x,a),(x,b),(x,c) \right\rbrace\] be a a group from the classical sense. Define \[A \times B= \left\lbrace \dfrac{0.9, 0.8}{(1,e)}, \dfrac{0.35, 0.25}{(1,a)}, \dfrac{0.10, 0.50}{(1,b)}, \dfrac{0.8, 0.6}{(1,c)}, \dfrac{0.7, 0.6}{(x,e)}, \dfrac{0.35, 0.25}{(x,a)}, \dfrac{0.10, 0.50}{(x,b)}, \dfrac{0.7, 0.6}{(x,c)} \right\rbrace \] and let \(T_m(x,y) = \min \{ x,y \}\) be a standard intersection \(t\)-norm for all \( x,y \in [0,1]. \) Then \[ A \times B \in TFMS(G \times H). \]

Proposition 1. Let \(A_{i}\in TFMS(G_{i})\) for \(i=1,2.\) Then \(A_{1}\times A_{2}\in TFMS(G_{1}\times G_{2}).\)

Proof. Let \((a_{1},b_{1}),(a_{2},b_{2})\in G_{1}\times G_{2}.\) Then \begin{align*} (CM_{A \times B})((a_{1},b_{1})(a_{2},b_{2}))&=(CM_{A \times B})(a_{1}a_{2},b_{1}b_{2}) =T(CM_{A}(a_{1}a_{2}),CM_{B}(b_{1}b_{2}))\\ &\geq T(T(CM_{A}(a_{1}),CM_{A}(a_{2})),T(CM_{B}(b_{1}),CM_{B}(b_{2})))\\ &=T(T(CM_{A}(a_{1}),CM_{B}(b_{1}),T(CM_{A}(a_{2}),CM_{B}(b_{2}))\\ &=T((CM_{A \times B})(a_{1},b_{1}),(CM_{A \times B})(a_{2},b_{2})). \end{align*} Also \begin{align*} (CM_{A \times B})(a_{1},b_{1})^{-1}&= (CM_{A \times B})(a_{1}^{-1},b_{1}^{-1})\\&= T(CM_{A}(a_{1}^{-1}),CM_{B}(b_{1}^{-1}))\geq T(CM_{A}(a_{1}),CM_{B}(b_{1})). \end{align*} Thus \(A_{1}\times A_{2}\in TFMS(G_{1}\times G_{2}).\)

Corollary 1. Let \( A \in TFMS(G)\) and \( B \in TFMS(H).\) Then \[A \times 1_{H},1_{G} \times B \in TFMS(G\times H).\]

Corollary 2. Let \(A_{i}\in TFMS(G_{i})\) for \(i=1,2,...,n\). Then \[A_{1}\times A_{2}\times... \times A_{n} \in TFMS(G_{1}\times G_{2}\times... \times G_{n}).\]

Proposition 2. Let \( A \in TFMS(G)\) and \( B \in TFMS(H)\) such that \(T\) be idempotent \(t\)-norm. Then for all \( \alpha \in [0,1]\) the following assertions hold:
(1) \( (A \times B)_{\star} =A_{\star} \times B_{\star}.\)
(2) \( (A \times B)^{\star} =A^{\star} \times B^{\star}.\)
(3) \( (A \times B)_{[\alpha]} =A_{[\alpha]} \times B_{[\alpha]}.\)
(4) \( (A \times B)_{(\alpha)} =A_{(\alpha)} \times B_{(\alpha)}.\)
(5) \( (A \times B)^{[\alpha]} =A^{[\alpha]} \times B^{[\alpha]}.\)
(6) \( (A \times B)^{(\alpha)} =A^{(\alpha)} \times B^{(\alpha)}.\)

Proof.

• (1) We know that \( (A \times B)_{\star} = \lbrace (x,y) \in G \times H \hspace{0.1cm}|\hspace{0.1cm} CM_{A \times B}(x,y)>0 \rbrace. \) Then \( (x,y) \in (A \times B)_{\star} \Longleftrightarrow CM_{A \times B}(x,y) >0 \Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) > 0\Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) > 0=T(0,0) \Longleftrightarrow CM_{A}(x)>0 \hspace{0.1cm} \textit{and} \hspace{0.1cm} CM_{B}(y)>0\Longleftrightarrow x\in A_{\star} \hspace{0.1cm} \textit{and} \hspace{0.1cm} y\in B_{\star} \Longleftrightarrow (x,y) \in A_{\star} \times B_{\star}\,. \) Hence \( (A \times B)_{\star} =A_{\star} \times B_{\star}.\)
• (2) As \( (A \times B)^{\star} = \lbrace (x,y) \in G \times H \hspace{0.1cm}|\hspace{0.1cm} CM_{A \times B}(x,y)= CM_{A \times B}(e_{G},e_{H}) \rbrace \) so \( (x,y) \in (A \times B)^{\star} \Longleftrightarrow CM_{A \times B}(x,y) =CM_{A \times B}(e_{G},e_{H})\Longleftrightarrow T(CM_{A}(x),CM_{B}(y))=T(CM_{A}(e_{G}),CM_{B}(e_{H}))\Longleftrightarrow CM_{A}(x)=CM_{A}(e_{G}) \hspace{0.1cm} \textit{and} \hspace{0.1cm} CM_{B}(y)=CM_{B}(e_{H})\Longleftrightarrow x\in A^{\star} \hspace{0.1cm} \textit{and} \hspace{0.1cm} y\in B^{\star} \Longleftrightarrow (x,y) \in A^{\star} \times B^{\star}\,. \) Thus \( (A \times B)^{\star} =A^{\star} \times B^{\star}.\)
• (3) Let \( (A \times B)_{[\alpha]} = \lbrace (x,y) \in G \times H \hspace{0.1cm}|\hspace{0.1cm} CM_{A \times B}(x,y) \geq \alpha \rbrace. \) Now \( (x,y) \in (A \times B)_{[\alpha]} \Longleftrightarrow CM_{A \times B}(x,y) \geq \alpha \Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) \geq \alpha\Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) \geq \alpha=T(\alpha,\alpha) \Longleftrightarrow CM_{A}(x)\geq \alpha \hspace{0.1cm} \textit{and} \hspace{0.1cm} CM_{B}(y)\geq \alpha\Longleftrightarrow x\in A_{[\alpha]} \hspace{0.1cm} \textit{and} \hspace{0.1cm} y\in B_{[\alpha]} \Longleftrightarrow (x,y) \in A_{[\alpha]} \times B_{[\alpha]}\,. \) Thus \( (A \times B)_{[\alpha]} =A_{[\alpha]} \times B_{[\alpha]}.\)
• (4) Since \( (A \times B)_{(\alpha)} = \lbrace (x,y) \in G \times H \hspace{0.1cm}|\hspace{0.1cm} CM_{A \times B}(x,y) > \alpha \rbrace\), so \( (x,y) \in (A \times B)_{(\alpha)} \Longleftrightarrow CM_{A \times B}(x,y) > \alpha \Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) > \alpha\Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) > \alpha=T(\alpha,\alpha) \Longleftrightarrow CM_{A}(x) > \alpha \hspace{0.1cm} \textit{and} \hspace{0.1cm} CM_{B}(y) > \alpha\Longleftrightarrow x\in A_{(\alpha)} \hspace{0.1cm} \textit{and} \hspace{0.1cm} y\in B_{(\alpha)} \Longleftrightarrow (x,y) \in A_{(\alpha)} \times B_{(\alpha)}\,. \) So \( (A \times B)_{(\alpha)} =A_{(\alpha)} \times B_{(\alpha)}.\)
• (5) Because \( (A \times B)^{[\alpha]} = \lbrace (x,y) \in G \times H \hspace{0.1cm}|\hspace{0.1cm} CM_{A \times B}(x,y) \leq \alpha \rbrace\), then \( (x,y) \in (A \times B)^{[\alpha]} \Longleftrightarrow CM_{A \times B}(x,y) \leq \alpha \Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) \leq \alpha(x,y) \in (A \times B)^{[\alpha]} \Longleftrightarrow CM_{A \times B}(x,y) \leq \alpha \Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) \leq \alpha\Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) \leq \alpha=T(\alpha,\alpha) \Longleftrightarrow CM_{A}(x) \leq \alpha \hspace{0.1cm} \textit{and} \hspace{0.1cm} CM_{B}(y) \leq \alpha\Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) \leq \alpha=T(\alpha,\alpha) \Longleftrightarrow CM_{A}(x) \leq \alpha \hspace{0.1cm} \textit{and} \hspace{0.1cm} CM_{B}(y) \leq \alpha\Longleftrightarrow x\in A^{[\alpha]} \hspace{0.1cm} \textit{and} \hspace{0.1cm} y\in B^{[\alpha]} \Longleftrightarrow (x,y) \in A^{[\alpha]} \times B^{[\alpha]}\,. \) Therefore \( (A \times B)^{[\alpha]} =A^{[\alpha]} \times B^{[\alpha]}. \)
• (6) Because of \( (A \times B)^{(\alpha)} = \lbrace (x,y) \in G \times H \hspace{0.1cm}|\hspace{0.1cm} CM_{A \times B}(x,y) < \alpha \rbrace\), then \( (x,y) \in (A \times B)^{(\alpha)} \Longleftrightarrow CM_{A \times B}(x,y) < \alpha \Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) < \alpha \Longleftrightarrow T(CM_{A}(x),CM_{B}(y)) < \alpha=T(\alpha,\alpha) \Longleftrightarrow CM_{A}(x) < \alpha \hspace{0.1cm} \textit{and} \hspace{0.1cm} CM_{B}(y) < \alpha\Longleftrightarrow x\in A^{(\alpha)} \hspace{0.1cm} \textit{and} \hspace{0.1cm} y\in B^{(\alpha)} \Longleftrightarrow (x,y) \in A^{(\alpha)} \times B^{(\alpha)}\,. \) Hence \( (A \times B)^{(\alpha)} =A^{(\alpha)} \times B^{(\alpha)}.\)

Proposition 3. Let \( A \in TFMS(G)\) and \( B \in TFMS(H)\) such that \(T\) be idempotent \(t\)-norm. Then for all \((x,y) \in G \times H\) the following assertions hold:
(1) \( CM_{A \times B}(e_{G},e_{H}) \geq CM_{A \times B}(x,y),\)
(2) \( CM_{A \times B}((x,y)^{n}) \geq CM_{A \times B}(x,y), \)
(3) \(CM_{A \times B}(x,y)=CM_{A \times B}(x^{-1},y^{-1}).\)

Proof. Using Proposition \ref{prop1} we get that \( A \times B \in TFMS(G \times H).\) Now Theorem \ref{thm1} gives us that assertions are hold.

Proposition 4. Let \( A \in TFMS(G)\) and \( B \in TFMS(H)\) such that \(T\) be idempotent \(t\)-norm. Then for all \( \alpha \in [0,1]\) the following assertions hold:
(1) \( (A \times B)_{\star}\) is a subgroup of \(G \times H,\)
(2) \( (A \times B)^{\star} \) is a subgroup of \(G \times H,\)
(3) \( (A \times B)_{[\alpha]}\) is a subgroup of \(G \times H,\)
(4) \( (A \times B)_{(\alpha)}\) is a subgroup of \(G \times H.\)

Proof. (1) Let \( (x_{1},y_{1}),(x_{2},y_{2}) \in (A \times B)_{\star}\). We need to prove that \( (x_{1},y_{1})(x_{2},y_{2})^{-1} \in (A \times B)_{\star}.\) As \( (x_{1},y_{1}),(x_{2},y_{2}) \in (A \times B)_{\star}\), so \(CM_{A \times B}(x_{1},y_{1}) >0\) and \(CM_{A \times B}(x_{2},y_{2}) >0.\) Now \begin{align*} CM_{A \times B} ((x_{1},y_{1})(x_{2},y_{2})^{-1}) &=CM_{A \times B} ((x_{1},y_{1})(x^{-1}_{2},y^{-1}_{2}))\\&= CM_{A \times B} (x_{1}x^{-1}_{2},y_{1}y^{-1}_{2})=T(CM_{A}(x_{1}x^{-1}_{2}),CM_{B}(y_{1}y^{-1}_{2}))\\&\geq T(T(CM_{A}(x_{1}),CM_{A}(x^{-1}_{2})),T(CM_{B}(y_{1}),CM_{B}(y^{-1}_{2})))\\&\geq T(T(CM_{A}(x_{1}),CM_{A}(x_{2})),T(CM_{B}(y_{1}),CM_{B}(y_{2})))\\&=T(T(CM_{A}(x_{1}),CM_{B}(y_{1})),T(CM_{A}(x_{2}),CM_{B}(y_{2})))\\&=T(CM_{A \times B}(x_{1},y_{1}),CM_{A \times B}(x_{2},y_{2}))>T(0,0)=0\,. \end{align*} Thus \( CM_{A \times B} ((x_{1},y_{1})(x_{2},y_{2})^{-1}) >0 \), which means that \((x_{1},y_{1})(x_{2},y_{2})^{-1} \in (A \times B)_{\star}.\) Hence \( (A \times B)_{\star}\) is a subgroup of \(G \times H.\)
(2) Let \( (x_{1},y_{1}),(x_{2},y_{2}) \in (A \times B)_{\star}\). We need to prove that \( (x_{1},y_{1})(x_{2},y_{2})^{-1} \in (A \times B)^{\star}.\) Because \( (x_{1},y_{1}),(x_{2},y_{2}) \in (A \times B)^{\star}\) then \( CM_{A \times B}(x_{1},y_{1})=CM_{A \times B}(x_{2},y_{2}) =CM_{A \times B}(e_{G},e_{H})\), which means that \( T(CM_{A}(x_{1}),CM_{B}(y_{1}))=T(CM_{A}(x_{2}),CM_{B}(y_{2}))=T(CM_{A}(e_{G}),CM_{B}(e_{H})) \), so \(CM_{A}(x_{1})=CM_{A}(x_{2})=CM_{A}(e_{G}) \) and \(CM_{A}(y_{1})=CM_{A}(y_{2})=CM_{A}(e_{H}).\) Thus \begin{align*} CM_{A \times B} ((x_{1},y_{1})(x_{2},y_{2})^{-1}) &=CM_{A \times B} ((x_{1},y_{1})(x^{-1}_{2},y^{-1}_{2}))\\ &= CM_{A \times B} (x_{1}x^{-1}_{2},y_{1}y^{-1}_{2})=T(CM_{A}(x_{1}x^{-1}_{2}),CM_{B}(y_{1}y^{-1}_{2})) \\ &\geq T(T(CM_{A}(x_{1}),CM_{A}(x^{-1}_{2})),T(CM_{B}(y_{1}),CM_{B}(y^{-1}_{2})))\\ &\geq T(T(CM_{A}(x_{1}),CM_{A}(x_{2})),T(CM_{B}(y_{1}),CM_{B}(y_{2})))\\ &=T(T(CM_{A}(e_{G}),CM_{A}(e_{G})),T(CM_{B}(e_{H}),CM_{B}(e_{H})))\\ &=T(CM_{A}(e_{G}),CM_{B}(e_{H}))=CM_{A \times B}(e_{G},e_{H})\\ &\geq CM_{A \times B} ((x_{1},y_{1})(x_{2},y_{2})^{-1})\,. \hspace{0.1cm} \textit{(Proposition \ref{prop2}(1))} \end{align*}
Thus \(CM_{A \times B} ((x_{1},y_{1})(x_{2},y_{2})^{-1})=CM_{A \times B}(e_{G},e_{H})\), so \( (x_{1},y_{1})(x_{2},y_{2})^{-1} \in (A \times B)^{\star}.\) Hence we obtain that \( (A \times B)^{\star} \) is a subgroup of \(G \times H.\)
(3) Let \( (x_{1},y_{1}),(x_{2},y_{2}) \in (A \times B)_{[\alpha]}\). We need to show that \( (x_{1},y_{1})(x_{2},y_{2})^{-1} \in (A \times B)_{[\alpha]}.\) As \( (x_{1},y_{1}),(x_{2},y_{2}) \in (A \times B)_{[\alpha]}\) so \(CM_{A \times B}(x_{1},y_{1}) \geq \alpha\) and \(CM_{A \times B}(x_{2},y_{2}) \geq \alpha.\) Now \begin{align*} CM_{A \times B} ((x_{1},y_{1})(x_{2},y_{2})^{-1}) &=CM_{A \times B} ((x_{1},y_{1})(x^{-1}_{2},y^{-1}_{2}))\\ &= CM_{A \times B} (x_{1}x^{-1}_{2},y_{1}y^{-1}_{2})=T(CM_{A}(x_{1}x^{-1}_{2}),CM_{B}(y_{1}y^{-1}_{2}))\\ &\geq T(T(CM_{A}(x_{1}),CM_{A}(x^{-1}_{2})),T(CM_{B}(y_{1}),CM_{B}(y^{-1}_{2})))\\ &\geq T(T(CM_{A}(x_{1}),CM_{A}(x_{2})),T(CM_{B}(y_{1}),CM_{B}(y_{2})))\\ &=T(T(CM_{A}(x_{1}),CM_{B}(y_{1})),T(CM_{A}(x_{2}),CM_{B}(y_{2})))\\ &=T(CM_{A \times B}(x_{1},y_{1}),CM_{A \times B}(x_{2},y_{2})) \geq T(\alpha,\alpha)=\alpha\,. \end{align*} Thus \( CM_{A \times B} ((x_{1},y_{1})(x_{2},y_{2})^{-1}) \geq \alpha\) which means that \((x_{1},y_{1})(x_{2},y_{2})^{-1} \in (A \times B)_{[\alpha]}.\) Hence \( (A \times B)_{[\alpha]}\) is a subgroup of \(G \times H.\)
(4) If \( (x_{1},y_{1}),(x_{2},y_{2}) \in (A \times B)_{(\alpha)},\) then \(CM_{A \times B}(x_{1},y_{1}) > \alpha\) and \(CM_{A \times B}(x_{2},y_{2}) > \alpha.\) Now \begin{align*} CM_{A \times B} ((x_{1},y_{1})(x_{2},y_{2})^{-1}) &=CM_{A \times B} ((x_{1},y_{1})(x^{-1}_{2},y^{-1}_{2}))\\ &= CM_{A \times B} (x_{1}x^{-1}_{2},y_{1}y^{-1}_{2})=T(CM_{A}(x_{1}x^{-1}_{2}),CM_{B}(y_{1}y^{-1}_{2}))\\ &\geq T(T(CM_{A}(x_{1}),CM_{A}(x^{-1}_{2})),T(CM_{B}(y_{1}),CM_{B}(y^{-1}_{2}))) \\ &\geq T(T(CM_{A}(x_{1}),CM_{A}(x_{2})),T(CM_{B}(y_{1}),CM_{B}(y_{2})))\\ &=T(T(CM_{A}(x_{1}),CM_{B}(y_{1})),T(CM_{A}(x_{2}),CM_{B}(y_{2})))\\ &=T(CM_{A \times B}(x_{1},y_{1}),CM_{A \times B}(x_{2},y_{2})) > T(\alpha,\alpha)=\alpha\,. \end{align*} Thus \( CM_{A \times B} ((x_{1},y_{1})(x_{2},y_{2})^{-1}) > \alpha\) which means that \((x_{1},y_{1})(x_{2},y_{2})^{-1} \in (A \times B)_{(\alpha)}.\) Hence \( (A \times B)_{(\alpha)}\) is a subgroup of \(G \times H.\)

Proposition 5. Let \( A \in TFMS(G)\) and \( B \in TFMS(H).\) If \( A\times B \in TFMS(G \times H),\) then at least one of the following statements hold:
(1) \(CM_{B}(e_{H})) \geq CM_{A}(x)\) for all \( x \in G, \)
2) \(CM_{A}(e_{G})) \geq CM_{B}(y)\) for all \( y \in G. \)

Proof. Suppose that none of the statements holds, then we can find \( a \in G \) and \(b \in H \) such that \(CM_{A}(a) >CM_{B}(e_{H})\) and \(CM_{B}(b) >CM_{A}(e_{G}).\) Now \begin{align*}CM_{A \times B}(a,b)&=T(CM_{A}(a),CM_{B}(b))\\& > T(CM_{B}(e_{H}),CM_{A}(e_{G}))\\&=T(CM_{A}(e_{G}),CM_{B}(e_{H}))=CM_{A \times B}(e_{G},e_{H})\,.\end{align*} Thus \(CM_{A \times B}(a,b) > CM_{A \times B}(e_{G},e_{H})\), which is contradiction with Proposition \ref{prop2}(1), hence at least one of the statements hold.

Proposition 6. Let \( A \in FMS(G)\) and \( B \in FMS(H)\) such that \( A\times B \in TFMS(G \times H)\) and \(CM_{A}(x) \leq CM_{B}(e_{H})\) for all \( x \in G. \) Then \( A \in TFMS(G).\)

Proof. As \(CM_{A}(x) \leq CM_{B}(e_{H})\) for all \( x \in G\), so \(CM_{A}(y) \leq CM_{B}(e_{H})\) and \(CM_{A}(xy) \leq CM_{B}(e_{H})=CM_{B}(e_{H}e_{H})\) for all \( y \in G.\) Then \begin{align*} CM_{A}(xy) &=T(CM_{A}(xy),CM_{B}(e_{H}e_{H}))\\&=CM_{A \times B}(xy,e_{H}e_{H})\\ &=CM_{A \times B}((x,e_{H})(y,e_{H}))\\ &\geq T(CM_{A \times B}(x,e_{H}),CM_{A \times B}(y,e_{H}))\\ &=T(T(CM_{A}(x),CM_{B}(e_{H})),T(CM_{A}(y),CM_{B}(e_{H})))\\&=T(CM_{A}(x),CM_{A}(y))\,. \end{align*} Thus \[CM_{A}(xy) \geq T(CM_{A}(x),CM_{A}(y)).\] Also since \(CM_{A}(x) \leq CM_{B}(e_{H})\) for all \( x \in G\) so \(CM_{A}(x^{-1}) \leq CM_{B}(e_{H}).\) Thus \begin{align*}CM_{A}(x^{-1})&=T(CM_{A}(x^{-1}),CM_{A}(e_{H}))\\&=T(CM_{A}(x^{-1}),CM_{A}(e_{H}^{-1}))\\&=CM_{A \times B}((x,e_{H})^{-1}) \\&\geq CM_{A \times B}(x,e_{H}) \\&=T(CM_{A}(x),CM_{A}(e_{H}))=CM_{A}(x)\end{align*} and then \( CM_{A}(x^{-1}) \geq CM_{A}(x) .\) Therefore \( A \in TFMS(G).\)

Proposition 7. Let \( A \in FMS(G)\) and \( B \in FMS(H)\) such that \( A\times B \in TFMS(G \times H)\) and \(CM_{B}(x) \leq CM_{A}(e_{G})\) for all \( x \in H. \) Then \( B \in TFMS(H).\)

Proof. The proof is similar to Proposition \ref{prop19}.

Corollary 3. Let \( A \in FMS(G)\) and \( B \in FMS(H)\) such that \( A\times B \in TFMS(G \times H).\) Then either \( A \in TFMS(G)\) or \( B \in TFMS(H).\)

Proof. Using Proposition \ref{prop18} we get that \(CM_{B}(e_{H})) \geq CM_{A}(x)\) for all \( x \in G \) or \(CM_{A}(e_{G})) \geq CM_{B}(y)\) for all \( y \in G. \) Then from Proposition \ref{prop19} and Proposition \ref{prop20} we have that either \( A \in TFMS(G)\) or \( B \in TFMS(H).\)

Proposition 8. Let \( A,C \in TFMS(G)\) and \( B,D \in TFMS(H).\) If \( A \) is conjugate to \( B\) and \( C \) is conjugate to \( D,\) then \( A \times C \) is conjugate to \( B \times D.\)

Proof. As \( A \) is conjugate to \( B\) so \( CM_{A}(x)=CM_{C}(gxg^{-1}) \) and as \( B \) is conjugate to \( D\) so \( CM_{B}(y)=CM_{D}(hyh^{-1}) \) for all \( x,g \in G \) and \( y,h \in H. \) Now \begin{align*} CM_{A \times B}(x,y)&=T( CM_{A}(x),CM_{B}(y))\\&=T(CM_{C}(gxg^{-1}),CM_{D}(hyh^{-1}))\\ &=CM_{C \times D}(gxg^{-1},hyh^{-1})\\&=CM_{C \times D}((g,h)(x,y)(g^{-1},h^{-1}))\\ &=CM_{C \times D}((g,h)(x,y)(g,h)^{-1})\,. \end{align*} Thus \( CM_{A \times B}(x,y)=CM_{C \times D}((g,h)(x,y)(g,h)^{-1}) \) which means that \( A \times C \) is conjugate to \( B \times D.\)

Proposition 9. Let \( A \in TFMS(G)\) and \( B \in TFMS(H).\) Then \( A\) and \( B\) are commutative if and only if \( A \times B \) is a commutative.

Proof. Let \(x_{1},y_{1} \in G\) and \(x_{2},y_{2} \in H\) such that \( x=(x_{1},x_{2}) \in G \times H\) and \( y=(y_{1},y_{2}) \in G \times H.\) Let \( A\) and \( B\) are commutative then \( CM_{A}(x_{1}y_{1}) =CM_{A}(y_{1}x_{1}) \) and \( CM_{B}(x_{2}y_{2}) =CM_{B}(y_{2}x_{2}).\) Which implies \begin{align*} CM_{A \times B}(xy)&=CM_{A \times B}((x_{1},x_{2})(y_{1},y_{2}))\\&=CM_{A \times B}(x_{1}y_{1},x_{2}y_{2})\\ &=T(CM_{A}(x_{1}y_{1}),CM_{ B}(x_{2}y_{2}))\\&=T(CM_{A}(y_{1}x_{1}),CM_{ B}(y_{2}x_{2}))\\ &=CM_{A \times B}(y_{1}x_{1},y_{2}x_{2})\\&=CM_{A \times B}((y_{1},y_{2})(x_{1},x_{2}))\\&=CM_{A \times B}(yx)\,. \end{align*} Thus \( CM_{A \times B}(xy)=CM_{A \times B}(yx) \) and then \( A \times B \) is a commutative. Conversely, suppose that \( A \times B \) is a commutative. Then \( CM_{A \times B}(xy)=CM_{A \times B}(yx) \Longleftrightarrow CM_{A \times B}((x_{1},x_{2})(y_{1},y_{2}))=CM_{A \times B}((y_{1},y_{2})(x_{1},x_{2})) \Longleftrightarrow CM_{A \times B}(x_{1}y_{1},x_{2}y_{2})=CM_{A \times B}(y_{1}x_{1},y_{2}x_{2}) \Longleftrightarrow T(CM_{A}(x_{1}y_{1}),CM_{B}(x_{2}y_{2}))=T(CM_{A}(y_{1}x_{1}) ,CM_{B}(y_{2}x_{2})) \Longleftrightarrow CM_{A}(x_{1}y_{1})=CM_{A}(y_{1}x_{1})\) and \(CM_{B}(x_{2}y_{2})=CM_{B}(y_{2}x_{2}) \) which gives us that \( A\) and \( B\) are commutative.

Definition 10. Let \( G \times H\) and \( I \times J\) be groups and \(f : G \times H \to I \times J\) be a homomorphism. Let \( A \times B \in FMS(G \times H)\) and \( C \times D \in FMS(I \times J).\) Define \( f(A \times B) \in FMS(I \times J)\) and \( f^{-1}(C \times D) \in FMS(G \times H)\) as: \begin{align*} f(CM_{A \times B})(i,j)&=(CM_{f(A \times B)})(i,j)= \left\{ \begin{array}{rl} \sup \lbrace CM_{A \times B}(g,h) \hspace{0.1cm}|\hspace{0.1cm} g \in G, h \in H , f(g,h)=(i,j) \rbrace &\text{if }f^{-1}(i,j) \neq \emptyset \\ 0 &\text{otherwise } \\ \end{array} \right. \end{align*} and \[f^{-1}(CM_{C \times D}(g,h))=CM_{f^{-1}(C \times D)}(g,h)=CM_{C \times D}(f(g,h))\] for all \( (g,h) \in G \times H. \)

Proposition 10. Let \( G \times H\) and \( I \times J\) be groups and \(f : G \times H \to I \times J\) be an epimorphism. If \( A \in TFMS(G), B \in TFMS( H)\) and \(A \times B \in TFMS(G \times H),\) then \( f(A \times B) \in TFMS( I \times J).\)

Proof. (1) Let \( X=(i_{1},j_{1}) \in I \times J \) and \( Y=(i_{2},j_{2}) \in I \times J\) such that \[ f^{-1}(XY)=f^{-1}((i_{1},j_{1})(i_{2},j_{2}))=f^{-1}(i_{1}i_{2},j_{1}j_{2}) \neq \emptyset. \] Then \begin{align*} f(A \times B)(XY)&=f(A \times B)((i_{1},j_{1})(i_{2},j_{2}))\\ &=f(A \times B)(i_{1}i_{2},j_{1}j_{2})\\ &=\sup \lbrace CM_{A \times B}(g_{1}g_{2},h_{1}h_{2}) \hspace{0.1cm}|\hspace{0.1cm} g_{1},g_{2} \in G, h_{1},h_{2} \in H , f(g_{1}g_{2},h_{1}h_{2})=(i_{1}i_{2},j_{1}j_{2}) \rbrace\\ &= \sup \lbrace CM_{A \times B}(g_{1}g_{2},h_{1}h_{2}) \hspace{0.1cm}|\hspace{0.1cm} g_{1},g_{2} \in G, h_{1},h_{2} \in H , (f(g_{1}g_{2}),f(h_{1}h_{2}))=(i_{1}i_{2},j_{1}j_{2}) \rbrace\\ &=\sup \lbrace CM_{A \times B}(g_{1}g_{2},h_{1}h_{2}) \hspace{0.1cm}|\hspace{0.1cm} g_{1},g_{2} \in G, h_{1},h_{2} \in H , f(g_{1}g_{2})=i_{1}i_{2},f(h_{1}h_{2})=j_{1}j_{2} \rbrace\\ &=\sup \lbrace T(CM_{A}(g_{1}g_{2}) ,CM_{B}(h_{1}h_{2}) ) \hspace{0.1cm}|\hspace{0.1cm} g_{1},g_{2} \in G, h_{1},h_{2} \in H , f(g_{1}g_{2})=i_{1}i_{2},f(h_{1}h_{2})=j_{1}j_{2} \rbrace\\ &=\sup \lbrace T(CM_{A}(g_{1}g_{2}) ,CM_{B}(h_{1}h_{2}) ) \hspace{0.1cm}|\hspace{0.1cm} g_{1},g_{2} \in G, h_{1},h_{2} \in H , f(g_{1}g_{2})=i_{1}i_{2},f(h_{1}h_{2})=j_{1}j_{2} \rbrace\\ &\geq \sup \lbrace T(T(CM_{A}(g_{1}),CM_{A}(g_{2})) ,T(CM_{B}(h_{1}),CM_{B}(h_{2})) ) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1}g_{2})=i_{1}i_{2},f(h_{1}h_{2})=j_{1}j_{2} \rbrace\\ &=\sup \lbrace T(T(CM_{A}(g_{1}),CM_{B}(h_{1})) ,T(CM_{A}(g_{2}),CM_{B}(h_{2})) ) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1}g_{2})=i_{1}i_{2},f(h_{1}h_{2})=j_{1}j_{2} \rbrace \end{align*} \begin{align*} &=\sup \lbrace T(T(CM_{A}(g_{1}),CM_{B}(h_{1})) ,T(CM_{A}(g_{2}),CM_{B}(h_{2})) )\hspace{0.1cm}|\hspace{0.1cm} f(g_{1})=i_{1},f(g_{2})=i_{2},f(h_{1})=j_{1},f(h_{2})=j_{2} \rbrace\\ &=\sup \lbrace T(CM_{A \times B}(g_{1},h_{1}), CM_{A \times B}(g_{2},h_{2})) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1})=i_{1},f(g_{2})=i_{2},f(h_{1})=j_{1},f(h_{2})=j_{2} \rbrace\\ &=T(\sup \lbrace CM_{A \times B}(g_{1},h_{1}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{1},h_{1})=(i_{1},j_{1}) \rbrace, \sup \lbrace CM_{A \times B}(g_{2},h_{2}) \hspace{0.1cm}|\hspace{0.1cm} f(g_{2},h_{2})=(i_{2},j_{2})\rbrace)\\ &=T(f(A \times B)(i_{1},j_{1}),f(A \times B)(i_{2},j_{2}))=T(f(A \times B)(X),f(A \times B)(Y))\,.\end{align*} Thus \[ f(A \times B)(XY) \geq T(f(A \times B)(X),f(A \times B)(Y)).\]
(2) Let \( X=(i,j) \in I \times J \) then \begin{align*} f(A \times B)(X^{-1})&=f(A \times B)((i,j)^{-1})\\&=f(A \times B)(i^{-1},j^{-1})\\ &= \sup \lbrace CM_{A \times B}(g^{-1},h^{-1}) \hspace{0.1cm}|\hspace{0.1cm} g\in G, h\in H , f(g^{-1},h^{-1})=(i^{-1},j^{-1}) \rbrace\\ &=\sup \lbrace CM_{A \times B}(g^{-1},h^{-1}) \hspace{0.1cm}|\hspace{0.1cm} g\in G, h\in H , (f(g^{-1}),f(h^{-1}))=(i^{-1},j^{-1}) \rbrace \\ &=\sup \lbrace CM_{A \times B}(g^{-1},h^{-1}) \hspace{0.1cm}|\hspace{0.1cm} g\in G, h\in H , f(g^{-1})=i^{-1},f(h^{-1}))=j^{-1} \rbrace\\ &=\sup \lbrace T(CM_{A}(g^{-1}) , CM_{ B}(h^{-1}) ) \hspace{0.1cm}|\hspace{0.1cm} g\in G, h\in H , f(g^{-1})=i^{-1},f(h^{-1}))=j^{-1} \rbrace\\ &\geq \sup \lbrace T(CM_{A}(g) , CM_{ B}(h) ) \hspace{0.1cm}|\hspace{0.1cm} g\in G, h\in H , f^{-1}(g)=i^{-1},f^{-1}(h)=j^{-1} \rbrace\\ &=\sup \lbrace T(CM_{A}(g) , CM_{ B}(h) ) \hspace{0.1cm}|\hspace{0.1cm} g\in G, h\in H , f(g)=i,f(h)=j \rbrace\\ &=\sup \lbrace CM_{A \times B}(g,h) \hspace{0.1cm}|\hspace{0.1cm} (g,h) \in G\times H , f(g,h)=(i,j)\rbrace\\ &=f(A \times B)(i,j)=f(A \times B)(X) \end{align*} and then \(f(A \times B)(X^{-1}) \geq f(A \times B)(X).\) Therefore \( f(A \times B) \in TFMS( I \times J).\)

Proposition 11. Let \( G \times H\) and \( I \times J\) be groups and \(f : G \times H \to I \times J\) be a homomorphism. If \( C \in TFMS(I) \), \( D \in TFMS(J) \) and \( C \times D \in TFMS(I \times J),\) then \( f^{-1}(C \times D) \in TFMS(G \times H).\)

Proof. (1) Let \( X=(g_{1},h_{1}) \in G \times H\) and \( Y=(g_{2},h_{2}) \in G \times H.\) Then \begin{align*} f^{-1}(CM_{C \times D})(XY)&=f^{-1}(CM_{C \times D})((g_{1},h_{1}) (g_{2},h_{2}))\\ &=f^{-1}((CM_{C \times D})(g_{1}g_{2},h_{1}h_{2}))\\ &=CM_{C \times D}(f(g_{1}g_{2},h_{1}h_{2}))\\ &=CM_{C \times D}(f(g_{1}g_{2}),f(h_{1}h_{2}))\\ &=T(CM_{C}(f(g_{1}g_{2})),CM_{D}(f(h_{1}h_{2})))\\ &= T(CM_{C}(f(g_{1})f(g_{2})),CM_{D}(f(h_{1})f(h_{2}))) \\ & \geq T(T(CM_{C}(f(g_{1})),CM_{C}(f(g_{2}))),T(CM_{D}(f(h_{1})),CM_{D}(f(h_{2})))\\ &= T(T(CM_{C}(f(g_{1})),CM_{D}(f(h_{1}))),T(CM_{C}(f(g_{2}),CM_{D}(f(h_{2})))\\ &=T(CM_{C \times D}(f(g_{1}),f(h_{1})),CM_{C \times D}(f(g_{2}),f(h_{2})))\\&=T(CM_{C \times D}(f(g_{1},h_{1})),CM_{C \times D}(f(g_{2},h_{2})))\\&=T(f^{-1}(CM_{C \times D})(g_{1},h_{1}),f^{-1}(CM_{C \times D})(g_{2},h_{2}))\\ &=T(f^{-1}(CM_{C \times D})(X),f^{-1}(CM_{C \times D})(Y))\,. \end{align*} Thus \[f^{-1}(CM_{C \times D})(XY) \geq T(f^{-1}(CM_{C \times D})(X),f^{-1}(CM_{C \times D})(Y)).\]
(2) Let \( X=(g,h) \in G \times H,\) then \begin{align*}f^{-1}(CM_{C \times D})(X^{-1})&=f^{-1}(CM_{C \times D})((g_{1},h_{1}) ^{-1})\end{align*} \begin{align*}&=CM_{C \times D}(f(g,h)^{-1})\\ &=CM_{C \times D}(f(g^{-1},h^{-1}))\\ &=CM_{C \times D}(f^{-1}(g),f^{-1}(h))\\ &=T(CM_{C}(f^{-1}(g)),CM_{D}(f^{-1}(h)))\\ &\geq T(CM_{C}(f(g)),CM_{D}(f(h)))\\ &=CM_{C \times D}(f(g),f(h))\\ &=CM_{C \times D}(f(g,h))\\ &=f^{-1}(CM_{C \times D})(g,h)\\ &=f^{-1}(CM_{C \times D})(X)\end{align*} and then \(f^{-1}(CM_{C \times D})(X^{-1}) \geq f^{-1}(CM_{C \times D})(X).\) Thus \( f^{-1}(C \times D) \in TFMS(G \times H).\)

Acknowledgments

We would like to thank the referees for carefully reading the manuscript and making several helpful comments to increase the quality of the paper.

Author Contributions

All author contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The author declares no conflict of interest.

1. Introduction

Fuzzy set theory proposed by Zadeh [1] has achieved a huge impact in many fields to handle uncertainty/vagueness. Due to the vast majority of imprecise and vague information in real-life problems, different extensions of fuzzy set have been developed by some researchers. Yager [2] applied the idea of multiset [3], which is an extension of set with repeated elements in a collection to propose fuzzy multiset. Consequently, a fuzzy multiset allows repetition of membership degrees of elements in multiset framework. In fact, fuzzy multiset generalizes fuzzy set [4].

The concept of intuitionistic fuzzy sets (IFS) was proposed and studied in [5, 6, 7] as a generalization of fuzzy sets. The main advantage of the IFS is its ability to cope with the hesitancy that may exist due to information impression. This is achieved by incorporating a second function, along with the membership function, \(\mu\) of the conventional fuzzy set, called non-membership function, \(\nu\). The idea of IFS has found expression in many cases like medical diagnosis, career placements, pattern recognition and other MCDM problems [8, 9, 10, 11, 12, 13].

However robust the notion of IFS is, there are circumstances where \(\mu+\nu\geq 1\) unlike the situation captured in IFS (where, \(\mu+\nu\leq 1\)). The shortcoming in IFS naturally led to the introduction of a concept, called Pythagorean fuzzy sets (PFSs) by Yager [14]. PFS is a tool to deal with vagueness considering the membership grade, \(\mu\) and non-membership grade, \(\nu\) satisfying the condition \(\mu+\nu\geq 1\). As a generalized set, PFS has close relationship with IFS. This idea can be used to characterize the uncertain information more sufficiently and accurately than IFS. PFSs have been applied in many areas, like the one discussed in [15].

In [16], the concepts of IFS and fuzzy multiset were combined to proposed intuitionistic fuzzy multisets (IFMS) as the generalization of IFS in multiset framework or the extension of fuzzy multisets by incorporating count non-membership functions, \(CN=\left\lbrace \nu^1, ..., \nu^n\right\rbrace\) in addition to the count membership functions, \(CM=\left\lbrace \mu^1, ..., \mu^n\right\rbrace\) captured in fuzzy multisets. Some operations and modal operators on IFMS have been studied in [17, 18]. Due to the resourcefulness of IFMS, it has been applied in many real-life problems as seen in [19, 20, 21, 22, 23, 24, 25, 26, 27].

The motivation of this paper follows from the ideas of PFSs [14] and IFMS [16]. The paper proposes Pythagorean fuzzy multisets (PFMSs), studies its properties and also, its application to course placements. PFMS is either the incorporation of IFMS in PFS setting or PFS in multiset framework.

The paper is organized by presenting some mathematical preliminaries such as fuzzy sets, fuzzy multisets, IFSs, IFMSs and PFSs in Section 2. Moreover, Section 3 covers the concept of PFMS and explicates the ideas of level sets, accuracy and score functions in the setting of PFMS. Also, the idea of cuts in PFMSs context is discussed with some results, and some modal operators on PFMSs are proposed with some deduced theorems. In Section 4, the application of PFMSs in course placements are discussed through composite relation defined on PFMSs. Finally, Section 5 summarises the paper and gives some useful conclusions.

2. Preliminaries

2.1. Fuzzy sets

Definition [1] Let \(X\) be a nonempty set. A fuzzy set \(A\) of \(X\) is characterized by a membership function \[\mu_A:X\to [0,1].\] That is, \begin{equation*} \mu_A(x) = \left\{ \begin{array}{ll} 1, & \textrm{if}\: x\in X\\ 0, & \textrm{if}\: x\notin X\\ (0,1) & \textrm{if}\: x\: \textrm{is partly in}\: X \end{array} \right. \end{equation*} Alternatively, a fuzzy set \(A\) of \(X\) is an object having the form \[A=\left\lbrace \left\langle x, \mu_A(x)\right\rangle\mid x\in X\right\rbrace \: \textrm{or}\: A=\left\lbrace \left\langle \dfrac{\mu_A(x)}{x} \right\rangle\mid x\in X\right\rbrace, \] where the function \[\mu_A(x): X\to [0,1]\] defines the degree of membership of the element, \(x\in X\).

2.2. Fuzzy multisets

Definition 2. [2] Assume \(X\) is a set of elements. Then, a fuzzy bag/multiset \(A\) drawn from \(X\) can be characterized by a count membership function \(CM_A\) such that \[CM_A:X\to Q,\] where \(Q\) is the set of all crisp bags or multisets from the unit interval \(I=[0,1]\).
A fuzzy multiset can also be characterized by a high-order function. In particular, a fuzzy multiset \(A\) can be characterized by a function \[CM_A:X\to N^I \: \textrm{or}\: CM_A:X\to [0,1]\to N,\] where \(I=[0,1]\) and \(N=\mathbb{N}\cup \left\lbrace 0\right\rbrace\).
It follows that \(CM_A(x)\) for \(x\in X\) is given as \[CM_A(x)=\left\lbrace \mu^1_A (x), \mu^2_A (x),...,\mu^n_A (x),...\right\rbrace,\] where \(\mu^1_A (x), \mu^2_A (x),...,\mu^n_A (x),...\in [0,1]\) such that \(\mu^1_A (x)\geq \mu^2_A (x)\geq...\geq \mu^n_A (x)\geq ...\), whereas in a finite case, we write \[CM_A(x)=\left\lbrace \mu^1_A (x), \mu^2_A (x),...,\mu^n_A (x)\right\rbrace,\] for \(\mu^1_A (x)\geq \mu^2_A (x)\geq...\geq \mu^n_A (x)\).
A fuzzy multiset \(A\) can be represented in the form \[A=\left\lbrace \left\langle \dfrac{CM_A(x)}{x}\right\rangle \mid x\in X\right\rbrace \: \textrm{or}\: A=\left\lbrace \left\langle x, CM_A(x)\right\rangle \mid x\in X\right\rbrace.\]

2.3. Intuitionistic fuzzy sets

Definition 3. [5] Let a nonempty set \(X\) be fixed. An IFS \(A\) of \(X\) is an object having the form \[A=\left\lbrace \left\langle x, \mu_A(x), \nu_A(x) \right\rangle\mid x\in X\right\rbrace\] or \[A=\left\lbrace \left\langle \dfrac{\mu_A(x), \nu_A(x)}{x} \right\rangle\mid x\in X\right\rbrace,\] where the functions \[\mu_A(x): X\to [0,1] \: \textrm{and}\: \nu_A(x): X\to [0,1]\] define the degree of membership and the degree of non-membership, respectively of the element \(x\in X\) to \(A\), which is a subset of \(X\), and for every \(x\in X\), \[0\leq \mu_A(x)+ \nu_A(x)\leq 1.\] For each \(A\) in \(X\), \[\pi_A(x)= 1-\mu_A(x)-\nu_A (x)\] is the intuitionistic fuzzy set index or hesitation margin of \(x\) in \(X\). The hesitation margin \(\pi_A (x)\) is the degree of non-determinacy of \(x\in X\), to \(A\) and \(\pi_A (x)\in [0,1]\). The hesitation margin is the function that expresses lack of knowledge of whether \(x\in X\) or \(x\notin X\). Thus, \[\mu_A(x)+\nu_A(x)+\pi_A(x)=1.\]

2.4. Intuitionistic fuzzy multisets

Definition 4. [16] Let \(X\) be a nonempty set. An IFMS \(A\) drawn from \(X\) is of the form \[A=\left\lbrace \left\langle \dfrac{CM_A(x)}{x}, \dfrac{CN_A(x)}{x}\right\rangle| x\in X \right\rbrace \] where \[CM_A(x)= \mu_A^1 (x),...,\mu_A^n (x), ...\] and \[CN_A(x)= \nu_A^1 (x),...,\nu_A^n (x), ...\] are the count membership and count non-membership degrees defined by the functions $$CM_A:X\to N^{[0,1]} \: \textrm{and}\: CN_A:X\to N^{[0,1]}$$ such that \(0\leq CM_A(x) + CN_A(x)\leq 1\), where \(N=\mathbb{N}\cup \left\lbrace 0\right\rbrace\). }

If the count membership functions and count non-membership functions have only \(n-\)terms (i.e. finite), then \(n\) is called the dimension of \(A\). Consequently \[A=\left\lbrace \left\langle \dfrac{\mu_A^1 (x),...,\mu_A^n (x)}{x},\dfrac{\nu_A ^1(x),...,\nu_A ^n(x)}{x}\right\rangle \mid x\in X \right\rbrace \] for \(i=1,...,n\). For each IFMS \(A\) of \(X\), \[CH_A(x)= 1-CM_A(x)-CN_A(x)\] is the intuitionistic fuzzy multisets index or count hesitation margin of \(x\) in \(A\), where \[CH_A(x)= \pi_A^1(x), ..., \pi_A^n.\] The hesitation margin \(\pi_A^i (x)\) for each \(i=1,...,n\) is the degree of non-determinacy of \(x\in X\) to \(A\) and \(\pi_A^i (x)\in [0,1]\). The count hesitation margin is the function that expresses lack of knowledge of whether \(x\in A\) or \(x\notin A\). Thus, \[\mu_A^i(x)+\nu_A ^i (x)+\pi_A^i (x)=1\] for each \(i=1,...,n\).

2.5. Pythagorean fuzzy sets

Definition 5. [14] Let \(X\) be a universal set. Then, an PFS \(A\) of \(X\) is a set of ordered pairs defined by \[A=\left\lbrace \left\langle x, \mu_A(x), \nu_A(x) \right\rangle\mid x\in X\right\rbrace\] or \[A=\left\lbrace \left\langle \dfrac{\mu_A(x), \nu_A(x)}{x} \right\rangle\mid x\in X\right\rbrace,\] where the functions \[\mu_A(x): X\to [0,1] \: \textrm{and}\: \nu_A(x): X\to [0,1]\] define the degree of membership and the degree of non-membership, respectively of the element \(x\in X\) to \(A\), which is a subset of \(X\), and for every \(x\in X\), \[0\leq (\mu_A(x))^2 + (\nu_A(x))^2 \leq 1.\] Supposing \((\mu_A(x))^2 + (\nu_A(x))^2 \leq 1\), then there is a degree of indeterminacy of \(x\in X\) to \(A\) defined by \(\pi_A(x)=\sqrt{1-[(\mu_A(x))^2 + (\nu_A(x))^2]}\) and \(\pi_A(x)\in[0,1]\). In what follows, \((\mu_A(x))^2 + (\nu_A(x))^2 + (\pi_A(x))^2=1\). Otherwise, \(\pi_A(x)=0\) whenever \((\mu_A(x))^2 + (\nu_A(x))^2 =1\). }

3. Pythagorean fuzzy multisets

Definition 6. Let \(X\) be a nonempty set. Then, an PFMS \(A\) drawn from \(X\) is of the form \[A=\left\lbrace \left\langle \dfrac{CM_A(x)}{x}, \dfrac{CN_A(x)}{x}\right\rangle| x\in X \right\rbrace \] or \[A=\left\lbrace \left\langle x, CM_A(x), CN_A(x)\right\rangle |x\in X\right\rbrace\] where \[CM_A(x)= \mu_A^1 (x),...,\mu_A^n (x)\] and \[CN_A(x)= \nu_A^1 (x),...,\nu_A^n (x)\] are the count membership and count non-membership degrees defined by the functions $$CM_A:X\to N^{[0,1]} \: \textrm{and}\: CN_A:X\to N^{[0,1]}$$ such that \(0\leq [CM_A(x)]^2 + [CN_A(x)]^2\leq 1\), where \(N=\mathbb{N}\cup \left\lbrace 0\right\rbrace\).

For each PFMS \(A\) of \(X\), \[CH_A(x)=\sqrt{1-[CM_A(x)]^2-[CN_A(x)]^2}\] is the count hesitation margin of \(x\) in \(A\), where \[CH_A(x)= \pi_A^1(x), ..., \pi_A^n.\] The count hesitation margin \(CH_A (x)\) is the degree of non-determinacy of \(x\in X\) to \(A\) and \(CH_A(x)\in [0,1]\). The count hesitation margin is the function that expresses lack of knowledge of whether \(x\in A\) or \(x\notin A\). Thus, \[[CM_A(x)]^2+[CN_A(x)]^2+[CH_A(x)]^2=1.\] We denote the set of all PFMS over \(X\) by \(PFMS(X)\). Table 1 explains the difference between IFMS and PFMS.

Table 1. IFMS and PFMS.

IFMS	PFMS
\(CM +CN\leq 1\)	\(CM +CN\leq 1\) or \(CM +CN\geq 1\)
\(0\leq CM +CN\leq 1\)	\(0\leq CM^2 +CN^2 \leq 1\)
\(CH =1-(CM +CN)\)	\(CH=\sqrt{1-[CM^2 +CN^2 ]}\)
\(CM +CN +CH=1\)	\(CM^2 +CN^2 +CH^2=1\)

Example Let \(A\) be an PFMS of \(X=\left\lbrace x,y\right\rbrace\) such that $$CM_A(x)=0.7,0.5,0.4$$ $$CN_A(x)=0.3, 0.5, 0.6$$ $$CM_A(y)=0.8, 0.6, 0.4$$ $$CN_A(y)=0.4, 0.5, 0.5.$$ That is $$A=\left\lbrace \dfrac{\left\langle 0.7,0.5,0.4\right\rangle,\left\langle 0.3, 0.5, 0.6\right\rangle}{x}, \dfrac{\left\langle 0.8, 0.6, 0.4\right\rangle,\left\langle 0.4, 0.5, 0.5\right\rangle}{y}\right\rbrace.$$ Then $$CH_A(x)=0.6481, 0.7071, 0.6928$$ $$CH_A(y)=0.4472, 0.6245, 0.7681.$$

For easy computational purpose, an PFMS can be converted to PFS by taking the mean values of the count membership degrees, count non-membership degrees and count hesitation margin, respectively. That is, an PFMS \(A\) in Example 1 becomes an PFS $$A=\left\lbrace \dfrac{\left\langle 0.5333,0.4667\right\rangle}{x}, \dfrac{\left\langle 0.6, 0.4667\right\rangle}{y}\right\rbrace.$$

Definition 7. Two PFMSs \(A\) and \(B\) are said to be equal or comparable if \[CM_A(x)=CM_B(x),\; CN_A(x)=CN_B(x)\] \(\forall x\in X\).

Definition 8. Let \(A,B\in PFMS(X)\), then \(A\) is contained in \(B\) denoted by \(A\subseteq B\) if $$CM_A(x)\leq CM_B(x) \: \textrm{and}\: CN_A(x)\geq CN_B(x) \: \forall x\in X.$$ We say \(A\) is properly contained in \(B\), that is, \(A\subset B\) if \(A\subseteq B\) and \(A\neq B\). It means \(CM_A(x)\leq CM_B(x)\) and \(CN_A(x)\geq CN_B(x)\) but \(CM_A(x)\neq CM_B(x)\) and \(CN_A(x)\neq CN_B(x) \: \forall x\in X\).

Definition 9. Let \(X\) and \(Y\) be nonempty sets and let \(f:X\to Y\) be a mapping. Suppose \(A\in PFMS(X)\) and \(B\in PFMS(Y)\), respectively. Then

• (i) the inverse image of \(B\) under \(f\), denoted by \(f^{-1} (B)\), is an PFMS of \(X\) defined by \[f^{-1} (B)=\left\lbrace \left\langle \dfrac{CM_{f^{-1} (B)}(x)}{x}, \dfrac{CN_{f^{-1} (B)}(x)}{x} \right\rangle\mid x\in X\right\rbrace,\] where \(CM_{f^{-1} (B)}(x)=CM_B(f(x))\) and \(CN_{f^{-1} (B)}(x)=CN_B(f(x))\\ \forall x\in X\).
• (ii) the image of \(A\) under \(f\), denoted by \(f(A)\), is an PFMS of \(Y\) defined by \[f(A)=\left\lbrace \left\langle \dfrac{CM_{f(A)}(y)}{y}, \dfrac{CN_{f(A)}(y)}{y}\right\rangle \mid y\in Y\right\rbrace,\] where \begin{equation*} CM_{f(A)} (y) = \left\{ \begin{array}{ll} \bigvee_{x\in f^{-1}(y)}CM_A(x), & f^{-1}(y)\neq \emptyset\\ 0, & \textrm{otherwise}, \end{array} \right. \end{equation*} and \begin{equation*} CN_{f(A)} (y) = \left\{ \begin{array}{ll} \bigwedge_{x\in f^{-1}(y)}CN_A(x), & f^{-1}(y)\neq \emptyset\\ 0, & \textrm{otherwise}, \end{array} \right. \end{equation*} for each \(y\in Y\). This is called the extension principle of PFMS.

Theorem 10. Let \(A\in PFMS(X)\). Suppose that \(CH_A(x)=0\), then the following hold:

• (i) \(|CM_A(x)|=\sqrt{|(CN_A(x)+1)(CN_A(x)-1)|}\).
• (ii) \(|CN_A(x)|=\sqrt{|(CM_A(x)+1)(CM_A(x)-1)|}\).

Proof. Suppose \(x\in X\) and \(A\in PFMS(X)\). Then we prove (i) and (ii). Since \(CH_A(x)=0\) for every \(x\in X\), we have
\((CM_A(x))^2 + (CN_A(x))^2 =1\) \\ \(\Rightarrow -(CM_A(x))^2= (CN_A(x))^2-1\)\\ \(\Rightarrow -(CM_A(x))^2= (CN_A(x)+1)(CN_A(x)-1)\)\\ \(\Rightarrow |(CM_A(x))^2|= |(CN_A(x)+1)(CN_A(x)-1)|\)\\ \(\Rightarrow |CM_A(x)|^2= |(CN_A(x)+1)(CN_A(x)-1)|\)\\ \(\Rightarrow |CM_A(x)|= \sqrt{|(CN_A(x)+1)(CN_A(x)-1)|}\),\\ which proves \((i)\). The proof of \((ii)\) is similar to that of \((i).\)

3.1. Some operations under PFMSs

Definition 11. For any two PFMSs \(A\) and \(B\) drawn from \(X\), the following operations hold.

• (i) Complement\[A^c=\left\lbrace \left\langle \dfrac{CN_A(x)}{x},\dfrac{CM_A(x)}{x}\right\rangle\mid x\in X\right\rbrace\]
• (ii) Union \[A\cup B=\left\lbrace \left\langle \dfrac{max(CM_A(x), CM_B(x))}{x},\dfrac{min(CN_A(x), CN_B(x))}{x}\right\rangle\mid x\in X\right\rbrace\]
• (iii) Intersection \[A\cup B=\left\lbrace \left\langle \dfrac{min(CM_A(x), CM_B(x))}{x},\dfrac{max(CN_A(x), CN_B(x))}{x}\right\rangle\mid x\in X\right\rbrace.\]

Definition 12. Let \(A,B\in PFMS(X)\). Then, the addition of \(A\) and \(B\) is defined as \[A\oplus B=\left\lbrace \left\langle \dfrac{\sqrt{(CM_A(x))^2+(CM_B(x))^2-(CM_A(x))^2(CM_B(x))^2}}{x}, \dfrac{CN_A(x)CN_B(x)}{x} \right\rangle |x\in X \right\rbrace,\] and the multiplication of \(A\) and \(B\) is defined as \[A\otimes B=\left\lbrace \left\langle \dfrac{CM_A(x)CM_B(x)}{x}, \dfrac{\sqrt{(CN_A(x))^2+(CN_B(x))^2-(CN_A(x))^2(CN_B(x))^2}}{x} \right\rangle |x\in X \right\rbrace.\]

Proposition 1. Let \(A,B, C\in PFMS(X)\), then the following properties follow.

• (i)Complementary law\[{(A^c) }^c=A\]
• (ii) Idempotent laws \begin{eqnarray*} A\cup A & = & A\\ A\cap A & = & A \end{eqnarray*}
• (iii)Commutative laws \begin{eqnarray*} A\cup B & = & B\cup A\\ A\cap B & = & B\cap A\\ A\oplus B & = & B \oplus A\\ A\otimes B & = & B\otimes A \end{eqnarray*}
• (iv) Associative laws \begin{eqnarray*} (A\cup B)\cup C & = & A\cup (B\cup C)\\ (A\cap B)\cap C & = & A\cap (B\cap C)\\ (A\oplus B) \oplus C & = & A\oplus (B\oplus C)\\ (A\otimes B)\otimes C & = & A\otimes (B\otimes C) \end{eqnarray*}
• (v)Distributive laws\begin{eqnarray*} A\cup ( B\cap C) & = & (A\cup B)\cap (A\cup C)\\ A\cap ( B\cup C) & = & (A\cap B)\cup (A\cap C)\\ A\oplus (B\cup C)& = & (A\oplus B)\cup (A\oplus C)\\ A\oplus (B\cap C) & = & (A\oplus B)\cap (A\oplus C)\\ A\otimes (B\cup C)& = & (A\otimes B)\cup (A\otimes C)\\ A\otimes (B\cap C) & = & (A\otimes B)\cap (A\otimes C) \end{eqnarray*} Distributive laws hold for both sides (right and left).
• (vi)DeMorgan laws \begin{eqnarray*}{(A\cup B)}^c & = & A^c \cap B^c\\ {(A\cap B)}^c & = & A^c \cup B^c\\ {(A\oplus B)}^c & = & A^c \otimes B^c\\ {(A\otimes B)}^c & = & A^c \oplus B^c \end{eqnarray*}
• (vii) Absorption laws\begin{eqnarray*} A\cap (A\cup B) & = & A\\ A\cup (A\cap B) & = & A \end{eqnarray*}

Proof. Straightforward, so we omit.

Theorem 13. Let \(A,B\in PFMS(X)\) such that \(A=B^c\) and \(B=A^c\), then

• (i) \((A^c\cup B)\cap (A\cup B^c)=(A^c\cap B^c)\cup (A\cap B)\),
• (ii) \((A^c\cap B)\cup (A\cap B^c)=(A^c\cup B^c)\cap (A\cup B)\).

Proof. Since \(A=B^c\) and \(B=A^c\), we show that the left hand side (LHS) is equal to the right hand side (RHS). Now, \begin{eqnarray*} (A^c\cup B)\cap (A\cup B^c) & = & (B\cup B)\cap (A\cup A)\\ & = & A\cap B. \end{eqnarray*} Similarly, \begin{eqnarray*} (A^c\cap B^c)\cup (A\cap B) & = & (B\cap A)\cup (A\cap B)\\ & = & A\cap B. \end{eqnarray*} Thus, LHS=RHS, and hence \((i)\) is proved. The proof of \((ii)\) is similar to \((i),\) so we omit.

Definition 14. Let \(A\in PFMS(X)\). Then, the level/ground set of \(A\) is defined by \[A_*=\left\lbrace x\in X| CM_A(x)>0,\: CN_A(x)<1\right\rbrace.\] Certainly, \(A_*\) is a subset of \(X\). }

Proposition 2. Suppose \(A\) and \(B\) are PFMSs of a non-empty set \(X\), then

• (i) \((A\cap B)_*=A_*\cap B_*\),
• (ii) \((A\cup B)_*=A_*\cup B_*\).

Proof. Straightforward, so we omit.

3.2. Accuracy and score functions of PFMS

Definition 15. Let \(A\in PFMS(X)\). Then the score function, \(s\) of \(A\) is defined by \(s(A)=\Sigma_{i=1}^n[(CM_A(x_i))^2-(CN_A(x_i))^2]\), where \(s(A)\in[-1,1]\).

Definition 16. Let \(A\in PFMS(X)\). Then the accuracy function, \(a\) of \(A\) is defined by \(a(A)=\Sigma_{i=1}^n[(CM_A(x_i))^2+(CN_A(x_i))^2]\) for \(a(A)\in[0,1]\).

Theorem 17. Let \(A\in PFMS(X)\). Then the following hold \(\forall x\in X\):

• (i) \(s(A)=0 \Leftrightarrow CM_A(x_i)=CN_A(x_i)\).
• (ii) \(s(A)=1 \Leftrightarrow |CN_A(x_i)|=\sqrt{|(1+CM_A(x_i))(1-CM_A(x_i))|}\).
• (iii \(s(A)=-1 \Leftrightarrow CM_A(x_i)=\sqrt{(CN_A(x_i)+1)(CN_A(x_i)-1)}\).

Proof.

• (i) Suppose \(s(A)=0\). Then \((CM_A(x_i))^2=(CN_A(x_i))^2\) implies \(CM_A(x_i)=CN_A(x_i)\) \(\forall x_i\in X\).
  Conversely, assume \(CM_A(x_i)=CN_A(x_i)\). Then \((CM_A(x_i))^2=(CN_A(x_i))^2\) \(\forall x_i\in X\). Thus \((CM_A(x_i))^2-(CN_A(x_i))^2=0\). Hence \(s(A)=0\).
• (ii) Suppose \(s(A)=1\). Then
  \(1-(CM_A(x_i))^2=-(CN_A(x_i))^2\)
  \(\Rightarrow (1+CM_A(x_i))(1-CM_A(x_i))=-(CN_A(x_i))^2\)
  \(\Rightarrow |(1+CM_A(x_i))(1-CM_A(x_i))|=|CN_A(x_i)|^2\)
  \(\Rightarrow |CN_A(x_i)|=\sqrt{|(1+CM_A(x_i))(1-CM_A(x_i))|}\) \(\forall x_i\in X\).
  Conversely, assume \(|CN_A(x_i)|=\sqrt{|(1+CM_A(x_i))(1-CM_A(x_i))|}\). So we get
  \(|CN_A(x_i)|^2=\sqrt{|1-(CM_A(x_i))^2|}\)
  \(\Rightarrow (CN_A(x_i))^2=1-(CM_A(x_i))^2\) or \(|CN_A(x_i)|^2=\sqrt{|1-(CM_A(x_i))^2|}\)
  \(\Rightarrow -(CN_A(x_i))^2=1-(CM_A(x_i))^2\). Take \(-(CN_A(x_i))^2=1-(CM_A(x_i))^2\) \(\Rightarrow\) \((CM_A(x_i))^2-(CN_A(x_i))^2=1\) \(\Rightarrow\) \(s(A)=1\).
• (iii) Suppose \(s(A)=-1\). Then
  \((CN_A(x_i))^2-1=(CM_A(x_i))^2\)
  \(\Rightarrow (CN_A(x_i)-1)(CN_A(x_i)+1)=(CM_A(x_i))^2\)
  \(\Rightarrow CM_A(x_i)=\sqrt{(CN_A(x_i)-1)(CN_A(x_i)+1)}\).
  Conversely, suppose \(CM_A(x_i)=\sqrt{(CN_A(x_i)-1)(CN_A(x_i)+1)}\). Then
  \((CM_A(x_i))^2=(CN_A(x_i))^2-1 \) \(\Rightarrow (CM_A(x_i))^2-(CN_A(x_i))^2=-1\) \(\Rightarrow s(A)=-1\).

Theorem 18. Let \(A\in PFMS(X)\). Then the following statements hold \(\forall x\in X\):

• i) \(a(A)=1 \Leftrightarrow CH_A(x_i)=0\).
• (ii) \(a(A)=0 \Leftrightarrow |CM_A(x_i)|=|CN_A(x_i)|\).

Proof.

• (i) Suppose \(a(A)=1\). So we have \((CM_A(x_i))^2+(CN_A(x_i))^2=1\), that is, \(CH_A(x_i)=0\) since \(CH_A(x_i)=\sqrt{1-[(CM_A(x_i))^2+(CN_A(x_i))^2]}\).
  Conversely, assume that \(CH_A(x_i)=0\). Then it follows that \[(CM_A(x_i))^2+(CN_A(x_i))^2=1 \Rightarrow a(A)=1.\]
• (ii)] Suppose \(a(A)=0\). Then \((CM_A(x_i))^2=-(CN_A(x_i))^2\) or \((CN_A(x_i))^2=-(CM_A(x_i))^2\) \(\Leftrightarrow\) \(|CM_A(x_i)|^2=|CN_A(x_i)|^2\) \(\Leftrightarrow \) \(|CM_A(x_i)|=|CN_A(x_i)|\).

3.3. Some properties of PFMSs

3.3.1. \((\alpha, \beta)-\)cuts of PFMS

Definition 19. Let \(A\in PFMS(X)\). Then for \(\alpha, \beta\in [0,1]\), the sets \(A_{[\alpha, \beta]}\) and \(A_{(\alpha, \beta)}\) defined by \[A_{[\alpha, \beta]}=\left\lbrace x\in X\mid CM_A(x)\geq \alpha, \: CN_A(x)\leq \beta \right\rbrace\] and \[A_{(\alpha, \beta)}=\left\lbrace x\in X\mid CM_A(x)> \alpha, \: CN_A(x)< \beta \right\rbrace\] are called strong and weak upper \(\alpha, \beta-\)cuts of \(A\).
Similarly, the sets \(A^{[\alpha, \beta]}\) and \(A^{(\alpha, \beta)}\) defined by \[A^{[\alpha, \beta]}=\left\lbrace x\in X\mid CM_A(x)\leq \alpha,\: CN_A(x)\geq \beta \right\rbrace\] and \[A^{(\alpha, \beta)}=\left\lbrace x\in X\mid CM_A(x)< \alpha, \: CN_A(x)> \beta \right\rbrace\] are called strong and weak lower \(\alpha, \beta-\)cuts of \(A\).

Remark 1. Let \(A\in PFMS(X)\) and take any \(\alpha,\beta\in [0,1]\) such that \(A_{[\alpha, \beta]}\) and \(A^{[\alpha, \beta]}\) exist. Then, it follows that

• (i) \(A_{(\alpha,\beta)}\subseteq A_{[\alpha,\beta]}\) and \(A^{(\alpha,\beta)}\subseteq A^{[\alpha,\beta]}\).
• (ii) \(A_{[\alpha,\beta]}=B_{[\alpha,\beta]}, A_{(\alpha,\beta)}=B_{(\alpha,\beta)}, A^{[\alpha,\beta]}=B^{[\alpha,\beta]}\) and \(A^{(\alpha,\beta)}=B^{(\alpha,\beta)}\) iff \(A=B\).

For the purpose of this work, we shall be restricted to strong cuts of PFMS since \(A_{(\alpha,\beta)}\subseteq A_{[\alpha,\beta]}\) and \(A^{(\alpha,\beta)}\subseteq A^{[\alpha,\beta]}\).

Proposition 3. Let \(A,B\in PFMS(X)\) and \(\alpha, \beta, \alpha_1, \alpha_2, \beta_1, \beta_2\in [0,1]\). Then we have

• (i) \(A_{[\alpha_1,\beta_1]}\subseteq A_{[\alpha_2,\beta_2]}\) iff \(\alpha_1\geq \alpha_2\) and \(\beta_1\leq \beta_2\),
• (ii)\(A\subseteq B\) iff \(A_{[\alpha,\beta]}\subseteq B_{[\alpha,\beta]}\).

Proof.

• (i) Let \(x\in A_{[\alpha_1,\beta_1]}\Rightarrow CM_A(x)\geq \alpha_1\) and \(CN_A(x)\leq \beta_1\). Since \(\alpha_1\geq \alpha_2 \Rightarrow CM_A(x)\geq \alpha_1\geq \alpha_2\). Also, \(\beta_1\leq \beta_2 \Rightarrow CN_A(x)\leq \beta_1\leq \beta_2\). Hence, \(A_{[\alpha_1,\beta_1]}\subseteq A_{[\alpha_2, \beta_2]}\).
  Conversely, for \(A_{[\alpha_1,\beta_1]}\subseteq A_{[\alpha_2,\beta_2]}\), it is clear that \(\alpha_1\geq \alpha_2\) and \(\beta_1\leq \beta_2\).
• (ii) We know that \(A\subseteq B\Rightarrow CM_A (x)\leq CM_B (x)\) and \(CN_A (x)\geq CN_B (x)\) \(\forall x\in X\). For \(x\in A_{[\alpha,\beta]}\) and \(x\in B_{[\alpha,\beta]}\) \(\Rightarrow\) \(CM_B (x)\geq CM_A (x)\geq \alpha \) and \(CN_B (x)\leq CN_A (x)\leq \beta\). So, \(A_{[\alpha,\beta]}\subseteq B_{[\alpha,\beta]}\).
  The converse is straightforward.

Corollary 1. Let \(A,B\in PFMS(X)\) and \(\alpha,\beta, \alpha_1, \alpha_2,\beta_1,\beta_2\in [0,1]\). Then the following hold.

• (i) \(A^{[\alpha_1,\beta_1]}\subseteq A^{[\alpha_2,\beta_2]}\) iff \(\alpha_1\geq \alpha_2\) and \(\beta_1\leq \beta_2\).
• (ii) \(A\subseteq B\) iff \(A^{[\alpha,\beta]}\subseteq B^{[\alpha,\beta]}\).

Proof. It follows from Proposition 3.

Proposition 4. Let \(A\in PFMS(X)\). For any \(\alpha_1,\beta_1\alpha_2,\beta_2\in [0,1]\) such that \(\alpha_1\leq \alpha_2\) and \(\beta_1\geq \beta_2\), we have

• (i) \(A_{(\alpha_2,\beta_2)}\subseteq A_{[\alpha_2,\beta_2]}\subseteq A_{(\alpha_1,\beta_1)}\) and
• (ii) \(A^{(\alpha_1,\beta_1)}\subseteq A^{(\alpha_2,\beta_2)}\subseteq A^{[\alpha_2,\beta_2]}\).

Proof. Combining Definition 19 and Remark 1, the proof follows.

Proposition 5. Let \(A,B\in PFMS(X)\) and \(\alpha,\beta\in [0,1]\). Then

• (i)\((A\cap B)_{[\alpha,\beta]} =A_{[\alpha,\beta]}\cap B_{[\alpha,\beta]}\),
• (ii)\((A\cup B)_{[\alpha,\beta]} =A_{[\alpha,\beta]}\cup B_{[\alpha,\beta]}\).

Proof.

• (i) If \(A,B\in PFMS(X)\Rightarrow A\cap B\subseteq A\) and \(A\cap B\subseteq B\). By Proposition 3, \((A\cap B)_{[\alpha,\beta]}\subseteq A_{[\alpha,\beta]}\) and \((A\cap B)_{[\alpha,\beta]}\subseteq B_{[\alpha,\beta]}\) \(\Rightarrow (A\cap B)_{[\alpha,\beta]}\subseteq A_{[\alpha,\beta]}\cap B_{[\alpha,\beta]}\).
  Again, suppose \(x\in A_{[\alpha,\beta]}\cap B_{[\alpha,\beta]}\Rightarrow x\in A_{[\alpha,\beta]}\) and \(x\in B_{[\alpha,\beta]}\), then \begin{eqnarray*}A_{[\alpha,\beta]}\cap B_{[\alpha,\beta]}&=& \left\lbrace x\in X\mid CM_A (x)\geq \alpha,\: CN_A (x)\leq \beta \right\rbrace \cap \left\lbrace x\in X\mid CM_B (x)\geq \alpha, \: CN_B (x)\leq \beta\right\rbrace \\ &=& \left\lbrace x\in X\mid min[CM_A (x)\geq \alpha,CM_B (x)\geq \alpha],\: max[CN_A (x)\leq \beta,CN_B (x)\leq \beta]\right\rbrace\\ & =& \left\lbrace x\in X\mid min[CM_A (x),CM_B (x)]\geq \alpha, \: max[CN_A (x),CN_B (x)]\leq \beta\right\rbrace\\ &=& \left\lbrace x\in X\mid CM_{A\cap B} (x)\geq \alpha,\: CN_{A\cap B} (x)\leq \beta\right\rbrace\\ &\subseteq& (A\cap B)_{[\alpha,\beta]}.\end{eqnarray*}
  Consequently, \(x\in A_{[\alpha,\beta]}\cap B_{[\alpha,\beta]}\Rightarrow x\in (A\cap B)_{[\alpha,\beta]}\). Hence \((A\cap B)_{[\alpha,\beta]} =A_{[\alpha,\beta]} \cap B_{[\alpha,\beta]}\).
• (ii) For \(A,B\in PFMS(X)\), it is clear that \(A\subseteq A\cup B\) and \(B\subseteq A\cup B\). By Proposition 3, implies \(A_{[\alpha,\beta]}\subseteq (A\cup B)_{[\alpha,\beta]}\) and \(B_{[\alpha,\beta]}\subseteq (A\cup B)_{[\alpha,\beta]}\), that is, \(A_{[\alpha,\beta]}\cup B_{[\alpha,\beta]}\subseteq (A\cup B)_{[\alpha,\beta]}\).
  Also, \(x\in (A\cup B)_{[\alpha,\beta]}\Rightarrow\) \(CM_{(A\cup B)}(x)\geq \alpha\) and \(CN_{(A\cup B)}(x)\leq \beta\), that is,
  \begin{eqnarray*}(A\cup B)_{[\alpha,\beta]} &=& \left\lbrace x\in X\mid CM_{(A\cup B)}(x)\geq \alpha, CN_{(A\cup B)}(x)\leq \beta\right\rbrace\\ &=& \left\lbrace x\in X\mid max[CM_A(x),CM_B(x)]\geq \alpha, min[CN_A(x),CN_B(x)]\leq \beta\right\rbrace \\ &=& \left\lbrace x\in X\mid max[CM_A(x), CM_B(x)]\geq \alpha, min[CN_A(x), CN_B(x)]\leq \beta\right\rbrace\\ &=& \left\lbrace x\in X\mid CM_A(x)\geq \alpha, CN_A(x)\leq \beta\right\rbrace \cup \left\lbrace x\in X\mid CM_B(x)\geq \alpha, CN_B(x)\leq \beta\right\rbrace\\ &\subseteq& A_{[\alpha,\beta]}\cup B_{[\alpha,\beta]}.\end{eqnarray*}
  Thus \(x\in A_{[\alpha,\beta]}\) and \(x\in B_{[\alpha,\beta]}\). Hence \((A\cup B)_{[\alpha,\beta]}=A_{[\alpha,\beta]}\cup B_{[\alpha,\beta]}\).

Corollary 2. Let \(A,B\in PFMS(X)\) and \(\alpha,\beta\in [0,1]\). Then

• (i) \((A\cap B)^{[\alpha,\beta]} =A^{[\alpha,\beta]}\cap B^{[\alpha,\beta]}\),
• (ii) \((A\cup B)^{[\alpha,\beta]} =A^{[\alpha,\beta]}\cup B^{[\alpha,\beta]}\).

Proof. Straightforward from Proposition 5.

Proposition 6. Suppose \(\left\lbrace A_i\right\rbrace_{i\in I}\in PFMS(X)\) and \(\alpha,\beta\in [0,1]\), then

• (i)]\((\bigcap_{i\in I}A_i)_{[\alpha,\beta]}=\bigcap_{i\in I}(A_i)_{[\alpha,\beta]}\),
• (ii)]\((\bigcup_{i\in I}A_i)_{[\alpha,\beta]}=\bigcup_{i\in I}(A_i)_{[\alpha,\beta]}\),
• (iii)]\((\bigcap_{i\in I}A_i)^{[\alpha,\beta]}=\bigcap_{i\in I}(A_i)^{[\alpha,\beta]}\),
• (iv)]\((\bigcup_{i\in I}A_i)^{[\alpha,\beta]}=\bigcup_{i\in I}(A_i)^{[\alpha,\beta]}\).

Proof. (i) Let \(C=\bigcap_{i\in I}A_i\), then \(CM_C(x)=\bigwedge_{i\in I}CM_{A_{i}}(x)\) and \(CN_C(x)=\bigvee_{i\in I}CN_{A_{i}}(x)\) \(\forall x\in X\). Thus \begin{eqnarray*} C_{[\alpha,\beta]} & = & \left\lbrace x\in X\mid CM_{C}(x)\geq \alpha, CN_{C}(x)\leq \beta \right\rbrace\\ & = & \left\lbrace x\in X\mid (\bigwedge_{i\in I}CM_{A_{i}}(x))\geq \alpha, (\bigvee_{i\in I}CN_{A_{i}}(x))\leq \beta\right\rbrace\\ & = & \left\lbrace x\in X\mid \bigwedge_{i\in I}CM_{A_{i}}(x)\geq \alpha, \bigvee_{i\in I}CN_{A_{i}}(x)\leq \beta \right\rbrace\\ & = & \bigcap_{i\in I}(A_i)_{[\alpha,\beta]}. \end{eqnarray*}Hence \((\bigcap_{i\in I}A_i)_{[\alpha,\beta]}=\bigcap_{i\in I}(A_i)_{[\alpha,\beta]}\).
(ii)-(iv) follow similarly.

Remark 2. Suppose \(A,B,C\in PFMS(X)\) such that \(B\subseteq C\). Then for \(\alpha,\beta\in [0,1]\), we have

• (i) \((A\cap B)_{[\alpha,\beta]}\subseteq (A\cap C)_{[\alpha,\beta]},\)
• (ii) \((A\cup B)_{[\alpha,\beta]}\subseteq (A\cup C)_{[\alpha,\beta]}\),
• (iii) \((A\cap B)^{[\alpha,\beta]}\subseteq (A\cap C)^{[\alpha,\beta]},\)
• (iv)\((A\cup B)^{[\alpha,\beta]}\subseteq (A\cup C)^{[\alpha,\beta]}\).

Proposition 7. Let \(f\) be a function from \(X\) to \(Y\), \(A\in PFMS(X)\) and \(B\in PFMS(Y)\), respectively. Then, for any \(\alpha,\beta\in [0,1]\), we have

• (i)\(f(A_{[\alpha,\beta]})\subseteq (f(A))_{[\alpha,\beta]}\),
• (ii)\(f^{-1}(B_{[\alpha,\beta]})=(f^{-1}(B))_{[\alpha,\beta]}\),
• (iii) \(f(A_{(\alpha,\beta)})\subseteq f(A_{[\alpha,\beta]})\subseteq (f(A))_{[\alpha,\beta]}\),
• (iv)\(f^{-1}(B_{(\alpha,\beta)})\subseteq f^{-1}(B_{[\alpha,\beta]})=(f^{-1}(B))_{[\alpha,\beta]}\).

Proof.

• (i) Let \(y\in f(A_{[\alpha,\beta]})\), then \(\exists \: x\in A_{[\alpha,\beta]}\) such that \(f(x)=y\) and \(CM_A(x)\geq \alpha\), \(CN_A(x)\leq \beta\). Consequently, we get \[CM_A(f^{-1}(y))\geq \alpha, \alpha\in [0,1] \: \textrm{implies}\: CM_{f(A)}(y)\geq \alpha, \alpha\in [0,1],\]
  Similarly, \[CN_A(f^{-1}(y))\leq \beta, \beta\in [0,1] \: \textrm{implies}\: CN_{f(A)}(y)\leq \beta, \beta\in [0,1],\] and so, \(y\in (f(A))_{[\alpha,\beta]}\). Hence, \(f(A_{[\alpha]})\subseteq (f(A))_{[\alpha]}\).
• (ii) For every \(x\), \(x\in f^{-1}(B_{[\alpha,\beta]})\Leftrightarrow f(x)\in B_{[\alpha,\beta]}\) \(\Leftrightarrow CM_B(f(x))\geq \alpha\) and \(CN_B(f(x))\leq \beta\). Thus \(CM_{f^{-1}(B)}(x)=CM_B(f(x))\geq \alpha\) and \(CN_{f^{-1}(B)}(x)=CN_B(f(x))\leq \beta\) that is, \(x\in (f^{-1}(B))_{[\alpha,\beta]}\). Hence, \(f^{-1}(B_{[\alpha,\beta]})=(f^{-1}(B))_{[\alpha,\beta]}\).
• (iii) Since \(A_{(\alpha,\beta)}\subseteq A_{[\alpha,\beta]}\), then \(f(A_{(\alpha,\beta)})\subseteq f(A_{[\alpha,\beta]})\). Hence, the result follows from (i).
• (iv) Also, \(B_{(\alpha,\beta)}\subseteq B_{[\alpha,\beta]}\) and so, \(f^{-1}(A_{(\alpha,\beta)})\subseteq f^{-1}(A_{[\alpha,\beta]})\) by the same reasons as in (iii). The proof is completed by (ii).

Corollary Suppose \(f\) is a function from \(X\) to \(Y\). If \(A\in PFMS(X)\) and \(B\in PFMS(Y)\), respectively, then for at least one \(\alpha, \beta\in [0,1]\),

• (i)\(f(A^{[\alpha,\beta]})\subseteq (f(A))^{[\alpha,\beta]}\),
• (ii)\(f^{-1}(B^{[\alpha,\beta]})=(f^{-1}(B))^{[\alpha,\beta]}\),
• (iii) \(f(A^{(\alpha,\beta)})\subseteq f(A^{[\alpha,\beta]})\subseteq (f(A))^{[\alpha,\beta]}\),
• (iv)\(f^{-1}(B^{(\alpha,\beta)})\subseteq f^{-1}(B^{[\alpha,\beta]})=(f^{-1}(B))^{[\alpha,\beta]}\).

Proof. Similar to Proposition 7.

3.3.2. Some modal operators on PFMS

Now, we propose and explicate some modal operators on PFMS, which transform every PFMS to fuzzy multiset. These modal operators are similar to the operators necessity and possibility defined in some modal logics.

Definition 20. Let \(A\in PFMS(X)\). Then we define the following operators:

• (i) the necessity operator \[\square A=\left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2}\right\rangle |x\in X\right\rbrace\]
• (ii) the possibility operator \[\lozenge A=\left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x)\right\rangle |x\in X\right\rbrace.\]

Remark 3. If \(A\) is an ordinary fuzzy multiset, then \(\square A=A=\lozenge A\). An ordinary fuzzy multiset \(A\) can also be written in PFMS setting as \[A=\left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2}\right\rangle |x\in X\right\rbrace\] or \[A=\left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x)\right\rangle |x\in X\right\rbrace.\]

Theorem 21. Let \(A\in PFMS(X)\). Then the following properties hold:

• (i) \(\overline{\square \bar{A}}=\lozenge A\)
• (ii) \(\overline{\lozenge \bar{A}}=\square A\)
• (iii) \(\square \square A=\square A\)
• (iv) \(\lozenge \lozenge A=\lozenge A\)
• (v)\(\square \lozenge A=\lozenge A\)
• (vi) \(\lozenge \square A=\square A\).

Proof. Let \(x\in X\). Using Definition 20, we have

• (i) \begin{eqnarray*} \overline{\square \bar{A}} & = & \overline{\square \overline{\left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace}}\\ & = & \overline{\square \left\lbrace \left\langle x, CN_A(x),CM_A(x) \right\rangle |x\in X\right\rbrace}\\ & = & \overline{ \left\lbrace \left\langle x, CN_A(x),\sqrt{1-(CN_A(x))^2} \right\rangle |x\in X\right\rbrace}\\ & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \lozenge A. \end{eqnarray*}
• (ii) \begin{eqnarray*} \overline{\lozenge \bar{A}} & = & \overline{\lozenge \overline{\left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace}}\\ & = & \overline{\lozenge \left\lbrace \left\langle x, CN_A(x),CM_A(x) \right\rangle |x\in X\right\rbrace}\\ & = & \overline{ \left\lbrace \left\langle x, \sqrt{1-(CM_A(x))^2}, CM_A(x) \right\rangle |x\in X\right\rbrace}\\ & = & \left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \square A. \end{eqnarray*}
• (iii) \begin{eqnarray*} \square \square A & = & \square \square \left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \square \left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \square A. \end{eqnarray*}
• (iv) \begin{eqnarray*} \lozenge \lozenge A & = & \lozenge \lozenge \left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \lozenge \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \lozenge A. \end{eqnarray*}
• (v) \begin{eqnarray*} \square \lozenge A & = & \square \lozenge \left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \square \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, \sqrt{1-(\sqrt{1-(CN_A(x))^2)})^2} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, \sqrt{1-(1-(CN_A(x))^2)} \right\rangle |x\in X\right\rbrace \end{eqnarray*} \begin{eqnarray*} & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, \sqrt{(CN_A(x))^2)} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \lozenge A.\end{eqnarray*}
• (vi) \begin{eqnarray*} \lozenge \square A & = & \lozenge \square \left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \lozenge \left\lbrace \left\langle x, CM_A(x)), \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(\sqrt{1-(CM_A(x))^2)})^2}, \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(1-(CM_A(x))^2)}, \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{(CM_A(x))^2}, \sqrt{1-(CM_A(x))^2)} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \square A. \end{eqnarray*}

Corollary 4. Let \(A\in PFMS(X)\). Then the following properties hold:

• (i)\(\overline{\square \bar{A}}=\square \lozenge A= \lozenge \lozenge A\)
• (ii) \(\overline{\lozenge \bar{A}}=\lozenge \square A= \square \square A\).

Proof. Straightforward from Theorem 21.

Theorem 22. Let \(A,B\in PFMS(X)\). Then the following properties hold:

• (i) \(\square(A\cap B)=\square A\cap \square B\)
• (ii) \(\lozenge(A\cap B)=\lozenge A\cap \lozenge B\)
• (iii) \(\square(A\cup B)=\square A\cup \square B\)
• (iv) \(\lozenge(A\cup B)=\lozenge A\cup \lozenge B\).

Proof. The results are straightforward from Definitions 11 and 20, so we omit the proofs.

Theorem 23. Let \(A,B\in PFMS(X)\). Then the following properties hold:

• (i) \(\overline{\square \overline{(A\cap B)}}=\lozenge (A\cap B)\)
• (ii) \(\overline{\lozenge \overline{(A\cap B)}}=\square (A\cap B)\)
• (iii) \(\overline{\square \overline{(A\cup B)}}=\lozenge (A\cup B)\)
• (iv) \(\overline{\lozenge \overline{(A\cup B)}}=\square (A\cup B)\)
• (v) \(\square \square (A\cap B)=\square (A\cap B)\)
• (vi) \(\lozenge \lozenge (A\cap B)=\lozenge (A\cap B)\)
• (vii) \(\square \square (A\cup B)=\square (A\cup B)\)
• (viii) \(\lozenge \lozenge (A\cup B)=\lozenge (A\cup B)\)
• (ix) \(\square \lozenge (A\cap B)=\lozenge (A\cap B)\)
• (x) \(\lozenge \square (A\cap B)=\square (A\cap B)\)
• (xi) \(\square \lozenge (A\cup B)=\lozenge (A\cup B)\)
• (xii) \(\lozenge \square (A\cup B)=\square (A\cup B)\).

Proof. Synthesizing Theorems 21 and 22, the proofs follow.

Corollary 5. Let \(A,B\in PFMS(X)\). Then the following properties hold:

• (i) \(\overline{\square \overline{(A\cap B)}}=\lozenge \lozenge (A\cap B)=\square \lozenge (A\cap B)\)
• (ii) \(\overline{\lozenge \overline{(A\cap B)}}=\square \square (A\cap B)=\lozenge \square (A\cap B)\)
• (iii) \(\overline{\square \overline{(A\cup B)}}=\lozenge \lozenge (A\cup B)=\square \lozenge(A\cup B)\)
• (iv) \(\overline{\lozenge \overline{(A\cup B)}}=\square \square(A\cup B)=\lozenge \square(A\cup B)\).

Proof. Straightforward from Theorem 23.

Theorem 24. Let \(A\in PFMS(X)\). Then \(\square A\subset A\subset \lozenge A\).

Proof. Recall that for \(A=\left\lbrace \left\langle x, CM_A(x), CN_A(x)\right\rangle |x\in X\right\rbrace\), we have \[\square A=\left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2}\right\rangle |x\in X\right\rbrace\] and \[\lozenge A=\left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x)\right\rangle |x\in X\right\rbrace.\] Also, \(A\subset B\Leftrightarrow A\subseteq B\) and \(A\neq B\), and \(A\subseteq B\Leftrightarrow CM_A(x)\leq CM_B(x)\) and either \(CN_A(x)\geq CN_B(x)\) (or \(CN_A(x)\leq CN_B(x)\)) \(\forall x\in X\). To prove that \(\square A\subset A\), it is sufficient to show that \[\sqrt{1-(CM_A(x))^2}\geq CN_A(x).\] From Definition 6, we have \begin{eqnarray*} (CM_A(x))^2+(CN_A(x))^2\leq 1 & \Rightarrow & (CN_A(x))^2\leq 1-(CM_A(x))^2\\ & \Rightarrow & CN_A(x)\leq \sqrt{1-(CM_A(x))^2}, \end{eqnarray*} that is \(\sqrt{1-(CM_A(x))^2}\geq CN_A(x)\) \(\forall x\in X\). Thus \(\square A\subset A\). Again, we show that \(A\subset \lozenge A\). To see this, it is enough to prove that \[CM_A(x)\leq \sqrt{1-(CN_A(x))^2}.\] By Definition 6, We get \begin{eqnarray*} (CM_A(x))^2+(CN_A(x))^2\leq 1 & \Rightarrow & (CM_A(x))^2\leq 1-(CN_A(x))^2\\ & \Rightarrow & CM_A(x)\leq \sqrt{1-(CN_A(x))^2}\; \forall x\in X. \end{eqnarray*} Hence, \(A\subset \lozenge A\), and the proof is complete.

4. Composite relation on PFMS and its application in course placements

4.1. Composite relation defined on PFMS

In what follows, we define the composite relation on PFMS.

Definition 25. Let \(X\) and \(Y\) be two non-empty sets. A Pythagorean fuzzy multi-relation (PFMR), \(R\) from \(X\) to \(Y\) is a PFMS of \(X\times Y\) characterised by the count membership function, \(CM_R\) and count non-membership function, \(CN_R\). A PF multi-relation or PFMR from \(X\) to \(Y\) is denoted by \(R(X\to Y)\).

Definition 26. Let \(A\in PFMS(X)\). Then the max-min-max composition of \(R(X\to Y)\) with \(A\) is a PFMS \(B\) of \(Y\) denoted by \(B=R\circ A\), such that its count membership and count non-membership functions are defined by \[CM_B(y)=\bigvee_{x}(min[CM_A(x), CM_R(x,y)])\] and \[CN_B(y)=\bigwedge_{x}(max[CN_A(x), CN_R(x,y)])\] \(\forall x\in X\) and \(y\in Y\), where \(\bigvee=\)maximum, \(\bigwedge=\)minimum.

Definition 27. Let \(Q(X\to Y)\) and \(R(Y\to Z)\) be two PFMRs. Then the max-min-max composition \(R\circ Q\) is a PFMR from \(X\) to \(Z\) such that its count membership and count non-membership functions are defined by \[CM_{R\circ Q}(x,z)=\bigvee_{y}(min[CM_Q(x,y), CM_R(y,z)])\] and \[CN_{R\circ Q}(x,z)=\bigwedge_{y}(max[CN_Q(x,y), CN_R(y,z)])\] \(\forall (x,z)\in X\times Z\) and \(\forall y\in Y\).

Remark 4. From Definitions 26 and 27, the max-min-max composition \(B\) or \(R\circ Q\) is calculated by \[B=CM_B(y)-CN_B(y)CH_B(y)\] \(\forall y\in Y\) or \[R\circ Q=CM_{R\circ Q}(x,z)-CN_{R\circ Q}(x,z)CH_{R\circ Q}(x,z)\] \(\forall (x,z)\in X\times Z\).

Proposition 8. If \(R\) and \(S\) are two PFMRs on \(X\times Y\) and \(Y\times Z\), respectively. Then

• (i) \((R^{-1})^{-1}=R\),
• (ii) \((S\circ R)^{-1}=R^{-1}\circ S^{-1}\).

4.2. Composite relation on PFMS in course placement

We apply the notion of PFMR as follows. Let \[S=\left\lbrace s_1,...,s_l\right\rbrace, C=\left\lbrace c_1,...,c_m\right\rbrace \: \textrm{and}\:
A=\left\lbrace a_1,...,a_n\right\rbrace\] be finite set of subjects related to the courses, finite set of courses and finite set of applicants, respectively. Assume there are two PFMRs, \(R(A\to S)\) and \(U(S\to C)\) such that \[R=\left\lbrace \left\langle (a,s), CM_R(a,s), CN_R(a,s)\right\rangle| (a,s)\in A\times S \right\rbrace\] and \[U=\left\lbrace \left\langle (s,c), CM_U(s,c), CN_U(s,c)\right\rangle| (s,c)\in S\times C \right\rbrace,\] where \(CM_R(a,s)\) signifies the grade to which the applicant, \(a\) passes the related subject requirement, \(s\), and \(CN_R(a,s)\) signifies the grade to which the applicant, \(a\) does not pass the related subject requirement, \(s\).
Similarly, \(CM_U(s,c)\) signifies the grade to which the related subject requirement, \(s\) determines the course, \(c\), and \(CN_U(s,c)\) signifies the grade to which the related subject requirement, \(s\) does not determine the course, \(c\).
The composition, \(T\) of \(R\) and \(U\) is given as \(T=R\circ U\). This describes the state in which the applicants, \(a_i\) with respect to the related subjects requirement, \(s_j\) fit the courses, \(c_k\). Thus: \[CM_T(a_i,c_k)=\bigvee_{s_j\in S}\left\lbrace min[CM_R(a_i,s_j), CM_U(s_j,c_k)]\right\rbrace\] and \[CN_T(a_i,c_k)=\bigwedge_{s_j\in S}\left\lbrace max[CN_R(a_i,s_j), CN_U(s_j,c_k)]\right\rbrace\] \(\forall a_i\in A\) and \(c_k\in C\), where \(i,j\) and \(k\) take values from \(1,...,n\). The career placement can be attained if the value of \(T\) given by \[T=CM_T(a_i,c_k)-CN_T(a_i,c_k) CH_T(a_i,c_k),\] as calculated from \(R\) and \(U\) for the placements of \(a_i\) into any \(c_k\) with respect to \(s_j\) is the greatest, greater than 0.5.

4.2.1. Case study

Let \(A=\left\lbrace \textrm{Eli, Ella, Avi, Joe, Jones}\right\rbrace\) be the set of applicants for the course placements, \[C=\left\lbrace \textrm{medicine, pharmacy, surgery, anatomy, physiology}\right\rbrace\] be the set of courses the applicants are competing for, and \[S=\left\lbrace \textrm{English Lang., Maths, Biology, Physics, Chemistry, Health Sci.}\right\rbrace\] be the set of subjects' requirement to the set of courses.
Suppose the PFMR, \(R(A\to S)\) is given in Table 3. This data in Pythagorean fuzzy multi-values are supposedly drawn after the applicants sat for a multiple choice qualification assessments on the listed subjects within a specified time, for two different times where the first and second assessments are closely related to checkmate the effect of test contingencies.
The first entries are the membership values signifying the Pythagorean fuzzy multi-values of the marks allocated to the questions the applicants answered, and the second entries are the non-membership values signifying the Pythagorean fuzzy multi-values of the marks allocated to the questions failed. Converting the PFMSs in Table 2 to PFSs for easy computation, the results in Table 3 are obtained.

Table 2. \(R(A\to S)\).

\(R\)	English	Maths	Biology	Physics	Chemistry	Health
Eli	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.5,0.4\right\rangle\)	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.5,0.3\right\rangle\)	\(\left\langle 0.5,0.5\right\rangle\)	\(\left\langle 0.6,0.2\right\rangle\)
	\(\left\langle 0.5,0.3\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.5,0.4\right\rangle\)	\(\left\langle 0.5,0.3\right\rangle\)	\(\left\langle 0.5,0.1\right\rangle\)
Ella	\(\left\langle 0.5,0.3\right\rangle\)	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.5,0.3\right\rangle\)	\(\left\langle 0.4,0.5\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)	\(\left\langle 0.7,0.1\right\rangle\)
	\(\left\langle 0.4,0.3\right\rangle\)	\(\left\langle 0.8,0.2\right\rangle\)	\(\left\langle 0.7,0.3\right\rangle\)	\(\left\langle 0.6,0.2\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)	\(\left\langle 0.7,0.3\right\rangle\)
Avi	\(\left\langle 0.7,0.3\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)	\(\left\langle 0.7,0.3\right\rangle\)	\(\left\langle 0.5,0.4\right\rangle\)	\(\left\langle 0.4,0.5\right\rangle\)	\(\left\langle 0.6,0.3\right\rangle\)
	\(\left\langle 0.6,0.2\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)	\(\left\langle 0.5,0.2\right\rangle\)	\(\left\langle 0.4,0.5\right\rangle\)	\(\left\langle 0.5,0.5\right\rangle\)	\(\left\langle 0.7,0.3\right\rangle\)
Joe	\(\left\langle 0.6,0.4\right\rangle\)	\(\left\langle 0.8,0.2\right\rangle\)	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)
	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.6,0.1\right\rangle\)	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.5,0.2\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)
Jones	\(\left\langle 0.8,0.1\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)	\(\left\langle 0.8,0.2\right\rangle\)	\(\left\langle 0.7,0.1\right\rangle\)	\(\left\langle 0.6,0.1\right\rangle\)	\(\left\langle 0.8,0.1\right\rangle\)
	\(\left\langle 0.6,0.2\right\rangle\)	\(\left\langle 0.5,0.1\right\rangle\)	\(\left\langle 0.5,0.4\right\rangle\)	\(\left\langle 0.7,0.1\right\rangle\)	\(\left\langle 0.4,0.2\right\rangle\)	\(\left\langle 0.8,0.1\right\rangle\)

Table 3. \(R(A\to S)\).

\(R\)	English	Maths	Biology	Physics	Chemistry	Health
Eli	\(\left\langle 0.55,0.30\right\rangle\)	\(\left\langle 0.60,0.30\right\rangle\)	\(\left\langle 0.60,0.30\right\rangle\)	\(\left\langle 0.50,0.35\right\rangle\)	\(\left\langle 0.50,0.40\right\rangle\)	\(\left\langle 0.55,0.15\right\rangle\)
Ella	\(\left\langle 0.45,0.30\right\rangle\)	\(\left\langle 0.70,0.25\right\rangle\)	\(\left\langle 0.60,0.30\right\rangle\)	\(\left\langle 0.50,0.35\right\rangle\)	\(\left\langle 0.70,0.20\right\rangle\)	\(\left\langle 0.70,0.20\right\rangle\)
Avi	\(\left\langle 0.65,0.25\right\rangle\)	\(\left\langle 0.70,0.20\right\rangle\)	\(\left\langle 0.60,0.25\right\rangle\)	\(\left\langle 0.45,0.45\right\rangle\)	\(\left\langle 0.45,0.50\right\rangle\)	\(\left\langle 0.65,0.30\right\rangle\)
Joe	\(\left\langle 0.60,0.35\right\rangle\)	\(\left\langle 0.70,0.15\right\rangle\)	\(\left\langle 0.60,0.30\right\rangle\)	\(\left\langle 0.55,0.25\right\rangle\)	\(\left\langle 0.65,0.25\right\rangle\)	\(\left\langle 0.70,0.20\right\rangle\)
Jones	\(\left\langle 0.70,0.15\right\rangle\)	\(\left\langle 0.60,0.15\right\rangle\)	\(\left\langle 0.65,0.30\right\rangle\)	\(\left\langle 0.70,0.10\right\rangle\)	\(\left\langle 0.50,0.15\right\rangle\)	\(\left\langle 0.80,0.10\right\rangle\)

PFMR, \(U(S\to C)\) is the School bench-mark for admission into the mentioned courses in Pythagorean fuzzy values. The data is in Table 4.

Table 4. \(U(S\to C)\).

\(U\)	medicine	pharmacy	surgery	anatomy	physiology
English	\(\left\langle 0.8,0.1\right\rangle\)	\(\left\langle 0.9,0.1\right\rangle\)	\(\left\langle 0.5,0.4\right\rangle\)	\(\left\langle 0.7,0.3\right\rangle\)	\(\left\langle 0.8,0.2\right\rangle\)
Maths	\(\left\langle 0.7,0.2\right\rangle\)	\(\left\langle 0.8,0.1\right\rangle\)	\(\left\langle 0.5,0.3\right\rangle\)	\(\left\langle 0.5,0.4\right\rangle\)	\(\left\langle 0.5,0.3\right\rangle\)
Biology	\(\left\langle 0.9,0.1\right\rangle\)	\(\left\langle 0.8,0.2\right\rangle\)	\(\left\langle 0.9,0.1\right\rangle\)	\(\left\langle 0.8,0.2\right\rangle\)	\(\left\langle 0.9,0.1\right\rangle\)
Physics	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.5,0.2\right\rangle\)	\(\left\langle 0.5,0.4\right\rangle\)	\(\left\langle 0.6,0.3\right\rangle\)	\(\left\langle 0.6,0.2\right\rangle\)
Chemistry	\(\left\langle 0.8,0.2\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)	\(\left\langle 0.7,0.3\right\rangle\)	\(\left\langle 0.8,0.2\right\rangle\)	\(\left\langle 0.7,0.2\right\rangle\)
Health	\(\left\langle 0.8,0.1\right\rangle\)	\(\left\langle 0.8,0.2\right\rangle\)	\(\left\langle 0.7,0.3\right\rangle\)	\(\left\langle 0.9,0.1\right\rangle\)	\(\left\langle 0.8,0.2\right\rangle\)

The values of \(CM_{R\circ U}(a_i,c_k)\) and \(CN_{R\circ U}(a_i,c_k)\) of the composition \(T=R\circ U\) follow:

Table 5. \(CM_{R\circ U}(a_i,c_k)\) and \(CN_{R\circ U}(a_i,c_k)\).

\(CM\), \(CN\)	medicine	pharmacy	surgery	anatomy	physiology
Eli	\(\left\langle 0.60,0.15\right\rangle\)	\(\left\langle 0.60,0.20\right\rangle\)	\(\left\langle 0.60,0.30\right\rangle\)	\(\left\langle 0.60,0.15\right\rangle\)	\(\left\langle 0.60,0.20\right\rangle\)
Ella	\(\left\langle 0.70,0.20\right\rangle\)	\(\left\langle 0.70,0.20\right\rangle\)	\(\left\langle 0.70,0.30\right\rangle\)	\(\left\langle 0.70,0.20\right\rangle\)	\(\left\langle 0.70,0.20\right\rangle\)
Avi	\(\left\langle 0.70,0.20\right\rangle\)	\(\left\langle 0.70,0.20\right\rangle\)	\(\left\langle 0.65,0.25\right\rangle\)	\(\left\langle 0.65,0.25\right\rangle\)	\(\left\langle 0.65,0.25\right\rangle\)
Joe	\(\left\langle 0.70,0.20\right\rangle\)	\(\left\langle 0.70,0.15\right\rangle\)	\(\left\langle 0.70,0.30\right\rangle\)	\(\left\langle 0.70,0.20\right\rangle\)	\(\left\langle 0.70,0.20\right\rangle\)
Jones	\(\left\langle 0.80,0.10\right\rangle\)	\(\left\langle 0.80,0.15\right\rangle\)	\(\left\langle 0.70,0.30\right\rangle\)	\(\left\langle 0.80,0.10\right\rangle\)	\(\left\langle 0.80,0.20\right\rangle\)

After computing the values of the count of hesitation margins which signify the marks loss due to the hesitation in answering questions within the specified time, \(T\) is calculated as show in Table 6.

Table 6. \(T=CM_T-CN_T CH_T\).

\(T\)	medicine	pharmacy	surgery	anatomy	physiology
Eli	0.4821	0.4451	0.3775	0.4821	0.4451
Ella	0.5629	0.5629	0.5056	0.5629	0.5629
Avi	0.5629	0.5629	0.4706	0.4706	0.4706
Joe	0.5629	0.5953	0.5056	0.5629	0.5629
Jones	0.7408	0.7129	0.5056	0.7408	0.6869

The course placements are carried out on the basis of which of the applicant has the greatest \(T\) such that \(T >0.5\). However, if an applicant is suitable to study more than one courses based on the value of \(T\), then the applicant would be allowed to make a personal choice within the range of the courses he/she has the greatest \(T\) such that \(T >0.5\).

From Table 6, the following placements are made: Eli is not suitable to read any of the courses; Ella is suitable to read any of medicine, pharmacy, anatomy and physiology; Avi is suitable to read either medicine or pharmacy; Joe is suitable to read pharmacy and Jones is suitable to read either medicine or anatomy. Suppose there is only one slot for medicine, it would be given to Jones. Also, if one applicant is to ready pharmacy, it would be Joe. Notwithstanding, Jones is very suitable to read any of the courses ahead of all the applicants.

5. Conclusion

We have proposed the idea of PFMSs as the generalization of PFSs such that each of membership degree, non-membership degree and hesitation margin of PFS, are allowed to repeat as count membership degrees, count non-membership degrees and count hesitation degrees. It was shown that PFMS is either the incorporation of IFMS into PFS setting or PFS in multiset framework. We have discussed some algebraic properties of PFMS and proposed the analogs of the modal logic operators "necessity" and "possibility" with some results. Also the ideas of level sets, cuts, accuracy and score functions were established in the setting of PFMS with a number of results. The notion of composite relation is proposed in PFMSs and applied to practical decision-making problem of course placements in higher institution. PFMS as a soft computing technique can find expression in image enhancement techniques for better image quality, other multi-criteria decision-making (MCDM) problems or multi-attribute decision-making (MADM) problems among other potential applications.

Acknowledgments

The author would like to thank the Editor -in-chief for his technical comments and the reviewers for their insightful contributions.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

The concept of Vehicle Routing Problem (VRP) was first proposed in [1] and a mathematical programming formulation and algorithm method for VRP were also developed in this study. This problem generalizes the famous and common Traveling Salesman Problem (TSP) which is one of the simplest routing problems. According to [2], the TSP involves finding the optimal/shortest route that connects all routes exactly once and return to the starting node from a given set of finite route and also measuring the distance between them. Due to the set of several available routes, VRP is a computationally difficult problem even though many algorithms (such as heuristic and exact algorithm) have been proposed over the years. The challenging task is how to produce a solution that will be fast and reliable.

The VRP is a combinatorial optimization and integer programming problem which finds optimal path in order to deliver goods and services to a finite set of customers. It can be described as a graph \(G(\mathcal{V}, \mathcal{E})\) where \(\mathcal{V}\) denote the set of nodes and \(\mathcal{E}\) denote the weighted edges between the nodes. The VRP has been a problem for several decades and one of the most studied problem in logistics engineering, applied mathematics and computer science, and which is described as finding optimal routes for a fleet of vehicles to serve some scattered customers from depot [1, 3, 4, 5, 6].

The Capacitated VRP (CVRP) involves finding set of optimal routes for some fleets of homogeneous vehicles with capacity constraints, moving from a central depot to service customer demands and return to same depot. Our basic assumption before solving CVRP includes; for the customer, the quantity of demand and the traveling cost from one customer to another are known, and for the vehicles, the exact number of vehicles and its capacity must be known.

The VRP consists of several variants which includes; TSP [7, 8, 9, 10], Multiple TSP [11, 12], CVRP [10, 13, 14], VRP with time windows (VRPTW) [15, 16], dynamic VRP (DVRP)[17, 18], pickup and delivery VRP (PDVRP)[19], periodic VRP (PVRP)[20] and so on. These variants have several application which include; DVRP is applicable in the courier services [21], Milk-collection problem [22], and so on.

In this paper, our goal is to find optimal set of routes for some vehicles delivering goods or services to some known locations. Further, the capacity of these homogeneous vehicles must not be exceeded. To achieve this goal, we shall;

1. formulate a mathematical model for the CVRP problem,
2. develop a reinforcement learning framework for solving CVRP, and
3. check the optimal gap between column generation, Google's operations research tools and reinforcement learning.

Our proposed algorithm is different from [5] and will be applied on Augerat et al.,[14] instances. The remainder of our paper is broken down as follows: Section 2 shows the literature review. Section 3 introduces the mathematical model of CVRP and algorithms to be used. The various techniques discussed are then applied on CVRP in Section 4. A summary of the findings and proposed future research directions are finally given in Section 5.

2. Literature review

Some researchers came up with significant ideas which has really contributed to solving VRP and transportation problem in general. In the first article [1], some trial problems were calculated but no practical application was made. Based on the first article [1], Clarke and Wright in [27] developed a saving-based heuristics algorithm that enables rapid selection of optimal or near-optimal route. Augerat et al., [14] applied three algorithm namely; constructive, randomized greedy and a very simple tabu search algorithms to separate capacity constraints from CVRP and some experimental results (from known instances and romdomly generated instances) with these algorithms were reported. Toth and Vigo [28] reviewed exact algorithms based on branch-and-bound approach and applied it on CVRP. Experimental results comparing the performance of different relaxations and algorithms on some instances were also presented. Fukasawa et al., [29] applied branch-and-cut-and-price algorithm to solve several instances of CVRP to optimality. Baldacci and Mingozzi [30] solved a CVRP problem using an exact method based on set-partitioning formulation of the problem. The proposed method was effective but has limited solving power of customers to about 100. Uchoa et al., [31] proposed and used new set of CVRP instances ranging from \(100\) to \(1000\) customers. Some exact and heuristic methods were also applied on the proposed instances.

Akhtar et al., [32] applied Backtracking Search Algorithm (BSA) on CVRP in order to minimize travel cost for waste collection routes distances. The obtained results showed \(36.80%\) distance reduction and reducing the fuel cost by \(4.77%\), thus the BSA algorithm was concluded to be a viable tool. Franco et al., [33] used Nearest Neighbor Algorithm (NNA) to perform a comparative study between Augerat et al., [14] instances to solve CVRP and determine which instance offers the best solution. In their work, there was \(15.7%\) improvement on \(opt-A-n33-k5\) instance. Rojas-Cuevas et al., [34] used Capacitated Vehicle Routing Problem for Carriers (CVRPfC) to solve distribution problem. The route planning were obtained through CVRPfC and the authors concluded that the CVRPfC can improve the competitiveness of the carriers by providing better fares to their customers. Ibrahim with his coauthors [7, 13] applied column generation to solve CVRP to optimality using Augerat et al., [14] instances. Kohl et al., [35] and Desrochers et al.,[1]author11} applied column generation to solve VRP with time windows. Some experimental results were presented and it was concluded that the algorithm are helpful in solving VRP with practical-sized VRPTW benchmark.

Reinforcement learning, pointer network and Neural combinatorial Optimization have also been applied in solving VRP. Bello et al., [25] proposed a neural networks and reinforcement learning to solve VRP with single vehicle. The algorithm (Neural combinatorial Optimization) achieved near-optimal results on \(2D\) euclidean graphs upto \(100\) nodes. Ishaya et al., [8] improved the algorithms given in [25], by adding 2-opt search on the algorithms to achieve a near-optimal solution and was able to solve upto 200 nodes. Nazari et al., [5] presented a reinforcement learning framework for solving the VRP. The model was applied on VRP and TSP and this approach perform very well on medium-sized problem in terms of the solution quality with computational time. Advantage of this method is that it scales well when the problem-size is increasing.

3. Mathematical Model and Methodology

3.1. Nomenclature definitions

All vehicles will originate and end at the depot, while each of the customer is visited exactly once. Let us define the following:

• \(\mathcal{C} = \{1, 2,\dots, m\}\) represent the set of m-customers to be considered,
• \(\mathcal{V} = \{{v_{0}},{v_{1}}, \dots, {v_{m}},{v_{m+1}}\}\) is the set of vertices in G. The vertices \(v_{0}= v_{m+1}\) represent the depot, and \(\{{v_{1}},\dots,{v_{m}}\}\) represent customers nodes,
• \(\mathcal{E} = \{({v_{i}}, {v_{j}}) \hspace{.2cm} \vert \hspace{.2cm} 0 \le i,j \le m, \hspace{.2cm}i\ne j \}\) is a set of \(\vert \mathcal{V} \vert \ast (\vert \mathcal{V} \vert - 1)\) directed routes/edges between the vertices. If in both directions the distance between two vertices are identical, we then add the \((i < j)\) restriction,
• K denote the fleet of available vehicles in a single depot. All vehicles considered are homogeneous \emph{i.e.,} equal capacities. We have \(n\)-vehicles,
• Q is the maximum capacity of a vehicle, which limits the number of customers to be visited before returning to the depot,
• \(\pi(a \vert s)\) is the probability density function,
• \(\mathcal{A}\) is the set of actions,
• \(Q(s,a;\pi)\) is the action value function,
• \(\vert x_1 - x_2 \vert + \vert y_1 - y_2 \vert\) is the manhattan distance,
• \(\pi\) is the policy,
• \(P(s,a,y)\) is the transitive probability,
• \(a_n\) is the alignment vector,
• C \(= (c_{ij})\) is the cost of traveling from nodes \(i\) to \(j\) and \(c_{ij}\ge 0\) is the corresponding distance of edges \((v_{i},v_{j})\), the diagonal of the matrix i.e \(c_{ii} = 0\) always. Depending on whether the VRP variant in consideration is symmetric or not, \(c_{ij} = c_{ji}\). The triangle inequality is assumed to hold generally, \emph{i.e.,} \(c_{ij} \le c_{ik}+c_{kj}\) and \((0 \le i,j,k \le m)\),
• \(R_i = (v_{0}^{i}, v_{1}^{i},v_{2}^{i}, v_{3}^{i} \dots, v_{k_i}^{i}, v_{k_{i+1}}^{i})\) is a vector of the route of vehicle \(i\) which start and end at the depot, with \(v_{0}^{i} = v_{k_{i+1}}^{i} = v_{0}, v_{j}^{i} \ne v_{\ell}^{i}, 0 \le j < \ell \le k_i\), and \(k_i\) is the length of route \(R_i\),
• \(\mathcal{S} = \{R_1,\dots, R_n\}\) is the set of route which represent the VRP solution instance,
• C\((R_{i}) = \sum_{j=0}^{k_i}\) C\((v_{j}^{i}, v_{j+1}^{i})\) is the cost of route \(R_i\),
• C\(\mathcal{(S)} = \sum_{i=1}^{n}\) C\((R_i)\) is the total cost of solution \(\mathcal{S}\) which satisfies \(R_i \cap R_j = \{v_0\}\hspace{.3cm} \forall R_i, R_j, (1\le i,j \le n, i\ne j)\) and \(\cup_{i=1}^{n}R_i = \mathcal{S}\) in order for each customer to be served once. The route vectors is treated here as a set.

The goal of the VRP is to minimize the C\(\mathcal{(S)}\) on the graph \(\mathcal{G}(\mathcal{V},\mathcal{E})\). Let \(\mathcal{G}\) is the graph which contains \(\vert \mathcal{E} \vert +2\) vertices, and the customers ranges from \((1,2,\dots, m)\). The starting and returning depots are denoted by \(0\) and \(m+1\) respectively. Earlier in this section, we introduced the vehicle routing problem which we have now defined. However, the problem is not all about visiting the customers, there is more to their demands. In the following definitions, we shall specify these additional demands of the customers:

• demand; \(d = (d_0, \dots, d_m, d_{m+1})\) with \(d_i> 0\) and \(m\) is the total number of customers which is a vector of the demands of customer, the demand of the depot is denoted by \(d_0\); \(d_0 = d_{m+1} = 0\) always.
• Let us define our decision variable; \(x_{ijk} = \left\{ \begin{array}{rcl} 1 & \mbox{iff vehicle \(k\) moves from node \(i\) to \(j\)} \\ \\ 0 & \mbox{otherwise.} \end{array}\right.\)
• q\(_{j}\) is the quantity of demand at node \(j\)

The problem definition will be based on the following assumptions;

• The capacity constraints of all the vehicles are observed.
• Each customer can be served by only one vehicle.
• Each and every route starts at vertex \(0\) and ends at vertex \((m+1)\).

The mathematical formulation of CVRP is stated as follow, starting with the objective function subjected to: where (1) minimize the total travel cost by vehicle, constraint (2) restrict a customer to be visited by exactly one vehicle, (3) and (4) is the path-flow of vehicles, (9) and (10) is the coupling, (5) and (6) ensures the capacity constraint is observed and constraints (7) and (8) indicate integrality constraints. Also, we encode \(0\) and \(f\) in the model to denotes begin and final nodes, respectively, for convenience.Therefore, \(0 = v_0\) and \(f = v_{m+1}.\)

Note that subtours are avoided in the solution with constraint (5) that is, cycling paths which do not pass through the depot. Constraints (5) and (6) advantage in this problem is that in terms of our customers, the formulation has a polynomial number of constraints.

However, the Linear Programming (LP) relaxation of this formulation (model) generate a lower bound which is known to be weak when compared to other models. Many researchers and authors emphasized on capacity constraints that produce a better lower bounds, although the constraints increases exponentially in terms of number of customer thereby requiring the application of branch and cut (BAC) technique[36].

3.2. Column generation (CG) technique

In order to solve our objective function (1), we shall apply CG technique. This technique is efficient in solving large linear program problems [37]. The idea of this technique is that LP problems are usually too large and difficult to consider all variables explicitly. CG considers variables with potential to improve our objective function and this strategy helps to reduce the number of variables. In CG, the problem is splitted into some iterative steps which are; Restricted Master Problem and Pricing Problem.

3.3. Restricted master problem (RMP)

We are going to use set-partitioning to re-write the objective function. Let us recall that in problem definition Section 3.1 we defined We shall introduce following variables here: \begin{align*} y_{k} = \left\{ \begin{array}{rcl} 1 & \mbox{if route k is in the solution} \\ \\ 0 & \mbox{otherwise.} \end{array}\right. \end{align*} and \begin{align*} a_{ik} = \left\{ \begin{array}{rcl} 1 & \mbox{if customer i use route k} \\ \\ 0 & \mbox{otherwise.} \end{array}\right. \end{align*} Also, we denote \(C_k\) as the total costs of driving a route, this include cost of overtime, fixed cost, early and late arrival penalties. Then, we formulate the objective function as: subject to: Given a restricted set of routes (say \({R}\)) (11) is the objective function, each and every customer is visited by a vehicle exactly once in constraint (12) and constraint (13) is the integrality constraint that restrict the value of \(y_k\) to \(0\) or \(1\). The master problem can be solved using simplex algorithm described in Algorithm 1.

3.4. The pricing-problem (subproblem)

In order to solve the CVRP with column generation approach, the pricing-algorithm decomposes into \(\vert \mathcal{V} \vert\) identical problems, where each of the problem is a shortest path problem with time windows and capacity constraints. Specifically, the pricing-problem is an elementary shortest path with capacity and time windows constraints (ESPWCTWC), the elementary shortest path implies that each customer cannot appear more than once in the shortest path. We can now formulate the subproblem as follow, such that When solving ESPWCTWC as the subproblem, \(\hat{C}_{ij}= C_{ij} - \pi_{i}\) is the reduced cost of using route \((i, j)\) and \(C_{ij}\) is non-negative integer.
Solving the subproblem as a linear mixed integer programming will potentially reduce the integrality gap between optimal integer solution and the relaxed version of the CVRP problem, since the subproblem does not have integrality constraint/property[37]. More literatures on column generation technique can be obtain in[29, 38, 39, 40, 41, 42, 43, 44, 45]. The column generation algorithm is given in Algorithm 2.

3.5. Google's operations research tool

The second technique to be used is the Operations Research Tools (Goole's OR-Tool). Goole's OR-Tool is an open source software suitable for solving optimization problems. The algorithm is suitable in solving routing problem, constraint programming, flows problem, integer programing and so on[46]. The advantage of this software to us is that it enables us to find optimal tour and its length for routing problem using python. Its computational time is usually very fast compared to other techniques. Its solutions are usually near-optimal when compared with exact method. The algorithm computes distance between two points; \((x_{1}, y_{1})\), \((x_{2}, y_{2})\), using the manhattan distance which sum up the absolute distance of \(x\) and \(y\) coordiantes respectively. This can be obtained mathematically as \( \vert x_1 - x_{2} \vert + \vert y_1 - y_2 \vert \). We convert the algorithm's formula for computing the cost of transportation by computing the distance between two coordinates; \((x_{1}, y_{1})\), \((x_{2}, y_{2})\), i.e., using the Euclidean formula \(\sqrt{(x_{1} - x_{2})^2 + (y_{1} - y_{2})^2} \). One major advantage is that the algorithm is computationally very fast. Application of this technique shall be given in Section 4.

3.6. Reinforcement learning

Reinforcement learning (RL) is the learning of what-to-do, how to map situations to actions in order to maximize a numerical reward signal. The learner (agent) is not explicitly told the actions to take, but instead discover which actions yield more rewards by trying different action. RL provides a mathematical framework suited to solving games. In RL, Markov Decision Process (MDP), tool for modeling artificial intelligence agents that interacts with environment that offers rewards on completion of some certain actions, is the central mathematical concept[47].

Markov decision processes (MDP)

MDP is defined as a stochastic process in discrete time. The mathematical space of MDP has a defined states \(S\), for each state a reward signal \(r: S \times \mathcal{A} \rightarrow \mathcal{R}\). The agent has some actions, \(\mathcal{A}\), and probability distribution \(P(y\vert s,a)\), with current state \(s \in S\) and action \(a \in \mathcal{A}\), the probability of moving into given state \(y\). In this environment, the goal is to find a policy, \(\pi : S \rightarrow \mathcal{A}\) in order to maximize the discounted sum of total rewards \(R\) over all time steps where \(t\) denotes current time and \(\sigma \in [0,1)\) is the discount factor. Solution to this problem is the stationary policy \(\pi^{*}\) and this may be stochastic giving a probability distribution over the actions \(\mathcal{A}\) and state \(S\), where the probability density function is given as \(\pi(a \vert s)\). Furthermore, we introduce the Action-Value function \(Q(s,a;\pi)\), the expected reward from state \(s\), action \(a\), with a given policy \(\pi\), Following \(\pi^{*}\) from any time step,t and onwards, the optimality principle yields a recursive relation which is given as follows, through the optimal action-value function, \(Q(s,a;\pi)\), the optimal deterministic policy can be stated. According to the principle of optimality dictates that the path generated by choosing action \(a\) must be followed in order to maximize the expected return in state \(s\) where \(\delta_{a,x}\) denotes the kroniker delta function. The expected value can be defined in terms of expected reward \(r(s,a)\), transitive probability \(P(s,a,y)\), and the state-value function \(V(x)\) as follows, Furthermore, the optimal policy, \(\pi\) can be expressed as a function of \(V^{*}\) or \(Q^{*}\), Now, the knowledge of \(V^{*}\) or \(Q^{*}\) are sufficient in finding the optimal policy \(\pi^{*}\)[48]. More explanation can be found in[48].

3.7. Sequence-to-sequence model

Sequence-to-sequence models[23, 24, 49] are handy in tasks for which mapping from one sequence to another is required[5]. Over the past several years, these models have been studied extensively in the field of neural machine translation, and there are many variants of these models. Generally speaking, the architecture for the different versions is almost the same, which consists of two Recurrent Neural Networks (RNN), called the encoder and decoder. An encoder network reads through the input sequence and store the output knowledge in a fixed size sequence of vectors, and a decoder converts the encoded information back to an input sequence.

3.8. Neural combinatorial optimization

Several methods have been developed in order to solve a combinatorial optimization problem using recent techniques in artificial intelligence. Vinyals et al.,[24] was the first who first proposed the concept of Pointer Network, model inspired by sequence-to-sequence models [5]. Because it is invariant to the encoder sequence length, the Pointer Network enables the model to be applicable to combinatorial optimization problems, where the length of output sequence is determined by the source sequence. Vinyals et al., [24] used the pointer network architecture in supervised fashion in order to find a near optimal tours for TSPs from heuristic solutions. This dependence limits the Pointer Network from obtaining better solutions order than the ones provided during the training.

3.9. The model

In this section, we introduce our model, which is the simplified version of the Pointer Network. We formally define the problem and our proposed framework for generic combinatorial optimization problem with a set of input \(X = {x^{j}, j \dot{=} 1,2,\dots, N}\). Some elements of each input is allowed to change between the decoding steps, which is the case in several combinatorial problems such as the VRP. The dynamic elements might be an artifact of the decoding step itself, or they can be imposed by the environment. We represent every input \(x^{j}\) by a sequence of tuples \(\{x_{n}^{j} \dot{=} (s^{j}, d_{n}^{j}), n = 0,1,2,\dots\}\), where \(s^{j}\) and \(d_{t}^{j}\) are the static and dynamic elements of input, respectively, which can also be a tuples. \(x_{n}^{j}\) can be viewed as a vector which describes at time n the state of input \(j\). We will represent the set of all input states at fixed time \(n\) with \(X_{n}\). Starting with an arbitrary input in \(X_{0}\) and pointer \(z_{0}\) refer to this input. At every decoding time \(n, z_{n+1}\) points to one of the available \(X_{n}\), which will determine the input for the next decoding step; and this process goes on and on until a termination condition is satisfied. The termination condition is specific on a particular problem, showing that the sequence generated satisfies the feasibilty constraints. For instance, for the CVRP considered in this work, the terminating condition is satisfied when there is no more demand to be satisfied.

This process will generate sequence of length \(N, Z = \{z_{n}, n=0,1,\dots,N\}\), with a different sequence-length (probably), when compared with the length of the input sequence \(M\). The reason is, the vehicle may have to return to the depot to refill several times. Furthermore, we use \(Z_{n}\) to denote the decoded sequence up to time \(n\). Our interest is to find a stochastic policy \(\pi\) that will generates the sequence \(Z\) in a way that minimizes a loss objective function while the problem constraints are satisfied. The optimal policy \(\pi^{*}\) will generate the optimal solution with probability \(1\). Our goal to to make the optimal gap between \(\pi\) and \(\pi^{*}\) close to zero.

Similar to[23], in order to decompose the probability of generating sequence \(Z\) we use the probability chain rule, as follows: and is a recursive update of the problem representation with \(h \) as the state transition function. The right hand side of Equation 26 is computed using the attention mechanism, i.e., where \(g\) is an affine function which outputs a vector with input-size, and \(h_{n}\) is the RNN state decoder that gives the summaries of previous information on decoded steps, \((z_{0}, z_{1}, \dots, z_{n})\)[5].

3.10. Attention mechanism

Attention mechanism is a differentiable structure for addressing different parts of the input. The attention mechanism employed in our method is illustrated in[5]. In words, \(a_{n}\) specifies how much every input data point might be relevant in the next decoding step n. Set the embedded input \(j\) as \(\bar{x}_{n}^{j} = (\bar{s}^{j}, \bar{d}_{n}^{j})\), and \(f_{n} \in \mathbb{R}^{D}\) is the memory state of the RNN cell at decoding step n. The alignment vector \(a_n\) is computed as; where \(p_{n}^{j} = q_{a}^{N} tanh(W_{a}[\bar{x}_{n}^{j}; f_{n}]).\) The ``;'' here means the concatenation of two vectors. The conditional probabilities is computed by combining the context vector \(cv_{n}\) as with the embedded inputs, and then normalizing the values with the softmax function, as follows; where \(\bar{x}_{n}^{j} = q_{c}^{N} tanh(W_{c} [\bar{x}_{n}^{j}; cv_{n}]).\) from (28) to (30) \(q_{a}\), \(q_{c}\), \(W_{a}\) and \(W_{c}\) are trainable variables[5].
For training the policy gradient, we utilize the REINFORCE approach (in algorithm 0) which can be found in[5,8, 25] for VRP. Consider \(\mathcal{N}\) problems with probability distribution \(\Phi_{\mathcal{N}}\).

4. Results

We show the applications of Column Generation, Google's OR-Tools and Reinforcement Learning on Capacitated Vehicle Routing Problem. Our results were compared with best known values for each instances. These techniques have been previously discussed explicitly in Section 3, and Augerat et al., [14] instances will be use to test these methodologies and their results shall be compared. For our experiments, we used HP Elitebook \(840\) PC, \(1.9\)GHz processor, core i\(5\) with \(8\)GB Memory. We started by showing optimal tour with tables and graphs for Column generation, Google's OR-Tools and comapre them with reinforcement learning. We used Augerat et al., [14] (set P) data to perform our experiment. The optimality gap here is computed as As we progress, the optimal gap for each technique will given and computed.

4.1. Column generation applied on CVRP

The Column Generation (CG) method is an exact method for solving the CVRP and the VRP in general, this technique has been explicitly explained in Section 3.2, and gives an optimal solution to a small-size problem but become inefficient on big-size problem.

Table 3 gives the summary of the comparison of this technique's primal and dual problem, since CG work on dual solution of the relaxed master problem, their optimal gap in percentage. The table 3 consists of seven columns; Instances (contains the instances written as \(P-n16-k8\) which means the data consists of \(16\) nodes with \(8\) fleets of vehicles),Cities (locations to be visited), Best value (from literature), Relaxed Master Problem (RMP), Column Generation (based on dual values), column generation computational time and optimality gap. From the table, the optimal gap for all the instances considered gives \(0.00%\). From the instances we consider in this research, only coordinates and demands for each nodes are given, CG compute the distance between two coordinates; \((x_{1}, y_{1})\), \((x_{2}, y_{2})\), using the euclidean formula;

Table 1. CG results.

Instances	Cities	Best value	RMP (primal)	CG result (cost)	CG time	gap
P-n16-k8	16	450 [50]	450	450.00	14.72	0.00%
P-n20-k2	20	220 [50]	220	220.00	38.64	0.00%
P-n22-k2	22	216 [50]	216	216.00	44.31	0.00%
P-n22-k8	22	603 [50]	603	603.00	19.66	0.00%
P-n40-k5	39	458 [50]	458	458.00	32.22	0.00%
P-n50-k10	49	696 [50]	696	696.00	38.42	0.00%
P-n101-k4	100	681 [50]	681	681.00	39.16	0.00%

To obtain the optimality gap between the two results, we use the formula in (31). These results were obtained using gurobi solver in python [51]. The optimal routes for the first four instance is given in Figure 1.

The route for instance \(p-n22-k8\) is shown in (33) and the routes is shown

4.2. Google's OR-tool applied on CVRP

We applied Google's OR-Tool on CVRP as follow, the optimal gap formula is given in (31)

Table 2. Goole's OR-Tool results.

Instances	Best known values	Google’s OR-Tool		Optimality gap
		result	time (s)
A-n32-k5	784*[50]	796	.04	1.53%
P-n16-k8	450*[50]	450	.50	0.00%
P-n20-k2	216*[50]	227	.02	5.09%
P-n22-k2	216*[50]	217	.02	0.46%
P-n22-k8	603*[50]	623	.51	3.32%
P-n40-k5	458*[50]	494	.10	7.86%
P-n50-k7	554*[50]	574	.08	3.61%
P-n70-k10	827*[50]	940	.17	13.66%
P-n101-k4	681*[50]	741	.35	8.81%

From Table 2, the Google's OR-Tool computation seem to be very very fast, and gives a near-optimal solution when compared with our column generation. The percentage of the optimality gap is quite small for some instances. It is observed that the instance p-n16-k8 gives a gap of \(0.00%\) which means the solution is optimal. The Google's OR-Tool does not give us optimal solution but a near-optimal solution. Figure \ref{fig:orvsbest} shows the plot with optimal gap.

The percentage of the optimal gap between the solutions plots are given in fifth column in Table 2. From our experiments, the instance \(P-n70-k10\) appear to have the largest gap from optimality which is \(13.66%\). However, large optimal gap between the solutions shows that the obtained solution is far from optimality. Instances \(P-n101-k4\), \(P-n40-k5\) and \(P-n20-k2\) with gap \(8.81%\), \(7.86%\) and \(5.09%\) respectively shows that their solutions are not 'optimal'. Optimality gap of \(0.00%\) was obtained on instance \(P-n16-k8\) which indicate an optimal solution was reached.
Similarly, the tour for instance \(p-n101-k4\) with cost \(741\) is given as follow:
\( R_1 = (v_{28}, v_{12}, v_{68}, v_{80}, v_{54}, v_{24}, v_{29}, v_{3}, v_{77}, v_{79}, v_{78}, v_{34}, v_{35}, v_{65}, v_{66}, v_{71}, v_{9}, v_{81}, v_{33}, v_{51}, v_{20}, v_{30}, v_{70}, v_{31}, v_{88}, v_{7}, v_{52} )\),
\( R_2 = ( v_{89}, v_{60}, v_{5}, v_{84}, v_{17}, v_{45}, v_{47}, v_{36}, v_{49}, v_{64}, v_{63}, v_{90}, v_{32}, v_{10}, v_{62}, v_{11}, v_{19}, v_{48}, v_{82}, v_{8}, v_{32}, v_{18} ) \),
\( R_3 = (v_{27}, v_{69}, v_{1}, v_{50}, v_{76}, v_{26}, v_{58}, v_{13}, v_{94}, v_{95}, v_{97}, v_{87}, v_{57}, v_{15}, v_{43}, v_{100}, v_{85}, v_{93}, v_{59}, v_{99}, v_{96}, v_{6} ) \),
\(R_4 = ( v_{92}, v_{37}, v_{98}, v_{91}, v_{61}, v_{16}, v_{86}, v_{44}, v_{38}, v_{14}, v_{42}, v_{2}, v_{41}, v_{22}, v_{23}, v_{67}, v_{67}, v_{39}, v_{25}, v_{55}, v_{4}, v_{56}, v_{75}, v_{74}, v_{72}, v_{73}, v_{21},\\ \;\;\;\;\;v_{40}, v_{53} )\,. \)

4.3. CG, Google's OR-tools and RL on CVRP

So far we have shown the results for each of our methodologies i.e CG in Table 3, Google's OR-Tool with our algorithm in Table 1. Now, we compute the final table which summarize the CG, Google's OR-Tool and RL techniques and shown in Table 1. We use Beam Search with width \(10\) (BS10) for the RL.

Table 3. Summary of CG, Google's OR-Tools and RL results.

Instance	best value	CG		Google’s OR-Tool		RL
		result (gap)	time(s)	result (gap)	time(s)	result (gap)	time(s)
P-n16-k8	450*[50]	450 (0.00%)	14.72	450 (0.00%)	0.50	451 (0.22%)	0.30
P-n20-k2	216*[50]	216  (0.00%)	38.64	227 (5.09%)	.20	220 (1.85%)	0.45
P-n22-k2	216*[50]	216 (0.00%)	44.31	217 (0.46%)	.20	218 (0.93%)	0.55
P-n22-k8	603*[50]	603 (0.00%)	19.66	623 (3.32%)	.51	615 (2.67%)	1.10

From our comparison Table 1 above, we see that the first technique; column generation solution gives us the best bound, of course we expect nothing less since the technique gives an optimal solution but limited to a small size problem. The second technique, the Google's OR-Tool solution gives a near-optimal solution and the advantage of this technique over the previous is, it solves more problems than CG. The optimal gap for each of the instances considered using this technique is \(< 15%\). Lastly, the reinforcement learning also gives a new optimal solution compared with the previous two techniques. Its solutions outperform the OR-tools and is closer to the exact solution, the optimality gap is quite smaller compared with the Google's OR-Tool optimality gap. This technique is quite good since it can handle a large-size problem.

The computational time for each of these techniques are also of importance, since we want an algorithm that will be fast in computations. Although the CG technique gives an optimal solution, it has high computational time when compared with OR-Tools and RL. The Google's OR-Tool and RL appear to have 'almost' the same computational time and lesser than CG time, which shows these two techniques perform their experiment faster than the CG. The Optimal gap between the CG and OR-Tools is \(0.00%\) on instance \(P-n16-k8\). The gap between the CG and RL appears to be better and smaller ib value when compared with the gap between CG and OR-Tools.

4.4. Google's OR-Tools and RL with CVRP

Google's OR-Tools and RL have been compared with CG previously on small-sized problem. Here, we shall compare these two techniques to solve bigger-size problems. Table 4 shows the path-length and computational time for these techiques and their optimal gap. The ``instances" column shows the size of the problems and their vehicle capacity, Q. From Table 4, both technique's result are shown with their computational times. Our RL outperforms the google's OR-Tools. The solutions here are scaled down to \([0,1] \times [0,1]\) for a better solution. We have shown results using various techniques discussed and their tour-length cost, optimal tour, optimality gap and computational time. Each of these technique is applied on the same data set and the same PC was used to perform these experiments.

Table 4. OR-Tools and RL result.

Instances	OR-Tools		RL		optimality gap
	path-length	time	path-length	time
vrp50, Q=150	31.50	0.29	30.88	02.25	2.01%
vrp70, Q=135	36.05	0.29	35.74	03.02	0.87%
vrp100, Q=400	55.62	0.29	55.05	06.23	1.04%

5. Conclusion

The three algorithms discussed here; column generation, google's OR-Tools and reinforcement learning algorithms are examined on a small-scale problem. Also, google's OR-Tools and reinforcement learning are examined on large scale data. An algorithm that will find a near-optimal solution have been developed. We find the optimal set of routes for a fleet of vehicles delivering goods or services to various locations. In order to achieve this aim, we formulated a mathematical formulation for the Capacitated Vehicle Routing Problem. We further to solved this formulation with the three techniques; Column generation, Google's Operational Research tool and Reinforcement Learning. We compared the objective values for these techniques with the ''best known values" and calculated for the ``optimality gap'' between this solution, taking the ''best known value" as the lower bound. From our experiment, our Reinforcement Learning outperformed the google's OR-Tools a little bit. Although the computational time for the google's OR-Tools is faster than the Reinforcement Learning time but these experimental times are very close and the reinforcement learning is able to solve a large data set.

In our future work, a ``time window" constraint can be added to this problem. Basically, this problem shall have multiple vehicles with capacity constraint and each location will have a time window. This means that each of the customer/location will require demand at a particular time window \([a_i , b_i ]\), where \(a_i\) is the opening time at a location and a vehicle must arrive on or before \(a_i\) and \(b_i\) is the closing time, a vehicle is not allowed to come after \(b_i\) . Hence, the delivery time, \(s_i\) to a location must be \(a_i \le s_i \le b_i\) . Customers with similar time window can be merged together as long as the capacity limit of the vehicle is not exceeded.

Acknowledgments

We appreciate African Institute for Mathematical Science, Senegal for full funding in order to carry out this research at AIMS.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

It is well known that the central binomial coefficients have the following expressions; $$\binom{2n}{n}=\sum_{k=0}^n{\binom{n}{k}}^2,~\binom{2n+1}{n}=\sum_{k=0}^n{\binom{n}{k}}\binom{n+1}{k}.$$

For \(0\leq k\leq n\), the Narayana numbers of types \(A\) are defined as; $$N(n,k)=\frac{1}{n}\binom{n}{k+1}\binom{n}{k}.$$

Let \(N_n(x)=\sum_{k=0}^{n-1}N(n,k)x^k\) be the Narayana polynomials of types \(A\) (see [1]). It is well known that \(N_n(x)\) is the rank-generating function of the lattice of non-crossing partition lattice with cardinality \(\frac{1}{n+1}\binom{2n}{n}\) (see[2]). Hence the Catalan numbers have the following expression; $$\frac{1}{n+1}\binom{2n}{n}=\sum_{k=0}^{n-1}\frac{1}{n}\binom{n}{k+1}\binom{n}{k}.$$

The Narayana numbers of type \(B\) are given as; $$M(n,k)=\binom{n}{k}^2.$$

Let \(M_n(x)=\sum_{k=0}^nM(n,k)x^k\). Reiner [2] showed that \(M_n(x)\) is the rank-generating function of a ranked self-dual lattice with the cardinality \(\binom{2n}{n}\).

Let \(P(n,k)={\binom{n}{k}}\binom{n+1}{k},\) and \(S=\mathbb{P}\times \mathbb{P}\). According to [3 Proposition 1], \(P(n,k)\) is the number of paths in \(A_1(n+1)\) having \(k+1\) steps, where \(A_1(n)\) is the set of all lattice paths running from \((0;-1)\) to \((n;n)\) that use the steps in \(S\) and that remain strictly above the line \(y=-1\) except initially.

The numbers \(N(n,k),M(n,k)\) and \(P(n,k)\) have been extensively studied. The readers are referred to[4] for details. In [5], Daboul et al., reveals that $$\frac{d^n}{dx^n}(e^{1/x})=(-1)^ne^{1/x}\sum_{k=1}^{n}{\binom{n}{k}\binom{n-1}{k-1}(n-k)!x^{-n-k}},$$ where the \(\binom{n}{k}\binom{n-1}{k-1}(n-k)!\) are the Lah numbers. Motivated by this result, in this paper we show that the numbers \(M(n,k),N(n,k)\) and \(P(n,k)\) can be generated by higher-order derivative of functions of \(e^x\). As an application, we obtain new recurrence relations for these classical combinatorial numbers.

2. Differential operators and Narayana polynomials

Let \(P_n(x)=\sum_{k=0}^nP(n,k)x^{n-k},~Q_n(x)=\sum_{k=0}^nP(n,k)x^{k},\) then \(Q_n(x)=x^nP_n(1/x)\). The first few \(N_n(x),M_n(x)\) and \(P_n(x)\) are listed as follows; \begin{align*} N_1(x)&=1,~N_2(x)=1+x,N_3(x)=1+3x+x^2,~N_4(x)=1+6x+6x^2+x^3,\\ M_1(x)&=1+x,~M_2(x)=1+4x+x^2,M_3(x)=1+9x+9x^2+x^3,\\ P_1(x)&=2+x,~P_2(x)=3+6x+x^2,P_3(x)=4+18x+12x^2+x^3. \end{align*} We define \(\overline{N}(n,k)=(n+1)!n!N(n,k)\) and \(\overline{M}(n,k)=n!^2M(n,k)\). By using the explicit formulas of \(\overline{N}(n,k)\) and \(\overline{M}(n,k)\), it is routine to verify the following lemma.

Lemma 1. For \(0\leq k\leq n+1\), we have \begin{align*} \overline{N}(n+1,k)&=((n+1)(n+2)+2nk+k^2+3k)\overline{N}(n,k)+(4n+2n^2-2(k^2-1)){\overline{N}}(n,k-1)\\&\,\,\,\,\,\,+(n(n-1)-(k-2)(2n-k+1))\overline{N}(n,k-2),\\ \overline{M}(n+1,k)&=((n+1)^2+2(n+1)k+k^2)\overline{M}(n,k)+(1+4n+2n^2-2k(k-1)){\overline{M}}(n,k-1)\\&\,\,\,\,\,\,+(n^2-(2n+2-k)(k-2))\overline{M}(n,k-2), \end{align*} with initial conditions \(\overline{N}(0,0)=\overline{M}(0,0)=1\) and \(\overline{N}(0,k)=\overline{M}(0,k)=0\) for \(k\neq 0\).

In the following discussion, let \(D=\frac{d}{dx}\).

Theorem 1. For \(n\geq 1\), we have

Proof. Note that \begin{eqnarray*} (De^xD)\left(\frac{1}{1-e^x}\right)&=&\frac{2e^{2x}}{(1-e^x)^{3}},\\(De^xD)^2\left(\frac{1}{1-e^x}\right)&=&\frac{12e^{3x}(1+e^x)}{(1-e^x)^{5}},\\ (De^xD)^3\left(\frac{1}{1-e^x}\right)&=&\frac{144e^{4x}(1+3e^x+e^{2x})}{(1-e^x)^{7}}. \end{eqnarray*} Hence the formula (1) holds for \(n=1,2,3\). Assume that the result holds for \(n\), where \(n\geq 3\). Let \(\overline{N}_n(x)=\sum_{k=0}^{n-1}\overline{N}(n,k)x^k\). Note that \begin{align*} &(De^xD)^{n+1}\left(\frac{1}{1-e^x}\right)=(De^xD)\left(\frac{e^{(n+1)x}\overline{N}_n(e^x)}{(1-e^x)^{2n+1}}\right). \end{align*} It follows that
\( \overline{N}_{n+1}(x)=((n+1)(n+2)+(4n+2n^2)x+n(n-1)x^2)\overline{N}_n(x)+(4x-6x^2+2x^3+2nx(1-x^2))D(\overline{N}_n(x))\\+x^2(1-x)^2D^2(\overline{N}_n(x)). \) Equating the coefficients of \(x^k\) in both sides, we immediately get the recurrence relation of \(\overline{N}(n,k)\) given in Lemma 1. Therefore, the result holds for \(n+1\). Similarly, note that \begin{eqnarray*} (e^xD^2)\left(\frac{1}{1-e^x}\right)&=&\frac{e^{2x}(1+e^x)}{(1-e^x)^{3}},\\(e^xD^2)^2\left(\frac{1}{1-e^x}\right)&=&\frac{4e^{3x}(1+4e^x+e^{2x})}{(1-e^x)^{5}},\\ (e^xD^2)^3\left(\frac{1}{1-e^x}\right)&=&\frac{36e^{4x}(1+9e^x+9e^{2x}+e^{3x})}{(1-e^x)^{7}}. \end{eqnarray*} Hence the formula (2) holds for \(n=1,2,3\). Assume it holds for \(n\), where \(n\geq 3\). Let \(\overline{M}_n(x)=\sum_{k=0}^{n}\overline{M}(n,k)x^k\). Note that \begin{align*} &(e^xD^2)^{n+1}\left(\frac{1}{1-e^x}\right)=(e^xD^2)\left(\frac{e^{(n+1)x}\overline{M}_n(e^x)}{(1-e^x)^{2n+1}}\right). \end{align*} It follows that
\( \overline{M}_{n+1}(x)=(1+x+n^2(1+x)^2+n(2+4x))\overline{M}_n(x)+(3x-4x^2+x^3+2nx(1-x^2))D(\overline{M}_n(x))\\+x^2(1-x)^2D^2(\overline{M}_n(x)). \) Equating the coefficients of \(x^k\) in both sides, we immediately get the recurrence relation of \(\overline{M}(n,k)\) given in Lemma 1. Therefore, the result holds for \(n+1\). Along the same lines, it is routine to derive (3). This completes the proof.

Note that \(P(n,n-k)={\binom{n}{n-k}}\binom{n+1}{n-k},\) then \(P(n+1,n+1-k)={\binom{n+1}{n+1-k}}\binom{n+2}{n+1-k}.\) It is easy to verify the following lemma;

Lemma 2. For \(0\leq k\leq n+1\), we have \( (n+1)(n+2)P(n+1,n+1-k)=[(n+2)^2+(2n+5)k+k(k-1)]P(n,n-k)\\+[2(n^2+3n+1)-6(k-1)-2(k-1)(k-2)]P(n,n-k+1)+[n^2-(2n-1)(k-2)+(k-2)(k-3)]P(n,n-k+2). \)

Theorem 2. For \(n\geq 1\), we have

Proof. Note that $$(D^2e^x)\frac{e^x}{(1-e^x)^2}=\frac{2e^{2x}(2+e^x)}{(1-e^x)^4},$$ $$(D^2e^x)^2\frac{e^x}{(1-e^x)^2}=\frac{12e^{3x}(3+6e^x+e^{2x})}{(1-e^x)^6}.$$ Hence the result holds for \(n=1,2\). Assume that the result holds for \(n\). Then from (4), we get the recurrence relation
\( (n+1)(n+2)P_{n+1}(x)=[n^{2}x^2+(2+n)^2+2x(1+3n+n^2)]P_n(x)+x(1-x)[(2n-1)x+2n+5]P_n'(x)+x^2(1-x)^2P_n''(x). \) Equating the coefficients of \(x^k\) in both sides, we get the recurrence relation of the numbers \(P(n,n-k)\), which is given in Lemma 2, as desired. Along the same lines, one can derive (5). This completes the proof.

By a change of variable \(y=e^x\), we end our paper by giving a corollary;

Corollary 1. For \(n\geq 1\), let \(D_y=\frac{d}{dy}\), we have

1. \((yD_yy^2D_y)^n\left(\frac{1}{1-y}\right)=\frac{n!(n+1)!y^{n+1}N_n(y)}{(1-y)^{2n+1}},\)
2. \((y^2D_yyD_y)^n\left(\frac{1}{1-y}\right)=\frac{n!^2y^{(n+1)}M_n(y)}{(1-y)^{2n+1}},\)
3. \((yD_yyD_yy)^n\left(\frac{1}{1-y}\right)=\frac{n!^2y^{n}M_n(y)}{(1-y)^{2n+1}},\)
4. \((yD_yyD_yy)^n\frac{y}{(1-y)^2}=\frac{n!(n+1)!y^{(n+1)}P_n(y)}{(1-y)^{2n+2}},\)
5. \((yD_yy^2D_y)^n\frac{y}{(1-y)^2}=\frac{n!(n+1)!y^{(n+1)}Q_n(y)}{(1-y)^{2n+2}}.\)

Proof. It's not hard to verify the equations hold when \(n=1,2\)$$(yD_yy^2D_y)\left(\frac{1}{1-y}\right)=\frac{2y^{2}}{(1-y)^{3}},$$ $$(yD_yy^2D_y)^2\left(\frac{1}{1-y}\right)=\frac{12y^{3}(1+y)}{(1-y)^{5}}.$$ Assume the result holds for \(m\), where \(m\geq3\). Setting \(y=e^x\), we get \begin{align*} (yD_yy^2D_y)(yD_yy^2D_y)^{m}\left(\frac{1}{1-y}\right)=(e^xD_ye^{2x}D_y)\frac{m!(m+1)!y^{n+1}N_m(y)}{(1-y)^{2m+1}}\\=(De^xD)\frac{m!(m+1)!e^{(m+1)x}N_m(e^x)}{(1-e^x)^{2m+1}}\\=\frac{(m+1)!(m+2)!e^{(m+2)x}N_{m+1}(e^x)}{(1-e^x)^{2m+3}}\\=\frac{(m+1)!(m+2)!y^{m+1}N_{m+1}(y)}{(1-y)^{2m+3}}. \end{align*} Along the same lines, we can get the other statements. This completes the proof.

Acknowledgments

The authors appreciate the careful review, corrections and helpful suggestions to this paper made by the referee.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

Distance-based graph invariants, called topological indices, are widely used in studying of structure of molecular graphs in organic chemistry. The Wiener index is a well-known topological index introduced as structural descriptor for acyclic organic molecules [1]. It is defined as the sum of distances between all unordered pairs of vertices of an undirected connected graph \(G\) with vertex set \(V(G)\): $$ W(G) \ = \sum_{\{u,v\} \subseteq V(G)} d(u,v), $$ where distance \(d(u,v)\) is the number of edges in the shortest path connecting vertices \(u\) and \(v\) in \(G\). The Wiener index is intensively studied in mathematical and theoretical chemistry and has found numerous applications in the modeling of physico-chemical, pharmacological and biological properties of organic molecules (see books [2,3,4,5,6,7,8,9,10] and reviews [11, 12, 13, 14, 15, 16, 17, 18, 19]). We will consider the Wiener index of hexagonal chains that include molecular graphs of catacondensed unbranched benzenoid hydrocarbons. Since this class of chemical compounds is attracting the great attention of theoretical chemists, the theory of the Wiener index of the respective molecular graphs has been developed for many years [20, 21]. Changes of the Wiener index of polycyclic structures under transformations of various kinds were investigated in [22, 23, 24, 25, 26, 27, 28, 29]. The structure of hexagonal chains of certain classes can be encoded by binary vectors. Operations on chains' binary codes generate new hexagonal chains. In this paper, relations between Wiener indices of chains for some operations on chains' codes are studied and illustrative numeric examples are presented.

2. Hexagonal chains and their segments

The classification of molecular graphs of benzenoid hydrocarbons is based on the kind of connection of hexagonal rings with one another [30]. A hexagonal chain is a connected plane graph in which every inner face is bounded by a hexagon. An inner face with its hexagonal bound is called a hexagonal ring (or simply ring ). Two hexagonal rings of a chain are either disjoint or have exactly one common edge (adjacent rings), and no three rings share a common vertex. A ring having exactly one adjacent ring is called terminal . A hexagonal chain has exactly two terminal rings. A ring adjacent to exactly two other rings has two vertices of degree 2. If these two vertices are adjacent, then the ring is angularly connected; if these two vertices are not adjacent, then it is linearly connected. A segment is a maximal subchain in which all rings are linearly connected. A segment including a terminal hexagon is a terminal segment . The number of hexagons in a segment is called its length . Denote by \({\cal G}_{n,{\ell}}\) the set of all hexagonal chains having \(n\) segments of length \(\ell\). All hexagonal chains of \({\cal G}_{5,3}\) are shown in Figure 1. Some properties of graphs of this class were studied in [31, 32, 33, 34, 35]. Hexagonal chains of \({\cal G}_{n,2}\) with minimal length of segments are known as fibonacenes [36, 37]. The number of hexagonal rings of \(G \in {\cal G}_{n,{\ell}}\) is equal to \(h=n(\ell-1)+1\) and the number of segments is \(n=(h-1)/(\ell-1)\). Since \(|\,{\cal G}_{n,2} | = |\,{\cal G}_{n,\ell}\,|\) for all \(\ell \ge 3\), the cardinality of \({\cal G}_{n,{\ell}}\) is equal to \(|\,{\cal G}_{n,{\ell}}\,| = 2^{n-3} + 2^{\lfloor {\frac{n-3}{2}} \rfloor}\), \(n\ge 2\) [37].

3. Representation of hexagonal chains

The structure of hexagonal chains is completely defined by a way of segment attachment. We consider a nonterminal segment \(S\) with two neighboring segments embedded into the regular hexagonal lattice on the plane and draw a line through the centers of the hexagons of \(S\). If the neighboring segments of \(S\) lie on different sides of the line, then \(S\) is called a zigzag segment. If these segments lie on the same side, then \(S\) is said to be a spiral segment. It is convenient to assume that the terminal segments are zigzag segments. For a hexagonal chain \(X\), denote by \(U(X)\) the set of its spiral segments, \(u_X=|U(X)|\).

Based on two types of segments, hexagonal chains of \({\cal G}_{n,\ell}\) can be represented by binary codes. We assume that all segments of a chain are sequentially numbered by \(0, 1,\dots, n-1\) beginning from a terminal segment. Since the terminal segments can be ignored when a chain is restored, codes of all hexagonal chains of \({\cal G}_{n,\ell}\) have a length of \(n-2\), \(n\ge 3\). Assume that every spiral segment of chain \(X\) corresponds to 1 in the code of \(X\) while every zigzag segment corresponds to 0. A binary code of hexagonal chain \(X\) and a hexagonal chain \(X\) induced by a binary word \(r\) will be denoted by \(r(X)\) and \(X(r)\), respectively. Note that molecular graphs of more general classes of benzenoid hydrocarbons can be also represented by binary codes [31, 38].

There are two extremal chains with respect to the type of segments. The zigzag hexagonal chain \(Z_{n,\ell} \in {\cal G}_{n,\ell}\) contains only zigzag segments, \(r(Z_{n,\ell})=(00..0)\). All segments of the spiral hexagonal chain \(O_{n,\ell} \in {\cal G}_{n,\ell}\) are spiral ones (with the exception of the terminal segments), \(r(O_{n,\ell})=(11..1)\). The zigzag and the spiral hexagonal chains of \({\cal G}_{5,3}\) are shown in Figure 1. Denote by \(e_i\) the binary vector \(e_i=(0...0\!\stackrel{i}{1}\!0...0)\) of length \(n-2\) for \(n \ge 3\), \(i=1,2,\dots,n-2\). These vectors form the standard basis for the vector space of dimension \(n-2\) over \(\mathbf{Z}_2\). Let \(C_i\) be the basis hexagonal chains corresponding to basis vectors \(e_i\), \(i=1,2,\dots,n-2\). Then code \(r(X)\) of \(X \in {\cal G}_{n,\ell}\) can be expressed as a linear combination of the basis vectors: \(r(X)= x_1 \, e_1 + x_2 \, e_2 + \dots + x_{n-2} \, e_{n-2}\). Since codes \(e_i\) and \(e_{n-i-1}\) are symmetrical, basis chains \(C_i\) and \(C_{n-i-1}\) are isomorphic and \(W(C_i) = W(C_{n-i-1})\).

Hexagonal chains \(O_{n,\ell}\) and \(Z_{n,\ell}\) are extremal graphs with respect to the Wiener index among all chains of \({\cal G}_{n,\ell}\) [13]: \(W(O_{n,\ell}) < W(G) < W(Z_{n,\ell})\) for all \(G \in {\cal G}_{n,\ell} \setminus \{O_{n,\ell}, Z_{n,\ell}\}\), where \begin{eqnarray*} W_{\min} & = & \big( 8 n^3 (\ell-1)^2 (2 \ell-3)+96 n^2 (\ell-1)^2-2 n (\ell-1)(2 \ell -75) + 81 \big) /3, \\ W_{\max} & = & \big( 16 n^3 (\ell-1)^3 + 72 n^2 (\ell-1)^2 + n(\ell-1)(12 \ell+134)+81 \big)/3. \end{eqnarray*}

The average of these extremal values is equal to \begin{eqnarray*} W_{\mathrm{avr}} & = & \left( W_{\min}+W_{\max} \right)/2 \\ & = & \left( 4 n^3 (\ell-1)^2 (4\ell-5) + 84 n^2 (\ell-1)^2 + 2 n(\ell-1) (2 \ell+71)+81 \right)/3. \end{eqnarray*}

4. Graph operations

Let \(X, Y \in {\cal G}_{n,\ell}\) with codes \(r(X)=x=(x_1,x_2,\dots,x_{n-2})\) and \(r(Y)=y=(y_1,y_2,\dots,y_{n-2})\). Define the following operations of these hexagonal chains:

• the complement of a hexagonal chain \(X\) is a new chain \(Y\) with code \(r(Y)=r(\overline{X}) =(\overline{x}_1,\overline{x}_2,\dots,\overline{x}_{n-2})\), that is, \(r(Y)\) is a bitwise complement of \(r(X)\). This operation changes the type of all segments of \(X\). Example: \((\overline{10011})=(01100)\);
• the unary operation \(t_{i,j}(X)\) changes the type of \(i\)-th and \(j\)-th segments to opposite. For example, if \(r(X)=(1000111)\), then hexagonal chain \(Y=t_{2,6}(X)\) has code \(r(Y)= (1100101)\);
• the sum modulo 2 of hexagonal chains \(X\) and \(Y\) is a new chain \(G=X+Y\) with code \(r(G)= x + y =(x_1+y_1,x_2+y_2,\dots,x_{n-2}+y_{n-2})\). The resulting chain inherits spiral segments of the initial chains except spiral segments in the same positions. Example: \((10011) + (10101) = (00110)\);
• the difference modulo 2 of hexagonal chains \(X\) and \(Y\) is a new chain \(G=X-Y\) with code \(r(G)= x - y =(x_1-y_1,x_2-y_2,\dots,x_{n-2}-y_{n-2})\). Despite of \(G=X-Y =X+Y\), we will distinguish between these operations. Example: \((10011) - (10101) = (00110)\);
• a hexagonal chain \(G=XY\) is called the product of chains \(X\) and \(Y\) with code \(r(G)= xy=(x_1y_1,x_2y_2,\dots,x_{n-2}\,y_{n-2})\). This operation is an analogue of bitwise logical operation ``AND''. The resulting chain has spiral segments if they are in the same positions of the initial chains. Example: \((10011) (10101) = (10001)\);
• operation ``\(\vee\)'' of hexagonal chains \(X\) and \(Y\) gives a new chain \(G=X \vee Y\) with code \(r(G)= x \vee y =(x_1 \vee y_1, x_2 \vee y_2,\dots,x_{n-2} \vee y_{n-2})\), where \(x_i \vee y_i\) is an analogue of bitwise logical operation ``OR''. All spiral segments of the initial chains are served in \(G\). Example: \((10011) \vee (10101) = (10111)\).

The binary relation \(X \le Y\) between chains \(X, Y \in {\cal G}_{n,\ell}\) is defined by conditions \(x_i \le y_i\), \(i=1,2,\dots,n-2\). This relation induces a partial order on \({\cal G}_{n,\ell}\). In the next section, changes of the Wiener index under introduced operations over chains' codes are examined.

5. Changes of the Wiener index

The following useful formula allows the calculation Wiener index of hexagonal chains through their binary codes [26].

Proposition 1. For a hexagonal chain \(X \! \in {\cal G}_{n,\ell}\) with a code \((x_1,x_2, \dots, x_{n-2})\), \begin{eqnarray*} W(X) & = & W_{\max} - 16(\ell-1)^2 \sum_{i=1}^{n-2} i(n-i-1) x_i. \end{eqnarray*}

The sum of Wiener indices of a hexagonal chain and its complement is twice the average value \(W_{\mathrm{avr}}\).

Proposition 2. If \(X \in {\cal G}_{n,\ell}\), then \(W(\overline{X})=W_{\min}+W_{\max} - W(X)\).

Proof. By Proposition 1, \begin{eqnarray*} W(X) + W(\overline{X}) & = & W_{\max} + W_{\max}-16(\ell-1)^2 \sum_{i=1}^{n-2} i(n-i-1)\cdot 1 \\ & = & W_{\max} + W_{\min}. \end{eqnarray*}

Proposition 2 allows to determine the structure of hexagonal chains with average value of the Wiener index. If \(r(X)=r(\overline{X})\), then chains \(X\) and \(\overline{X}\) are obviously isomorphic. This implies equality \(W(X)=W_{\mathrm{avr}}\). There are also non-isomorphic chains \(X\) and \(\overline{X}\) with property \(W(X)= W(\overline{X})= W_{\mathrm{avr}}\) [26]. Consider hexagonal chains without units in the same positions of their codes.

Proposition 3. Let \(X, Y \in {\cal G}_{n,\ell}\) and \(XY=Z_{n,\ell}\). If \(G=X+Y\), then $$ W(G) = W(X) + W(Y) - W_{\max}. $$

Proof. Let \(x=r(X)\) and \(y=r(Y)\). By Proposition 1, we can write $$ W(X) + W(Y) - W_{\max} = W_{\max}-16(\ell-1)^2 \sum_{i} i(n-i-1) (x_i+y_i) = W(G). $$

As an illustration, consider chains \(X, Y \in {\cal G}_{7,3}\) and \(G=X+Y\) shown in Figure 2. These graphs have codes \(r(X)=(10010)\), \(r(Y)=(00101)\), and \(r(G)=(10111)\). By computer calculations, their Wiener indices are \(W(X)=19327\), \(W(Y)=19263\), \(W(G)=18431\), and \(W_{\max}=20159\). By Proposition 3, \(W(G)=19327 + 19263 - 20159=18431\).

Proposition 4. Let \(X, Y \in {\cal G}_{n,\ell}\) and \(Y \le X\). If \(G=X-Y\), then $$ W(G) = W(X) - W(Y) + W_{\max}. $$

Proof. If \(x=r(X)\) and \(y=r(Y)\), then by Proposition 1 \begin{eqnarray*} W(Y) - W(X) - W_{\max} & = & \mbox{} -W_{\max} + 16(\ell-1)^2 \sum_{i} i(n-i-1) (x_i-y_i) \\ & = & \mbox{} -W(G). \end{eqnarray*} Since \(y_i \le x_i\), \((x_i-y_i) \ge 0\) for every \(i\).

Hexagonal chains \(X, Y \in {\cal G}_{7,3}\) and \(G=X-Y\) shown in Figure 3 illustrate Proposition 4. Codes of these graphs are \(r(X)=(11001)\), \(r(Y)=(01001)\), and \(r(G)=(10000)\). Computer calculations give the following Wiener indices: \(W(X)=19007\), \(W(Y)=19327\), and \(W(G)=19839\). By Proposition 4, \(W(G)=19007 - 19327 + 20159=19839\). The next result answers on the following question: how many times do we need to apply operation \(t_{i,j}\) to a hexagonal chain \(X\) such that \(W(X)=W(t_{i,j}(X))\)? For asymmetrical chains, it is sufficient to apply the operation once time.

Corollary 2. Let \(X \in {\cal G}_{n,\ell}\) be an asymmetrical hexagonal chain with a code \((x_1, x_2,\dots,x_{n-2})\), i.e., \(x_i \not = x_{n-i-1}\) for some \(i \in \{1,2,\dots,\lceil(n-2)/2\rceil\}\). If \(Y=t_{i,n-i-1}(X)\), then \(W(Y)=W(X)\).

Proof. Assume that \(x_i=0\) and \(x_{n-i-1}=1\). Then \(Y=(X + C_i) - C_{n-i-1}\). By Propositions 4 and 3, we can write \(W(Y)= W(X + C_i) - W(C_{n-i-1}) + W_{\max} = W(X) + W(C_i) - W_{\max} - W(C_{n-i-1}) + W_{\max} = W(X)\).

Let \(\mu(X)\) be the number of pairs of non-equal \(i\)-th and \((n-i-1)\)-th components of chain's code \(r(X)\), \(i=1,2,\dots,\lceil (n-2)/2 \rceil\). Repeating Corollary 1, one can construct a family of \(2^{\mu(X)}\) chains having the same Wiener index (some chains may be isomorphic).

Decomposition of the Wiener index of chains of \({\cal G}_{n,\ell}\) into the sum of Wiener indices of basis hexagonal chains was reported in [26]. It can be also derived using graph operations.

Corollary 3. Let \(X \in {\cal G}_{n,\ell}\) and \(r(X)=(x_1,x_2,\dots,x_{n-2})\). Then $$ W(X) = x_1 W(C_1) + x_2 W(C_2) + \dots + x_{n-2} W(C_{n-2}) - (u-1)W_{\max}, $$ where \(u = x_1 + x_2 + \dots + x_{n-2}\) is the number of units of \(x\).

Proof. Let \(i_1, i_2,\dots,i_u\) be positions of units in \(x\). Then the corresponding hexagonal chain \(X\) can be represented as \(X=C_{i_1} + C_{i_2} + \dots + C_{i_u}\). Applying Proposition 3, we get \(W(X) = W(C_{i_1}) + W(C_{i_2}) + \dots+ W(C_{i_u}) - (u-1) W_{\max} = x_1 W(C_1) + x_2 W(C_2) + \dots + x_{n-2} W(C_{n-2}) - (u-1)W_{\max}\).

Let us consider operations of hexagonal chains with arbitrary codes.

Proposition 5. Let \(X, Y \in {\cal G}_{n,\ell}\). If \(G=X+Y\), then $$ W(G) = W(X) + W(Y) - 2 W(XY) + W_{\max}. $$

Proof. It is easy to verify that the hexagonal chain \(G\) can be represented as \(G=X+Y=(X-XY) + (Y-XY)\), where \((X-XY)(Y-XY)=Z_{n,\ell}\) and \(X \ge XY\), \(Y \ge XY\). By Propositions 3 and 4, \begin{eqnarray*} W(G) & = & W(X-XY) + W(Y-XY) - W_{\max} \\ & = & W(X) - W(XY) + W_{\max} + W(Y) - W(XY) + W_{\max} - W_{\max} \\ & = & W(X) + W(Y) - 2 W(XY) + W_{\max}. \end{eqnarray*}

Hexagonal chains \(X, Y \in {\cal G}_{7,3}\) and \(G=X+Y\) shown in Figure 4 illustrate Proposition 5. These chains have codes \(r(X)=(11100)\), \(r(Y)=(00110)\), \(r(XY)=(00100)\), and \(r(G)=(11010)\). Their Wiener indices are \(W(X)=18751\), \(W(Y)=19071\), \(W(XY)=19583\), and \(W(G)=18815\). By applying Proposition 5, \(W(G)= 18751 + 19071 - 2\cdot 19583 + 20159=18815\).

Proposition 6. Let \(X, Y \in {\cal G}_{n,\ell}\). If \(G=X-Y\), then $$ W(G) = W(X) - W(Y) + 2W(\overline{X}Y) - W_{\max}. $$

Proof. Hexagonal chain \(G\) can be decomposed as \(G=X-Y=(X+\overline{X}Y) - (Y-\overline{X}Y)\), where \((X+\overline{X}Y) \ge (Y-\overline{X}Y)\), \(X(\overline{X}Y)=Z_{n,\ell}\) and \(Y \ge \overline{X}Y\). By Propositions 3 and 4, \begin{eqnarray*} W(G) & = & W(X+\overline{X}Y) - W(Y-\overline{X}Y) + W_{\max} \\ & = & W(X) + W(\overline{X}Y) - W_{\max} - (W(Y) - W(\overline{X}Y) + W_{\max}) + W_{\max} \\ & = & W(X) - W(Y) + 2 W(\overline{X}Y) - W_{\max}. \end{eqnarray*}

Consider again hexagonal chains \(X\) and \(Y\) shown in Figure 4. Chains \(\overline{X}Y\) and \(X-Y\) have codes \(r(\overline{X}Y)=(00010)\), \(r(X-Y)=(00101)\), and Wiener indices \(W(\overline{X}Y)=19647\), \(W(X-Y)=18815\). By Proposition \ref{XYminusGen}, we obtain \(W(G)= 18751 - 19071 + 2 \cdot 19647 - 20159=18815\).

Proposition 7. Let \(X, Y \in {\cal G}_{n,\ell}\). If \(G=X \vee Y\), then $$ W(G) = W(X) + W(Y) - W(XY). $$

Proof. Using Proposition 1, we get \begin{eqnarray*} W(X)\!+\!W(Y)\!-\!W(XY) & = & W_{\max}\!-\!16(\ell-1)^2 \sum_{i} i(n-i-1) (x_i+y_i-x_i y_i), \\ W(X \vee Y) &= & W_{\max}-16(\ell-1)^2 \sum_{i} i(n-i-1) (x_i \vee y_i). \end{eqnarray*} Comparison of these equalities completes the proof.

Hexagonal chains \(X, Y \in {\cal G}_{9,3}\) and \(G=X \vee Y\) of Figure 5 illustrate Proposition 7. The structure of graph \(X+Y\) is presented for comparison. These chains have codes \(r(X)=(1010101)\), \(r(Y)=(0100011)\), \(r(XY)=(0000001)\), and \(r(G)=(1110111)\), and their Wiener indices are equal to \(W(X)=37111\), \(W(Y)=37943\), \(W(XY)=39479\), and \(W(G)=35575\). By Proposition 7, \(W(G)=37111+37943-39479=35575\).

In conclusion, we note that the considered approach may be useful in studying of topological indices for sets of hexagonal chains induced by algebraic constructions.

Acknowledgments

This work is supported by the Russian Foundation for Basic Research (project numbers 19--01--00682 and 17--51--560008) and the Iranian National Science Foundation (INSF) under the contract No. 96004167.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

For all graph theoretical terms and notations not defined here the reader is referred to [1]. We only consider simple finite loopless undirected graphs. For a graph \(G=(V,E)\) with \(\vert V\vert=n\) vertices, an edge is a pair of two connected vertices \(x,y\), we denote it by \(xy,xy\in E\); when two vertices \(x,y\) are not connected this pair form the non-edge \(xy,xy\not\in E\). In \(G\) a \(2\)-factor is a subset of edges \(F\subset E\) such that every vertex is incident to exactly two edges of \(F\). Since \(G\) is finite a \(2\)-factor consists of a collection of vertex disjoint cycles spanning the vertex set \(V \). When the collection consists of an unique cycle the \(2\)-factor is connected, so it is a hamiltonian cycle.

We intend to determine, for any integer \(n\ge 3\), a graph \(G=(V,E),n=\vert V\vert\) with a minimum number of edges such that for every non-edge \(xy\) it is always possible to include the non-edge \(xy\) into a connected \(2\)-factor, i.e., the graph \(G_{xy}=(V,E\cup \{xy\})\) has a hamiltonian cycle \(H,xy\in H\). In other words for any non-edge \(xy\) of \(G\) there exits a hamiltonian path between \(x\) and \(y\).

This problem is related to the minimal \(2\)-factor extension studied in [2] in which the \(2\)-factors are not necessary connected. It is also related to the problem of finding minimal graphs for non-edge extensions in the case of perfect matchings (\(1\)-factors) studied in [3]. Another problem of hamiltonian extension can be found in [4].

Definition 1. Let \(G=(V,E)\) be a graph and \(xy\not\in E\) an non-edge. If \(G_{xy}=(V,E\cup \{xy\})\) has a hamitonian cycle that contains \(xy\) we shall say that \(xy\) has been extended (to a connected \(2\)-factor, to an hamiltonian cycle).

Definition 2. A graph \(G=(V,E)\) is connected 2-factor expandable or hamiltonian expandable (shortly expandable) if every non-edge \(xy\not\in E\) can be extended.

Definition 3. An expandable graph \(G=(V,E)\) with \(\vert V \vert=n\) and a minimum number of edges is a minimum expandable graph. The size \(\vert E\vert\) of its edge set is denoted by \(Exp_h(n)\).

The case where the \(2\)-factor is not constrained to be hamiltonian is studied in [2]. In this context \(Exp_2(n)\) denotes the size of a minimum expandable graph with \(n\) vertices. It follows that \(Exp_h(n)\ge Exp_2(n)\). We use the following notations. For \(G=(V,E)\), \(N(v)\) is the set of neighbors of a vertex \(v\), \(\delta(G)\) is the minimum degree of a vertex. A vertex with exactly \(k\) neighbors is a \(k\)-vertex. When \(P=v_i,\ldots,v_j\) is a sequence of vertices that corresponds to a path in \(G\), we denote by \({\bar P}=v_j,\ldots,v_i\) its mirror sequence (both sequences correspond to the same path). We state our result.

Theorem 1. The minimum size of a connected \(2\)-factor expandable graph is: $$Exp_h(3)=2,Exp_h(4)=4,Exp_h(5)=6; Exp_h(n)= \lceil {3\over 2}n\rceil,n\ge 6$$

Proof. For \(n\ge 3\) we have \(Exp_h(n)\ge Exp_2(n)\). In [2] it is proved that the three graphs given by Figure 1 are minimum for \(2\)-factor extension. They are also minimum expandable for connected \(2\)-factor extension.

Now let \(n\ge 6\). From [2] we know the following when \(G\) a minimum expandable graph for the \(2\)-factor extension:

• \(G\) is connected;
• if \(\delta(G)=1\) then \(Exp_2(n)\ge {3\over 2}n\);
• for \(n\ge 7\), if \(u,v\) are two \(2\)-vertices such that \(N(u)\cap N(v)\ne\emptyset\) then \(Exp_2(n)\ge {3\over 2}n\);

The graph given by Figure 2 is minimum for \(2\)-factor extension (see [2]). One can check that it is expandable for connected \(2\)-factor extension. So we have \(Exp_h(6)=9={3\over 2}n\).

Suppose that \(G\) is a minimum expandable graph with \(n\ge 7\) and \(\delta(G)=2\). Let \(v\in V\) with \(d(v)=2\), \(N(v)=\{u_1,u_2\}\). If \(u_1u_2\not\in E\) then \(u_1u_2\) cannot be expanded into a hamiltonian cycle. So \(u_1u_2\in E\). If \(d(u_1)=2\) then \(u_2\in N(u_1)\cap N(v)\) and \(Exp_h(n)\ge {3\over 2}n\). So from now one we may assume \(d(u_1),d(u_2)\ge 3\). Suppose that \(d(u_1)=d(u_2)=3\). Let \(N(u_1)=\{v,u_2,v_1\}, N(u_2)=\{v,u_1,v_2\}\). If \(v_1\ne v_2\) then \(u_1v_2\) is not expandable. If \(v_1= v_2\) then \(vv_1\) is not expandable. From now we can suppose that \(d(u_1)\ge 3,d(u_2)\ge 4\). Moreover \(v\) is the unique \(2\)-vertex in \(N(u_2)\). It follows that every \(2\)-vertex \(u\in V\) can be matched with a distinct vertex \(u_2\) with \(d(u_2)\ge 4\). Then \(\Sigma_{v\in V}d(v)\ge 3n\) and thus \(m\ge {3\over 2}n\).

When \(\delta(G)\ge3\) we have \(\Sigma_{v\in V}d(v)\ge 3n\). Thus for any expandable graph we have \(\vert E\vert=m\ge {3\over 2}n,n\ge 7\).

For any even integer \(n\ge 8\) we define the graph \(G_n=(V,E)\) as follows. Let \(n=2p\), \(V=A\cup B\) where \(A=\{a_1,\ldots,a_p\}\) and \(B= \{b_1,\ldots,b_p\}\). \(A\) (resp. \(B\)) induces the cycle \(C_A=(A,E_A)\) with \(E_A=\{a_1a_2,a_2a_3,\ldots,a_pa_1\}\) (resp. \(C_B=(B,E_B)\) with \(E_B=\{b_1b_2,b_2b_3,\ldots,b_pb_1\}\). Now \(E=E_A\cup E_B\cup E_C\) with \(E_C=\{a_2b_2,a_3b_3,\ldots,a_{p-1}b_{p-1},a_1b_p,a_pb_1\}\). Note that \(G_n\) is cubic so \(m= {3\over 2}n\). (see \(G_{10}\) in Fig. 3)

We show that \(G_n\) is expandable. First we consider a non-edge \(a_ia_j,p\ge j>i\ge 1\). Note that the case of a non-edge \(b_ib_j\) is analogous. We have \(j\ge i+2\) and since \(a_1a_p\in E\) from symmetry we can suppose that \(j< p\). Let \(P=a_j,a_{j-1},\ldots,a_{i+1},b_{i+1},b_{i+2},\ldots,b_{j+1},a_{j+1},a_{j+2},b_{j+2},\ldots,c_j\) where \(c_j\) is either \(a_p\) or \(b_p\) and let \(Q=a_i,b_i,b_{i-1},a_{i-1},\ldots,c_i\) where \(c_i\) is either \(a_1\) or \(b_1\). From \(P\) and \(Q\) one can obtain an hamiltonian cycle containing \(a_ib_j\) whatever \(c_i\) and \(c_j\) are.

Now we consider a non-edge \(a_ib_j\). Without loss of generality we assume \(j\ge i\). Suppose first that \(j=i\), so either \(i=1\) or \(i=p\). Without loss of generality we assume \(i=j=1\): \(a_1,b_p,b_{p-1},\ldots, b_2,a_2,a_3,\ldots, a_p,b_1,a_1\) is an hamiltonian cycle. Now assume that \(j>i\): Let \(P_j=b_j,b_{j-1},\ldots,b_{i+1},a_{i+1},a_{i+2},\ldots,a_{j_+1},b_{j+1},b_{j+2}, a_{j+2},\ldots, c_p\) where either \(c_p=a_p\) or \(c_p=b_p\), \(P_i=a_i,b_i,b_{i-1},a_{i-1},a_{i-2},\ldots,c_1\) where either \(c_1=a_1\) or \(c_1=b_1\). If \(c_p=a_p\) and \(c_1=a_1\) then \(P_j,b_1,b_p,P_i,a_j\) is an hamiltonian cycle. If \(c_p=a_p\) and \(c_1=b_1\) then \(P_j,a_1,b_p,P_i,a_j\) is an hamiltonian cycle. The two other cases are symmetric.

For any odd integer \(n=2p+1\ge 7\) we define the graph \(G_n=(V,E)\) as follows. We set \(V=A\cup B\cup\{v_{n}\}\) where \(A=\{a_1,\ldots,a_p\}\) and \(B= \{b_1,\ldots,b_p\}\). \(A\cup\{v_n\}\) (resp. \(B\cup\{v_n\}\)) induces the cycle \(C_A=(A\cup\{v_n\},E_A)\) with \(E_A=\{a_1a_2,a_2a_3,\ldots,a_pv_n,v_na_1\}\) (resp. \(C_B=(B\cup\{v_n\},E_B)\) with \(E_B=\{b_1b_2,b_2b_3,\ldots,b_pv_n,v_nb_1\}\). Now \(E=E_A\cup E_B\cup E_C\) with \(E_C=\{a_ib_i\vert 1\le i\le p\}\cup\{a_1v_n,b_1v_n,a_pv_n,b_pv_n\}\). Note that \(m= \lceil{3\over 2}n\rceil\). (see \(G_{7}\) and \(G_{11}\) in Figure 3)

We show that \(G_n\) is expandable. First, we consider a non-edge \(a_ia_j,p\ge j>i\ge 1\) (the case of a non-edge \(b_ib_j\) is analogous). \(a_i,a_{i+1},\ldots,a_{j-1},b_{j-1},b_{j-2},b_{j-3},\ldots,b_i,b_{i-1}, a_{i-1},a_{i-2},b_{i-2},\ldots,v_n,c_p,d_p,d_{p-1},c_{p-1},\ldots, c_j,d_j\), where \(d_j=a_j\) and for any \(k,j\le k\le p,\) the ordered pairs \(c_k,d_k\) correspond to either \(a_k,b_k\) or \(b_k,a_k\), is an hamiltonian cycle. Second, let a non-edge \(a_ib_j,p\ge j>i\ge 1\). We use the same construction as above taking \(d_j=b_j\).

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

Let us begin with some notation. Let \( {b}\) be a row vector and \({a}\) be a finite row vector; \(({b})_j\) denotes \(j\)th element of vector \({b}\), \(|{b}|\) denotes length of \( {b}\); \({a} ◡ {b} ≝\left( ({a})_1 ,\ldots , ({a})_{|{a}|} \ , ({b})_1 , \ldots \right) \), \(({b})_{l}^{m} ≝\left( ({b})_j \right) _{j=l}^{m}\), \(|{b}|_x ≝ \sum_{j=1}^{|{b}|} ({b})_j x^{j-1}\), \({1}_{m} ≝\left( 1 \right)_{j=1}^{m}\), \({0}_{m} ≝\left( 0 \right)_{j=1}^{m}\), \([ l, m ]_j ≝ ({{1}}_\infty \times ( {{0}}_{l} ◡ {{1}}_{m} ) )_j\), where \(\times\) denotes the Kronecker product, \( l \in \mathbb{N}\), \(m \in \mathbb{N} \cup \{\infty\}\). Note that the last function can be expressed by the ceiling function: \([ l, m ]_j = \lceil (j + m)/(l + m)\rceil - \lceil j/(l + m)\rceil\).

Also, we use ordinary notation to denote the corresponding entrywise operations. For example, \({a}^2\) expresses the Hadamard square: \({a}^2 =\left( {({a})_j}^2 \right) _{j=1}^{|{a}|}\).

It should be noted that there are many papers on sequences generated by linear difference equations with variable coefficients. See, for examle, [1], [2], [3]. The simple approach illustrated here involves constructing for each such sequence a corresponding recursive vector sequence, which can be explicitly expressed using the following property of Hadamard product: \(({a}{c})\ast ({b}{d})=({a} \ast {b})({c} \ast {d})\), where \(|{c}| = |{a}|\), \(|{d}| = |{b}|\) and \(\ast \in \{ ◡ , \times \}\).

2. Linear recurrences

First we use this property with respect to concatenation.

Theorem 1. If \(x_1\), \(x_2\) are arbitrary numbers, \(a_n\), \(b_n\) are arbitrary number sequences and \(x_n = a_n x_{n-1} + b_n x_{n-2}\) for \(n\geqslant 3\), then $$ x_n = \sum_{j=1}^{f_n} ((x_1 - x_2)({f})_j + x_2) \biggl( \prod_{\substack{3 \leqslant k \leqslant n \\ ({f}_k)_j = 1}} b_k \biggr) \biggl( \prod_{\substack{3 \leqslant k \leqslant n \\ ({f}_k + {f}_{k+1})_j = 0}} a_k \biggr) \text{,} $$ where \(f_n\) is \(n\)th Fibonacci number, \({f} =\left( 0,1,0,0,\ldots \right) \) is infinite Fibonacci word; \({f}_k\) is obtained from \({f}\) by replacing each entry of zero with \(f_{k-1}\) zeros and each entry of one with \(f_{k-2}\) ones.

Proof. Define vectors: \({x}_1 =\left( x_1 \right)\), \({x}_2 =\left( x_2 \right)\), \({x}_n = (b_n {x}_{n-2}) ◡ (a_n {x}_{n-1})\) for \(n\geqslant 3\) and \({y}_1 = {x}_1\), \({y}_n = {y}_{n-1} ◡ {x}_n\), \({p}_n = {1}_{f_{n+1}-1} ◡ (b_n {1}_{f_{n-2}}) ◡ (a_n {1}_{f_{n-1}}) \) for \(n\geqslant 2\).
Let \(\Lambda_k {b} ≝ {b} ◡ ({b})_k^{|{b}|}\). We have \({y}_n = {p}_n \Lambda_{f_{n-1}} {y}_{n-1}\) for \(n\geqslant 3\), from which it follows that \({y}_n = {y}_{2,n} \prod_{k=3}^n{p}_{k,n}\), where \({y}_{2,n} = \Lambda_{f_{n-1}}\Lambda_{f_{n-2}}\ldots\Lambda_{f_2}{y}_2\), \({p}_{k,n} = \Lambda_{f_{n-1}}\Lambda_{f_{n-2}}\ldots\Lambda_{f_k}{p}_k\).
Partition \({y}_{2,n} = {y}_1' ◡ \ldots ◡ {y}_n'\) such that \(|{y}_i'| = f_i\), then \({y}_1' = {x}_1\), \({y}_2' = {x}_2\), \({y}_n' = {y}_{n-2}' + {y}_{n-1}'\) for \(n\geqslant 3\). Similarly partition \({p}_{k,n} = {p}_{k,1}' ◡ \ldots ◡ {p}_{k,n}'\) such that \(|{p}_{k,i}'| = f_i\), then \({p}_{k,i}'={1}_{f_i}\) for \(1 \leqslant i\leqslant k-1\), \({p}_{k,k}'=(b_k {1}_{f_{k-2}}) ◡ (a_k {1}_{f_{k-1}})\), \({p}_{k,i}' = {p}_{k,i-2}'◡ {p}_{k,i-1}'\) for \(k+1 \leqslant i\leqslant n\).
Note that \(({x}_n)_{-} = ({y}_n')_{-} \prod_{k=3}^n({p}_{k,n}')_{-}\), where by \(({a})_{-}\) we denote the vector composed of elements of \({a}\) in reverse order. Now \(({y}_n')_{-}\) and \(({p}_{k,n}')_{-}\) can be expressed in terms of infinite generalized Fibonacci words: \(({y}_n')_{-} = (x_1 - x_2) ( {f} )_1^{f_n} + x_2 {1}_{f_n}\), \(({p}_{k,n}')_{-} = ( {f}_{k+1} + b_k {f}_k + a_k ({1}_{f_n}-{f}_{k+1}-{f}_k))_1^{f_n}\). Finally using \(x_n = |({x}_n)_{-}|_1\) we get the result.

Remark 1. It is easy to check that the result of Theorem 1 can be reformulated as follows:

The same sequence can be expressed with help of the Kronecker product.

Theorem 2. If \(x_1\), \(x_2\) are arbitrary numbers, \(a_n\), \(b_n\) are arbitrary number sequences and \(x_n = a_n x_{n-1} + b_n x_{n-2}\) for \(n\geqslant 3\), then $$ x_n = \sum_{\substack{2^{n-2} + 1 \leqslant j \leqslant 2^{n-1} \\ \vartheta(2^{n-1}-j+1)= 1} } ((x_2-x_1)[ 1,1 ]_j + x_1) \prod_{\substack{0 \leqslant k \leqslant n-3 \\ [ 3\cdot2^k,2^k ]_j =1 }}a_{k+3} \prod_{\substack{0 \leqslant k \leqslant n-3 \\ [ 2^k,2^k ]_j =0 }}b_{k+3} \text{,} $$ where \(\vartheta (n) ≝ \prod_{k=0}^{\infty} (1-[ 3\cdot2^k,2^k ]_n)\).

Proof. Define vectors: \({r}_1 =\left( x_1 , x_2 \right)\), \({r}_n = ({1}_2 \times {r}_{n-1})({h}_n \times {1}_{2^{n-2}})\) for \(n\geqslant 2\), where \({h}_n =\left( 0,1,b_{n+1},a_{n+1}\right)\). It can be easily shown that \(|({r}_n)_{2^{n-1}+1}^{2^n}|_1 = x_{n+1}\). Solving the recurrence equation we get: \(r_n = ({1}_{2^{n-1}} \times {r}_1) \prod_{k=2}^{n} ({1}_{2^{n-k}} \times {h}_k \times {1}_{2^{k-2}})\). Taking into account that \({h}_k =\left( 0,1,1,1 \right)\left( b_{k+1},1,b_{k+1},1\right)\left( 1,1,1, a_{k+1}\right) \) and doing some calculations we get the result.

The following lemma allows us to generalize the result to the nonhomogeneous case.

Lemma 1. If \({x}_1\) is arbitrary vector, \({b}_n\) is \(0,1\)-vector sequence, \({a}_n\) and \({c}_n\) are such that \(|{a}_n| = |{c}_n| = |{x}_1| \prod_{i=2}^n |{b}_i|\); \({x}_n = {a}_n ({b}_n \times {x}_{n-1})+{c}_n\) for \(n \geqslant 2\), then $$ {x}_n=({b}_{n,2} \times {x}_1) \prod_{k=2}^n({b}_{n,k+1} \times {a}_k) + \sum_{i=2}^n({b}_{n,i+1} \times {c}_i)\prod_{k=i+1}^n({b}_{n,k+1} \times {a}_k) \text{,} $$ where \({b}_{n,k} = {b}_n \times {b}_{n-1} \times \cdots \times {b}_k\), if \(k\leqslant n\) and \({b}_{n,k} = {1}_1\), if \(k > n\).

The proof is straightforward. Vectors \({r}_n'\) for similar nonhomogeneous sequence \(x_n' = a_n x_{n-1}' + b_n x_{n-2}' + c_n\) such that \(|({r}_n')_{2^{n-1}+1}^{2^n}|_1 = x_{n+1}'\), are defined as follows: \({r}_1' =\left( x_1 , x_2 \right)\), \({r}_n' = ({1}_2 \times {r}_{n-1}')({h}_n \times {1}_{2^{n-2}}) + c_n ( {0}_{2^n-1} ◡ {1}_1 )\) for \(n\geqslant 2\). To use the lemma we should, of course, do substitutions \({a}_n={h}_n \times {1}_{2^{n-2}}\) and \({b}_{n,k} = {1}_{2^{n-k+1}}\).

Theorem 3. If \(w_0\) is arbitrary number, \(a_{n,j}\) is arbitrary number sequence and \(w_n = \sum_{j=0}^{n-1}a_{n,j}w_j\) for \(n\in \mathbb{N}\), then $$ \sum_{j=0}^nw_j = w_0 \sum_{{v}\in \mathbb{V}_n}\prod_{k=1}^{|{v}|}a_{({v})_k , ({0}_1 ◡ {v})_k }\text{,} $$ where set \(\mathbb{V}_n\) consists of all vectors \({v}\) such that \(1 \leqslant({v})_{i-1} < ({v})_i \leqslant n\) for \(2 \leqslant i \leqslant |{v}|\).

Proof. Define vectors: \({w}_0 =\left( w_0 \right)\), \({w}_1 =\left( w_0 , a_{1,0} w_0 \right)\), \({w}_n = {w}_{n-1} ◡ ({w}_{n-1} {q}_n)\) for \(n\geqslant 2\), where \(({q}_n)_1 = a_{n,0}\), \(({q}_n)_{2^{k-1}+1}^{2^k} = a_{n,k} {1}_{2^{k-1}}\) for \(1 \leqslant k\leqslant n-1\). From the recurrence equation it follows that if equality \(|{w}_l|_1 = \sum_{j=0}^l w_j\) is true for \(l=n-1\geqslant 1\), then it is true for \(l=n\); it is true for \(l=1\), so we conclude that it is true for any \(l \in \mathbb{N}\). Solving the recurrence equation, we get: \({w}_n = w_0 \prod_{k=1}^n ({1}_{2^{n-k}} \times ({1}_{2^{k-1}} ◡ {q}_k)) \) for \(n \in \mathbb{N} \). Noting that \({q}_k =\left( a_{k,\lceil \log_2 j\rceil} \right)_{j=1}^{2^{k-1}}\) we have $$ |{w}_n|_1 = w_0 \sum_{j=1}^{2^n}\prod_{k=1}^n([2^{k-1},2^{k-1}]_j (a_{k,\lceil \log_2 ( 1+ (j-1)\bmod2^{k-1})\rceil} -1) +1) \text{.} $$ The quantity \([2^k,2^k]_j\) equals the value of \(k\)th digit of number \((j-1)\) in binary numeral system. If \(k_i\) is serial number of \(i\)th digit \(1\), then \(\lceil \log_2 (1+{(j-1)\bmod 2^{k_i}})\rceil = 1+k_{i-1}\). Therefore, \(|{w}_n|_1 = w_0 \sum_{j}\prod_{i}a_{k_i + 1,k_{i-1}+1}\), assuming \(k_{0}=-1\). Here \((k_i +1)\) ranges over \({v} \in \mathbb{V}_n\): \(k_i +1 = ({v})_i\).

In the same way vectors \({w}_n'\) for nonhomogeneous sequence \(w_n' = c_n + \sum_{j=0}^{n-1}a_{n,j}w_j'\) such that \(|{w}_l'|_1 = \sum_{j=0}^l w_j\), are defined as follows: \({w}_0' =\left( w_0 \right)\), \({w}_1' =\left( w_0 , c_1 + a_{1,0} w_0 \right)\), \({w}_n' = {w}_{n-1}' ◡ ({w}_{n-1}' {q}_n) + c_n ({0}_{2^n-1} ◡ {1}_1) \) for \(n\geqslant2\).

3. Quadratic map

It is well-known that in many cases iterations of a polynomial of degree 2 in the general case, i.e. solutions of quadratic map, can be expressed by iterations of a polynomial of degree 2 with one parameter.

Theorem 4. Let \(p^{(0)}(x)=x\), \(p^{(n)}(x)=p^{(n-1)}(p(x))\) \((\) for \(n \in \mathbb{N}\) \()\) be iterations of polynomial \(p(x)=\lambda (x+1)x\), then $$ p^{(n)}(x)=\sum_{k=1}^{2^n}x^k\sum_{i=(k-1)\omega_n}^{k \omega_n -1} \prod_{j=1}^n \lambda ^{\mu_{i,j-1}}\binom{\mu_{i,j-1}}{\mu_{i,j} - \mu_{i,j-1}}\text{,} $$ where \(\omega_n = 2^{\frac{n(n-1)}{2}}\), \(\mu_{i,j} = \lceil \frac{1+ i \bmod \omega_{j+1}}{\omega_j} \rceil\).

Proof. Any polynomial \(p^{(n)}(x)\) can be expressed as follows: \(p^{(n)}(x) = \sum_{k=1}^{2^n} g_{n,k}x^k\), where \(g_{n,k}=g_{n,k}(\lambda)\) are polynomials defined by equalities: \(g_{0,1}=1\), \(g_{n,k}=\sum_{i=1}^{2^{n-1}}q_{k,i} g_{n-1,i}\) (for \( 1 \leqslant k \leqslant2^n\)), where \(q_{k,i}=\lambda^i \binom{i}{k-i}\). Let \({p}_0 =\left( 1 \right)\), \({p}_n = ({1}_{2^n} \times {p}_{n-1}) ({q}_n \times {1}_{\omega_{n-1}})\) for \(\in \mathbb{N}\), where $${q}_n = (q_{1,j})_{j=1}^{2^{n-1}} ◡ (q_{2,j})_{j=1}^{2^{n-1}} ◡ \ldots ◡ (q_{2^n,j})_{j=1}^{2^{n-1}} =\left( q_{\lceil i/2^{n-1}\rceil, 1+ (i-1)\bmod 2^{n-1} }\right)_{i=1}^{2^{2n-1}} \text{.} $$ Then \(|({p}_n)_{1+(k-1)\omega_n}^{k\omega_n}|_1 = g_{n,k}\). Solving the equation, we get: $$ {p}_n = \prod_{k=1}^n \biggl( {1}_{2^{\frac{(n+k+1)(n-k)}{2}}} \times {q}_k \times {1}_{\omega_{k-1}} \biggr) \text{.} $$ The last step is to check that \({1}_{\infty} \times {q}_k \times {1}_{\omega_{k-1}} =\left( {q}_{\mu_{j,k},\mu_{j,k-1}} \right)_{j=0}^{\infty} \).

Remark 2. Using generating polynomial \(|{q}_k|_t=\lambda(1+t^m) ((\lambda t^{m+1} + \lambda t^{2m+1})^m -1)/(\lambda t^{m+1} + \lambda t^{2m+1}-1)\), where \(m=2^{k-1}\) and taking in account formula \(|{a} \times {b}|_t =|{a}|_{t^{|{b}|}} |{b}|_t\) we can represent \(p^{(n)}(x)\) as Hadamard's product of \(n\) functions. The polynomial \(|{q}_k|_t\) can be derived using polynomials \(\varphi_{k,m} = \sum_{j=1}^m \binom{j}{k-j} (\lambda t)^{j-1}\) as follows. Taking in account that \(\varphi_{k,m} = \lambda t (\varphi_{k-1,m}+\varphi_{k-2,m})-(\lambda t)^m \binom{m+1}{k-m-1}\) for \(k \geqslant 3\) and \(m \geqslant 1\), we can write \begin{align} \lambda^{-1} |{q}_k|_t & = \sum_{k=1}^{2 m}t^{(k-1)m}\varphi_{k,m} \notag = 1+t^m + \lambda t^{m+1}\sum_{k=1}^{2m-1}t^{(k-1)m}\varphi_{k,m} + \lambda t^{2 m+1}\sum_{k=1}^{2m-2}t^{(k-1)m}\varphi_{k,m} -\lambda^m \sum_{k=2}^{2m}t^{k m}\binom{m+1}{k-m-1}\notag \\ & = 1+t^m + \lambda t^{m+1}(\lambda^{-1}|{q}_k|_t - t^{m(2m-1)}\varphi_{2m,m}) + \lambda t^{2 m+1}(\lambda^{-1}|{q}_k|_t - t^{m(2m-1)}\varphi_{2m,m}- t^{m(2m-2)}\varphi_{2m-1,m} ) \notag \\ & -\lambda^m (t^{m(m+1)}(1+t^m)^{m+1}-(m+t^m+1)t^{2 m^2 +m}). \notag \end{align} From here taking in account \(\varphi_{2m,m}=(\lambda t)^{m-1}\) and \(\varphi_{2m-1,m}=m (\lambda t)^{m-1}\) we immediately get \(|{q}_k|_t\).

Let's consider another episode. Let \(s^{(0)}(x)=x\), \(s^{(n)}(x)=s^{(n-1)}(s(x))\) (for \(n \in \mathbb{N} \)) be iterations of polynomial \(s(x)= s_\lambda(x)=\lambda (x^2-1)+1\) and \(\lambda \neq 0\). Define vectors: \({s}_1 =\left( x-1, 1\right)\), \({s}_n = \lambda {s}_{n-1}^{\langle 2 \rangle} - (\lambda -1) ({0}_{2^{2^{n-1}}-1} ◡ {1})\) for \(n \geqslant 2\), where triangular brackets indicate Kronecker degree. Obviously, \(|{s}_n|_1 = s(|{s}_{n-1}|_1) = s^{(n)}(x)\). Solving the equation, we get: where \({l}_n = {1}_{2^{2^n}-1} ◡\left( \lambda^{-1} \right)\), \({r}_{\lambda,n} = \prod_{i=1}^{n}{l}_i^{\langle 2^{n-i}\rangle}\), \({r_\lambda} = \prod_{i=1}^{\infty}{l}_i^{\langle \infty \rangle}\). Therefore, we have where \({h}_n = \log_2\bv 2,1 \right)^{\langle 2^{n-1} \rangle}\). And it can be easily shown that $$({h}_n)_j = 2^{n-1}-\sum_{k=0}^{\infty}[2^k,2^k]_j \text{ and } (\log_2 {r}_{2,n})_j = \sum_{k,i=0}^{\infty}[2^{2^i k}(2^{2^i}-1),2^{2^i k}]_j $$ by using simple formula $$(\log_2 ({1}_{m-1} ◡\left( 2 \right))^{\langle n \rangle})_j = \sum_{k=0}^{n-1}[m^k(m-1),m^k]_j \text{ (for } j\leqslant m^n \text{).} $$ Substituting \(2\) for \(x\) in (2), we have $$ s_\lambda^{(n)} (2) = \lambda^{2^{n-1}-1}\sum_{k=1}^{2^{n-1}}\kappa_{n,k}{\lambda}^{-k}\text{,} $$ where \(\kappa_{n,k}\) denotes the number of elements in \(\log_2 {r}_{2,n-1}\) that equal to \(k\). This function can be defined recursively as follows: $$\kappa_{n,0} = 3^{2^{n-1}} , \kappa_{1,k} = \delta_{k,1} , \kappa_{n,k} =\delta_{k,2^{n-1}} - \delta_{k,2^{n-2}} + \sum_{i=0}^k \kappa_{n-1,k-i} \kappa_{n-1,i} \text{.} $$ Evaluating \(\kappa_{n,k}\) we have: $$\kappa_{n,1} = 2^{n-1} 3^{2^{n-1}-1}, \kappa_{n,2} = -\kappa_{n,1} (\frac{1}{12}-\sum_{i=0}^{n-1}\kappa_{i+1,1}({\kappa_{i,1}}^2 + \delta_{k,2^{n-1}} - \delta_{k,2^{n-2}}) $$ and so on.

Remark 3. Replacing \(2^{n-1}\) by \(n\) in the last expression of (1), we get new sequence of vectors: \({s}_n' = \lambda^{n-1} {s}_1^{\langle n \rangle} ({r})_1^{2^n}\). Let \(f_n(x) = |{s}_n'|_1\). Conjecture: $$ f_n(x)= \begin{cases} f_{n/2}(s(x)) \text{, if } n \text{ is even} \\ \lambda x f_{n-1}(x) \text{, if } n \text{ is odd}\\ x \text{, if } n=1. \end{cases} $$

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

Let \(G=(V,E)\) be a simple, connected graph with \(n=|V|\) vertices. The spectrum of \(G\) consists of the eigenvalues of its \((0,1)\)-adjacency matrix \(A\), ordered as \(\lambda_1>\lambda_2\geq\dots\geq\lambda_n\). Let \(x_1,\dots,x_n\) be an orthonormal basis of eigenvectors of \(A\), such that \(x_i\) is an eigenvector of \(\lambda_i\) for \(i=1,\dots,n\). The adjacency matrix \(A\) then has a spectral decomposition \(A=Q\Lambda Q^\top\), where \(\Lambda=\mathrm{diag}(\lambda_1,\dots,\lambda_n)\) and \(Q=[x_1\ x_2\ \dots\ x_n]\) is the matrix with eigenvectors listed in columns.

As in Nikiforov [1], let \(w_k\) denote the number of walks with \(k\) vertices, hence of length \(k-1\), in a graph \(G\). Since \(w_k\) is the sum of entries of \(A^{k-1}\), the spectral decomposition yields \(A^{k-1}=Q\Lambda^{k-1}Q^\top\), so

Apparently, only those eigenvalues \(\lambda_i\) for which \(\sum_{j=1}^n x_{i,j}\) is not zero affect the value of \(w_k\). Such eigenvalues are called the main eigenvalues. The spectral radius \(\lambda_1\) of a connected graph is always a main eigenvalue, due to its strictly positive eigenvector \(x_1\) [2]. Regular graphs, for which \(x_1\) is proportional to the all-one vector \(\mathbf{j}\), have exactly one main eigenvalue, as all their other eigenvectors are orthogonal to \(\mathbf{j}\).

It follows from (1) that \(\lambda_1=\lim_{k\to\infty} \sqrt[2k]{w_{2k+1}}\), as \(\lambda_1\) has the largest absolute value among the eigenvalues of \(A\) by the Perron-Frobenius theorem [2].

Nikiforov proved in [1] that the inequality \(\lambda_1^r\geq w_{s+r}/w_s\) holds for all odd \(s>0\) and all \(r>0\). He further showed that \(\lambda_1^r\) can be smaller than \(w_{s+r}/w_s\) for even \(s\) and odd \(r\) on the example of complete bipartite graphs, and then posed the following problem.

Problem 1.[1] Let \(G\) be a connected bipartite graph. Is it true that $$ \lambda_1^r\geq\frac{w_{s+r}}{w_s} $$ for every even \(s\geq 2\) and even \(r\geq 2\)?

Several counterexamples have been found since the problem was posed. Nikiforov himself offered the complete tripartite graph \(K_{2t,2t,t}\) as a counterexample for \(s=r=2\). Elphick and Réti [3] produced an infinite family of unicyclic graphs as counterexamples for \(s=r=2\) and further showed that the path \(P_4\) is a counterexample for even \(s\geq 2\) and arbitrary \(r\). One of the reviewers of [4] provided the following more general result.

Theorem 1.[4] Let \(G\) be a connected graph with two main eigenvalues \(\lambda_1\) and \(\lambda_i\), such that \(0>\lambda_i>-\lambda_1\). If \(s\geq 2\) and \(r\geq 2\) are even, then $$ \lambda_1^r < \frac{w_{s+r}}{w_s}. $$

While Cvetković [5] posed the problem of characterizing graphs with a given number of main eigenvalues already in 1978, the first results on graphs with two main eigenvalues started to appear only after a seminal paper by Hagos [6] was published in 2002. Hagos showed that a graph has exactly \(k\) main eigenvalues if and only if \(k\) is the maximum number such that \(\mathbf{j}\), \(A\mathbf{j}\), \dots, \(A^{k-1}\mathbf{j}\) are linearly independent. For \(k=2\) this means that there exist \(\alpha\) and \(\beta\) such that

and that \(G\) is not regular. Graph \(G\) satisfying (2) is also called a 2-walk \((\alpha,\beta)\)-linear graph and its main eigenvalues are [6, Corollary 2.5]

Various constructions of graphs with two main eigenvalues have been described in a number of papers [7, 8, 9, 10, 11, 12, 13, 14], and graphs satisfying the requirements of Theorem 1 can be found in most of these papers.

Our purpose here is to generalize a set of counterexamples presented in [4], which enables one to show that the ratio \(\frac{w_{s+r}}{\lambda_1^r w_s}\) can be significantly larger than 1. Since this cannot be shown by using main eigenvalues and their eigenvectors only, we will resort to combinatorial counting of walks in such graphs.

Let us recall the definition of equitable partition of vertices. Let \(\pi=\{\pi_1,\dots,\pi_k\}\) be a partition of the vertex set of a graph \(G\), and for each \(v\in\pi_i\) denote by \(d^{(j)}_v\) the number of neighbors of \(v\) in \(\pi_j\). The partition \(\pi\) is called {\em equitable} if for all \(i\) and \(j\), the value \(d^{(j)}_v\) has the same value, denote it by \(d^{i\to j}\), for all \(v\in\pi_i\). The {\em quotient matrix} of such partition \(\pi\) is the matrix \(Q_{\pi}=\left(d^{i\to j}\right)\).

Definition 1. For positive integers \(p,q\) and \(r\), let \(\mathcal{G}_{p,q,r}\) be the set of graphs which have an equitable vertex partition with the quotient matrix \(\left[\begin{smallmatrix} p & q \\ r & 0 \end{smallmatrix}\right]\).

We can now state the main results of the paper.

Theorem 2. Let \(G\in\mathcal{G}_{p,q,r}\) and let \(A\cup B\) be an equitable partition of \(G\) with the quotient matrix \(\left[\begin{smallmatrix} p & q \\ r & 0 \end{smallmatrix}\right]\). Then for any \(k\in\mathbb{N}\),

Let \((F_n)_{n\geq 0}\) be the sequence of Fibonacci numbers and \(\varphi=(1+\sqrt5)/2\) be the golden ratio.

Theorem 3. In each of the following cases, there exists a separate sequence of connected bipartite graphs \((G_p)_{p\in\mathbb{N}}\) such that:
a) \( \displaystyle\lim_{p\to\infty} \displaystyle\frac{w_{s+r}(G_p)}{\lambda_1^r(G_p) w_s(G_p)} =\displaystyle\frac{s+r-2}{s-2}, \) for even \(s\geq 4\) and even \(r\geq 2\);
b) \( \displaystyle\lim_{p\to\infty} \displaystyle\frac{w_{s+r}(G_p)}{\lambda_1^r(G_p) w_s(G_p)} =\displaystyle\frac{s+r}{s}, \) for even \(s\geq 2\) and even \(r\geq 2\);
c) \( \displaystyle\lim_{p\to\infty} \displaystyle\frac{w_{s+r}(G_p)}{\lambda_1^r(G_p) w_s(G_p)} =\displaystyle\frac{s+r+2}{s+2}, \) for even \(s\geq 2\) and even \(r\geq 2\);
d) \( \displaystyle\lim_{p\to\infty} \displaystyle\frac{w_{s+r}(G_p)}{\lambda_1^r(G_p) w_s(G_p)} =\displaystyle\frac{F_{s+r}}{\varphi^r F_s}, \) for even \(s\geq 2\) and even \(r\geq 2\);
e) \( \displaystyle\lim_{p\to\infty} \displaystyle\frac{w_{s+r}(G_p)}{\lambda_1^r(G_p) w_s(G_p)} =\displaystyle\frac{F_{s+r-2}}{\varphi^r F_{s-2}}, \) for even \(s\geq 4\) and even \(r\geq 2\).

Theorem 2 is proved in Section 2, while the various parts of Theorem 3 are proved in Section 3.

2. Numbers of walks in graphs from \(\mathcal{G}_{p,q,r}\)

Proof. [Proof of Theorem 2]

Let \(A\cup B\) be an equitable partition of \(G\) with the quotient matrix \(Q=\left[\begin{smallmatrix} p & q \\ r & 0 \end{smallmatrix}\right]\). Due to \(Q_{2,2}=0\), from each vertex in \(B\) a walk can continue only to one of its \(r\) neighbors in \(A\), while from each vertex in \(A\) a walk can continue to either one of its \(p\) neighbors in \(A\) or one of its \(q\) neighbors in \(B\).

We can now classify the \(k\)-walks in \(G\) according to the \(k\)-sequence of letters \(A\), \(B\) indicating to which set the vertices along a walk belong. For a given \(k\)-sequence of letters \(A\) and \(B\), the number of the corresponding \(k\)-walks can be determined by choosing the first vertex of a walk and then by considering pairs of successive letters:

• each pair \(AA\) yields \(p\) choices for the second \(A\) after the vertex corresponding to the first \(A\) is chosen;
• each pair \(AB\) yields \(q\) choices for \(B\) after the vertex for \(A\) is chosen;
• each pair \(BA\) yields \(r\) choices for \(A\) after the vertex for \(B\) is chosen.

For example, the sequence \(BAABA\) encodes \(|B|\cdot r\cdot p\cdot q\cdot r=pqr^2|B|\) walks with five vertices, while \(AABAABA\) encodes \(|A|\cdot p\cdot q\cdot r\cdot p\cdot q\cdot r=p^2q^2r^2|A|\) walks with seven vertices.

The fact that a feasible type sequence does not contain the pair \(BB\) means that each letter \(B\) may occupy either a single position between any two consecutive letters \(A\), or a single position prior to the first \(A\) or after the last \(A\). Since the number of walks with \(k\) vertices with a given letter sequence is influenced by the first and the last letter of the sequence, we will count them separately, working out in detail the first possibility only.

Hence suppose that a given letter sequence starts and ends with the letter \(A\) and that it contains \(l\) letters \(B\) (and consequently \(k-l\) letters \(A\)). There are \(k-l-1\) feasible positions for letters \(B\) between consecutive letters \(A\), so the number of such letter sequences is \({k-l-1\choose l}\). The initial letter \(A\) yields \(|A|\) choices for the initial vertex of a \(k\)-walk. Each letter \(B\) appearing in the type sequence produces one pair \(AB\) and one pair \(BA\), which together yield \(qr\) choices for two corresponding vertices along a \(k\)-walk. This leaves a total of \(k-1-2l\) pairs \(AA\) remaining in the type sequence, each of which yields \(p\) choices for the corresponding vertex in a \(k\)-walk. Hence each letter sequence starting and ending with \(A\) corresponds to a total of \(p^{k-2l-1}(qr)^l|A|\) walks with \(k\) vertices, and the number of \(k\)-walks corresponding to all such letter sequences is equal to $$ \sum_{l\geq 0} {k-l-1\choose l} p^{k-2l-1}q^lr^l|A|. $$ Following the similar argument, we get that:

• the number of \(k\)-walks corresponding to letter sequences starting with \(A\) and ending with \(B\) is equal to $$ \sum_{l\geq 1} {k-l-1\choose l-1} p^{k-2l}q^lr^{l-1}|A|; $$
• the number of \(k\)-walks corresponding to letter sequences starting with \(B\) and ending with \(A\) is equal to $$ \sum_{l\geq 1} {k-l-1\choose l-1} p^{k-2l}q^{l-1}r^l|B|; $$
• the number of \(k\)-walks corresponding to letter sequences starting and ending with \(B\) is equal to $$ \sum_{l\geq 2} {k-l-1\choose l-2}p^{k-2l+1}q^{l-1}r^{l-1}|B|. $$

Summing up these four cases we see that the total number of \(k\)-walks in \(G\) is: Note that the number of edges with one vertex in \(A\) and another in \(B\) can be counted as both \(q|A|\) and \(r|B|\), which yields \(q|A|=r|B|\), so (5) becomes: Upper limits for the three sums in (6) can be determined from the corresponding binomial coefficients:

• nonzero summands in the first sum are obtained for \(k-l-1\geq l\), i.e., for \(l\leq \left\lfloor\frac{k-1}2\right\rfloor\);
• nonzero summands in the second sum are obtained for \(k-l-1\geq l-1\), i.e., for \(l\leq \left\lfloor\frac{k}2\right\rfloor\);
• nonzero summands in the third sum are obtained for \(k-l-1\geq l-2\), i.e., for \(l\leq \left\lfloor\frac{k+1}2\right\rfloor\).

Putting these upper limits in (6) yields (4).

3. Nikiforov's ratio when \(q\) and \(r\) are powers of \(p\)

The key to proving various parts of Theorem 3 is to turn the expression for the number of walks \(w_k\) into a polynomial of a single variable by letting \(q\) and \(r\) to be the powers of \(p\): \(q=p^c\) and \(r=p^d\) for some nonnegative integers \(c\) and \(d\). In such case we have In addition, the quotient matrix \(Q=\left[\begin{smallmatrix} p & q \\ r & 0 \end{smallmatrix}\right]\) determines a divisor of any graph \(G\in\mathcal{G}_{p,q,r}\). As graphs in \(\mathcal{G}_{p,q,r}\) are not regular when \(p+q\neq r\), by [15, Theorem 3.9.9] any of them has two main eigenvalues that are equal to the eigenvalues of \(Q\): Now we can determine the limit of the Nikiforov's ratio \(\frac{w_{s+r}}{\lambda_1^r w_s}\) by discussing possible cases. Since the Nikiforov's problem assumes both \(s\) and \(r\) to be even, we will assume that \(k\) in (7) is even, i.e., $$k=2k',$$ in order to simplify discussion.
Case 1: \(c+d-2>0\). In this case $$ \lim_{p\to\infty} \frac{\lambda_1}{p^{(c+d)/2}}=1. $$ The highest exponents appearing in the three sums of (7) are $$ 2k'+(c+d-2)(k'-1)-1, 2k'+(c+d-2)k'-d, 2k'+(c+d-2)k'+1-2d, $$ respectively. We can now distinguish the following subcases. Subcase 1(a): \(c=0\), hence \(d>2\). In this case, the highest exponent in (7) is \(2k'+(c+d-2)(k'-1)-1\), appearing in the first sum for \(l=k'-1\), and the corresponding coefficient is equal to $$ {2k'-(k'-1)-1\choose k'-1}=k'=\frac{k}2. $$ Hence in this subcase \begin{align*} \lim_{p\to\infty} \frac{w_{s+r}}{\lambda_1^r w_s} =\lim_{p\to\infty} \frac{w_{s+r}}{p^{(s+r)+(c+d-2)(\frac{s+r}2-1)-1}} \frac{p^{r(c+d)/2}}{\lambda_1^r} \frac{p^{s+(c+d-2)(\frac{s}2-1)-1}}{w_s} =\frac{s+r}2\cdot1\cdot\frac2{s} =\frac{s+r}s. \end{align*} There remains to construct a sequence \((G_p)_{p\geq 1}\) of connected bipartite graphs with equitable partition \(A_p\cup B_p\) that corresponds to this case. The simplest choice is to let, for each \(p\geq 1\), the vertex set \(A_p\) to consist of vertices \(\{a_0,\dots,a_{p^d-1}\}\cup\{a'_0,\dots,a'_{p^d-1}\}\) and the vertex set \(B_p\) to consist of vertices \(b\) and \(b'\) only. The subgraph induced by \(A_p\) should be \(p\)-regular, say by making the vertex \(a_i\) adjacent to vertices \(a'_i,\dots,a'_{i+p-1}\) for \(i=0,\dots,p^d-1\), where addition is done modulo \(p\), while the vertex \(b\) is adjacent to vertices in \(\{a_0,\dots,a_{p^d-1}\}\) and the vertex \(b'\) is adjacent to vertices in \(\{b_0,\dots,b_{p^d-1}\}\). \(A_p\cup B_p\) is then an equitable vertex partition with the quotient matrix \(\left[\begin{smallmatrix} p & 1 \\ p^d & 0 \end{smallmatrix}\right]\), as requested, thus proving part b) of Theorem 3.
Subcase 1(b): \(c=1\), hence \(d>1\). In this case, the highest exponent in (7) is \(2k'+(c+d-2)(k'-1)-1=2k'+(c+d-2)k'-d\), appearing in the first sum for \(l=k'-1\) and in the second sum for \(l=k'\). Hence the corresponding coefficient is equal to $$ {2k'-(k'-1)-1\choose k'-1} + {2k'-k'-1\choose k'-1} = k'+1 = \frac{k+2}2. $$ Then in this subcase \begin{align*} \lim_{p\to\infty} \frac{w_{s+r}}{\lambda_1^r w_s} =\lim_{p\to\infty} \frac{w_{s+r}}{p^{(s+r)+(c+d-2)(\frac{s+r}2-1)-1}} \frac{p^{r(c+d)/2}}{\lambda_1^r} \frac{p^{s+(c+d-2)(\frac{s}2-1)-1}}{w_s} =\frac{s+r+2}2\cdot1\cdot\frac2{s+2} =\frac{s+r+2}{s+2}. \end{align*} There remains to construct a sequence \((G_p)_{p\geq 1}\) of connected bipartite graphs with equitable partition \(A_p\cup B_p\) that corresponds to this case. The simplest choice is to let, for each \(p\geq 1\), the vertex set \(A_p\) to consist of vertices \(\{a_1,\dots,a_{p^d}\}\cup\{a'_1,\dots,a'_{p^d}\}\) and the vertex set \(B_p\) to consist of vertices \(\{b_1,\dots,b_p\}\cup\{b'_1,\dots,b'_{p}\}\). The subgraph induced by \(A_p\) should be \(p\)-regular, which can be done in the same way as in Subcase 1.a. The subgraph induced by the vertices in \(\{a_1,\dots,a_{p^d}\}\cup\{b_1,\dots,b_p\}\) should be isomorphic to the complete bipartite graph \(K_{p^d,p}\), as well as the subgraph induced by the vertices in \(\{a'_1,\dots,a'_{p^d}\}\cup\{b'_1,\dots,b'_p\}\). \(A_p\cup B_p\) is then an equitable vertex partition with the quotient matrix \(\left[\begin{smallmatrix} p & p \\ p^d & 0 \end{smallmatrix}\right]\), as requested, thus proving part c) of Theorem 3.

Subcase 1(c): \(d=0\), hence \(c>2\).
In this subcase, the highest exponent in (7) is \(2k'+(c+d-2)k'+1-2d\), appearing in the third sum for \(l=k'\). The corresponding coefficient is equal to $$ {2k'-k'-1\choose k'-2}=k'-1=\frac{k-2}2. $$ Hence \begin{align*} \lim_{p\to\infty} \frac{w_{s+r}}{\lambda_1^r w_s} =\lim_{p\to\infty} \frac{w_{s+r}}{p^{(s+r)+(c+d-2)\frac{s+r}2+1-2d}} \frac{p^{r(c+d)/2}}{\lambda_1^r} \frac{p^{s+(c+d-2)\frac{s}2+1-2d}}{w_s} =\frac{s+r-2}2\cdot1\cdot\frac2{s-2} =\frac{s+r-2}{s-2}. \end{align*} There remains to construct a sequence \((G_p)_{p\geq 1}\) of connected bipartite graphs with equitable partition \(A_p\cup B_p\) that corresponds to this case. The simplest choice is to let, for each \(p\geq 1\), \(A_p\) to be the vertex set of a complete bipartite graph \(K_{p,p}\), and then to attach \(q=p^c\) pendant vertices to each vertex of \(K_{p,p}\), with these \(p^c|A|\) pendant vertices forming the vertex set \(B_p\). \(A_p\cup B_p\) is then an equitable vertex partition with the quotient matrix \(\left[\begin{smallmatrix} p & p^c \\ 1 & 0 \end{smallmatrix}\right]\), as requested, thus proving part a) of Theorem 3.
Subcase 1(d): \(d=1\), hence \(c>1\).
In this subcase, the highest exponent in (7) is \(2k'+(c+d-2)k'-d=2k'+(c+d-2)k'+1-2d\), appearing in the second sum for \(l=k'\) and in the third sum also for \(l=k'\). The corresponding coefficient is thus equal to $$ {2k'-k'-1\choose k'-1} + {2k'-k'-1\choose k'-2} = k' = \frac k2. $$ Hence \begin{align*} \lim_{p\to\infty} \frac{w_{s+r}}{\lambda_1^r w_s} =\lim_{p\to\infty} \frac{w_{s+r}}{p^{(s+r)+(c+d-2)\frac{s+r}2+1-2d}} \frac{p^{r(c+d)/2}}{\lambda_1^r} \frac{p^{s+(c+d-2)\frac{s}2+1-2d}}{w_s} =\frac{s+r}2\cdot1\cdot\frac2{s} =\frac{s+r}{s}. \end{align*} Constructing a sequence of connected, bipartite graphs corresponding to this subcase would yield another proof of part b) of Theorem 3.
Subcase 1(e): \(c,d\geq 2\). In this subcase, the highest exponent in (7) is \(2k'+(c+d-2)k'-d\), appearing in the second sum for \(l=k'\). The corresponding coefficient is equal to $$ {2k'-k'-1\choose k'-1} = 1, $$ so $$ \lim_{p\to\infty} \frac{w_{s+r}}{\lambda_1^r w_s} =\lim_{p\to\infty} \frac{w_{s+r}}{p^{(s+r)+(c+d-2)\frac{s+r}2-d}} \frac{p^{r(c+d)/2}}{\lambda_1^r} \frac{p^{s+(c+d-2)\frac{s}2-d}}{w_s} =1\cdot1\cdot1=1, $$ which is not helpful for the Nikiforov's problem.
Case 2: \(c+d-2=0\), i.e., \((c,d)\in\{(0,2),(1,1),(2,0)\}\). In this case $$ \lim_{p\to\infty} \frac{\lambda_1}{p} = \frac{1+\sqrt5}2. $$ The highest exponents appearing in the three sums of (7) are $$ 2k'-1, 2k'-d, 2k'+1-2d, $$ respectively. To calculate the corresponding coefficients in this case we will rely on the following summation formula \cite[Formula 1.61]{Gould}: $$ \sum_{l=0}^{\left\lfloor\frac k2\right\rfloor} {k-l\choose l}2^k\left(\frac z4\right)^l =\frac{x^{k+1}-y^{k+1}}{x-y}, $$ where \(x=1+\sqrt{z+1}\) and \(y=1-\sqrt{z+1}\). In particular, for \(z=4\) we have \(x=1+\sqrt5\) and \(y=1-\sqrt5\), so $$ \sum_{l=0}^{\left\lfloor\frac k2\right\rfloor} {k-l\choose l} =\frac{(1+\sqrt5)^{k+1}-(1-\sqrt5)^{k+1}}{2^{k+1}\sqrt5} =\frac{\varphi^{k+1}-\psi^{k+1}}{\sqrt5} =F_{k+1}, $$ where \(\varphi=x/2=(1+\sqrt5)/2\), \(\psi=y/2=(1-\sqrt5)/2\) and \((F_n)_{n\geq 0}\) is the usual Fibonacci sequence. Subcase 2(a): \((c,d)=(0,2)\). The highest exponent is \(2k'-1\), appearing in the first sum for all values of \(l=0,\dots,k'-1\). Hence the corresponding coefficient is equal to $$ \sum_{l=0}^{k'-1} {2k'-1-l\choose l}= F_{k}, $$ so $$ \lim_{p\to\infty} \frac{w_{s+r}}{\lambda_1^r w_s} = \lim_{p\to\infty} \frac{w_{s+r}}{p^{s+r-1}} \frac{p^r}{\lambda_1^r} \frac{p^{s-1}}{w_s} = \frac{F_{s+r}}{\varphi^r F_s}. $$ Since \(s\) and \(r\) are even and \(-\varphi< \psi< 0\), we have that $$ F_{s+r}=\frac{\varphi^{s+r}-\psi^{s+r}}{\sqrt5} >\frac{\varphi^r(\varphi^s-\psi^s)}{\sqrt5}=\varphi^r F_s, $$ so the limit above is larger than 1, although it tends to 1 when \(s\) tends to infinity. There remains to construct a sequence \((G_p)_{p\geq 1}\) of connected bipartite graphs with equitable partition \(A_p\cup B_p\) that corresponds to this case. We can use the same construction as in Subcase 1(a), setting for each \(p\geq 1\) the vertex set \(A_p\) to consist of vertices \(\{a_0,\dots,a_{p^2-1}\}\cup\{a'_0,\dots,a'_{p^2-1}\}\) and the vertex set \(B_p\) to consist of vertices \(b\) and \(b'\) only. The subgraph induced by \(A_p\) should be \(p\)-regular, while the vertex \(b\) is adjacent to vertices in \(\{a_0,\dots,a_{p^2-1}\}\) and the vertex \(b'\) is adjacent to vertices in \(\{b_0,\dots,b_{p^2-1}\}\). \(A_p\cup B_p\) is then an equitable vertex partition with the quotient matrix \(\left[\begin{smallmatrix} p & 1 \\ p^2 & 0 \end{smallmatrix}\right]\), as requested, thus proving part d) of Theorem 3.
Subcase 2(b): \((c,d)=(1,1)\). The highest exponent is \(2k'-1\), appearing in all three sums for all values of \(l\). Hence the corresponding coefficient is equal to \begin{align*} & \sum_{l=0}^{k'-1} {2k'-1-l\choose l} + \sum_{l=1}^{k'} {2k'-1-l\choose l-1} + \sum_{l=2}^{k'} {2k'-1-l\choose l-2} \\ =& \sum_{l=0}^{k'-1} {2k'-1-l\choose l} +\sum_{l=0}^{k'-1} {2k'-2-l\choose l} % l=1+l' +\sum_{l'=0}^{k'-2} {2k'-3-l\choose l} \\ % l=2+l' =& F_{2k'} + F_{2k'-1} + F_{2k'-2} = 2F_{k}, \end{align*} due to \(F_{2k'-1}+F_{2k'-2}=F_{2k'}\). Thus $$ \lim_{p\to\infty} \frac{w_{s+r}}{\lambda_1^r w_s} = \lim_{p\to\infty} \frac{w_{s+r}}{p^{s+r-1}} \frac{p^r}{\lambda_1^r} \frac{p^{s-1}}{w_s} = \frac{F_{s+r}}{\varphi^r F_s}. $$ Constructing a sequence of connected bipartite graphs corresponding to this subcase would yield another proof of part d) of Theorem 3.
Subcase 2(c): \((c,d)=(2,0)\). The highest exponent is \(2k'+1\), appearing in the third sum for all values of \(l=2,\dots,k'\), with the corresponding coefficient equal to $$ \sum_{l=2}^{k'} {2k'-1-l\choose l-2} =\sum_{l'=0}^{k'-2} {2k'-3-l\choose l}=F_{k-2}. % l=2+l' $$ Hence $$ \lim_{p\to\infty} \frac{w_{s+r}}{\lambda_1^r w_s} = \lim_{p\to\infty} \frac{w_{s+r}}{p^{s+r+1}} \frac{p^r}{\lambda_1^r} \frac{p^{s+1}}{w_s} = \frac{F_{s+r-2}}{\varphi^r F_{s-2}}. $$ There remains to construct a sequence \((G_p)_{p\geq 1}\) of connected bipartite graphs with equitable partition \(A_p\cup B_p\) that corresponds to this case. We can use the same construction as in Subcase 1.c, setting for each \(p\geq 1\), \(A_p\) to be the vertex set of a complete bipartite graph \(K_{p,p}\), and then to attach \(q=p^2\) pendant vertices to each vertex of \(K_{p,p}\), with these \(p^2|A|\) pendant vertices forming the vertex set \(B_p\). \(A_p\cup B_p\) is then an equitable vertex partition with the quotient matrix \(\left[\begin{smallmatrix} p & p^2 \\ 1 & 0 \end{smallmatrix}\right]\), as requested, thus proving part ) of Theorem 3.
Case 3: \(c+d-2< 0\), i.e., \((c,d)\in\{(0,0),(0,1),(1,0)\}\). In this case $$ \lim_{p\to\infty} \frac{\lambda_1}{p} = 1. $$ The highest exponents appearing in the three sums of (7) are $$ k-1, k+c-2, k+2c-3, $$ respectively, obtained for the least feasible values of \(l\). We can now distinguish the following subcases. Subcase 3(a): \(c=0\). The highest exponent is \(k-1\), appearing in the first sum for \(l=0\). The corresponding coefficient is \({k-1\choose 0}=1\), so $$ \lim_{p\to\infty} \frac{w_{s+r}}{\lambda_1^r w_s} = \lim_{p\to\infty} \frac{w_{s+r}}{p^{s+r-1}} \frac{p^r}{\lambda_1^r} \frac{p^{s-1}}{w_s} = 1\cdot1\cdot1=1, $$ which is not helpful for the Nikiforov's problem.
Subcase 3(b): \(c=1\), hence \(d=0\). The highest exponent is \(k-1\), appearing in the first sum for \(l=0\), the second sum for \(l=1\) and the third sum for \(l=2\), so the corresponding coefficient is \({k-1\choose 0}+{k-2\choose 0}+{k-3\choose 0}=3\). Hence $$ \lim_{p\to\infty} \frac{w_{s+r}}{\lambda_1^r w_s} = \lim_{p\to\infty} \frac{w_{s+r}}{p^{s+r-1}} \frac{p^r}{\lambda_1^r} \frac{p^{s-1}}{w_s} = 3\cdot1\cdot\frac13=1, $$ which again does not help with the Nikiforov's problem.
This closes the discussion of cases when \(q\) and \(r\) are various powers of \(p\). As a result, we see that it yields five different limit values that are larger than 1 for the Nikiforov's ratio \(\frac{w_{s+r}}{\lambda_1^r w_s}\) for even \(s\) and even \(r\). For each of these limit values, one of many possible sequences of connected bipartite graphs that achieves the limit has been constructed.

Acknowledgments

LF was supported by NSFC (No. 11671402, 11871479), Hunan Provincial Natural Science Foundation (2016JJ2138, 2018JJ2479) and Mathematics and Interdisciplinary Sciences Project of CSU. DS was partly supported by Grant ON174033 of the Serbian Ministry of Education, Science and Technological Development.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

We consider only simple and connected graphs. In a graph \( G \), the eccentricity [1, 2, 3] of a vertex \( v \) is one of the first distance-based, invariants. The \textit{eccentricity} of a vertex \(v\) in a connected graph \(G\) is the distance function as \begin{equation} ecc_{G}(v)= \max\limits_{u \in V(G)} d(u,v) . \end{equation} Total eccentricity of a graph \(G\) is the sum of all vertex eccentricities \begin{equation} Ecc(G)=\sum\limits_{z \in V(G)} ecc_{G}(z). \end{equation}

Fathalikhani et al. [4] computed total eccentricity of some graph operations, and found bound for that of tensor product. Eccentricity, center and radius computations on the cover graphs of distributive lattices were presented by Suzuki and McDermind [5]. Extremal trees, unicyclic, bicyclic graphs and extremal conjugated trees with respect to total eccentricity index were presented by Farooq et al. in [6].

If a graph \(G\) contains a vertex \(x\), such that \(G-x\) is a tree, then it is called apex tree. The vertex \( x \), is called an apex vertex of \(G\). A tree is always an apex tree, therefore a non-trivial apex tree is an apex tree which itself is not a tree. For any natural number \(k\ge1\), a graph \(G\) is called a \(k\)-apex tree if there exists a set \(X\) of \(k\) elements, and subset of \(V(G)\) such that \(G-X\) is a tree, while for any \(Y\subseteq V(G)\) with \(|Y|< k\), \(G-Y\) is not a tree. Any vertex from the set \(X\) is called a \(k\)-apex vertex. \( T_k(n) \) denotes the set of all \( k \)-apex trees of order \( n \), for \(k\ge1\) and \(n\ge3\). Sharp upper bound for the Randic (connectivity) index of \(k\)-apex trees for \(k \ge 2\) were discovered by Akhter et al. [7]. Extremal first reformulated Zagreb index of \(k\)-apex trees was discovered by Akhter et al. in [8]. In this paper we have found minimal total eccentricity of \(k\)-apex trees and corresponding graphs. The join of two vertex-disjoint graphs \(G\) and \(H\) is the graph \(G+H\) with \(V(G+ H) =V(G) \cup V(H)\) and the edges of \(G+H\) are all edges of graphs \(G\) and \(H\) and the edges obtained by joining each vertex of \(G\) with each vertex of \(H\). Let \(G \in T_k(n)\) and \(X\) be the set of \( k \)-apex vertices, by the graph \(X_G\) we mean the subgraph of \(G\) whose vertex set is \(X\). The complete graph with vertex set \(X\) will be denoted by \(KX_G\). The tree obtained from \(G\) by deleting apex vertices will be denoted by \(T_G\). We shall denote the star graph whose vertex set is \(V(G)-X\) by \(S_G\). If \(G \in T_k(n)\), then \(X_G+T_G\) and \(KX_G+S_G\) are also \(k\)-apex trees. A tree of \(n\) vertices will be denoted by \(T_n\). We shall denote the eccentricity of a vertex \(v\) in a graph \(G\) by \(ecc_G(v)\).

2. Minimal total eccentricity of \(k\)-apex trees

The following is a fundamental lemma, and is used to obtain many useful results.

Lemma 1. If \(u,v \in V(G)\) are not adjacent, then \(Ecc(G+uv) \le Ecc(G) .\)

The proof of above lemma is obvious.

Lemma 2. If \(G \in T_k(n)\), \(k \ge 1\) and \(n-k \ge 3\) then \(Ecc(G) \ge Ecc(KX_G+S_G) .\)

Proof. As \(V(G) = V(X_G+T_G) \) and \(G\) is a subgraph of \(X_G+T_G\) so by Lemma 1

As graph \(KX_G+T_G\) is either same as \(X_G+T_G\) or it is obtained by adding edges in \(X_G+T_G\) therefore by Lemma 1 If \(T_G\) is not a star then for every vertex \(v \in V(G)-X\), \(ecc_{KX_G+T_G}(v) \ge 2\) and if \(T_G=S_G\), then \(ecc_{KX_G+S_G}(v) \le 2\), therefore Combining inequalities (1), (2) and (3) we have the result. $$Ecc(G) \ge Ecc(KX_G+S_G) .$$

The following lemma gives a relation between eccentricities of join of graphs and eccentricities of graphs.

Lemma 3. If \(K_n\) and \(S_m\) are vertex disjoint graphs for \(n \ge 2\) and \(m \ge 2\), then \( Ecc(K_n+S_m)=Ecc(K_n)+Ecc(S_m) .\)

Proof. For any vertex \(v \in V(K_n)\), we have \( ecc_{K_n+S_m}(v)=ecc_{K_n}(v)=1\) and for any vertex \(u \in V(S_m)\), we have \( ecc_{K_n+S_m}(u)= ecc(S_m).\) Thus $$ Ecc(K_n+S_m)= \sum\limits_{v \in V(K_n+S_m)} ecc_{K_n+S_m}(v) .$$ As graphs \(K_n\) and \(S_m\) are vertex disjoint, therefore $$ Ecc(K_n+S_m)= \sum\limits_{v \in V(K_n)} ecc_{K_n+S_m}(v) + \sum\limits_{v \in V(S_m} ecc_{K_n+S_m} (v). $$ Since for any vertex \(v \in K_n\), \(ecc_{K_n+Sm}(v)=ecc_{K_n}(v)\) and for any vertex \(u \in (S_m)\), \(ecc_{K_n+S_m}(u)=ecc_{S_m}(u)\) therefore $$ Ecc(K_n+S_m)= \sum\limits_{v \in V(K_n)} ecc_{K_n}(v) + \sum\limits_{v \in V(S_m)} ecc_{S_m} (v) $$ and hence $$ Ecc(K_n+S_m) = Ecc(K_n) + Ecc(S_m) .$$

Theorem 1. If \(G \in T_k(n) \), \(k \ge 1\) and \(n-k \ge 3\), then $$Ecc(G) \ge 2n-k-1$$ and equality holds if there are \(k+1\) vertices of eccentricity \(1\) and all other vertices are of eccentricity \(2\).

Proof. For any graph \(G \in T_k(n)\) and for given conditions on \(k\) and \(n\), by Lemma 2, we have $$Ecc(G) \ge Ecc(KX_G+S_G).$$ As for each \(v \in X\), \(ecc_{KX_G+S_G} (v) = 1\), for one vertex in \(V(G)-X\), eccentricity is one and for all other \(n-k-1\) vertices in \(V(G)-X\), eccentricity is \(2\), therefore $$Ecc(G) \ge 2n-k-1 .$$

Theorem 2. If \(G \in T_k(n) \), \(k \ge 1\) and \(n-k = 2\), then $$Ecc(G) \ge n$$ and equality holds if all vertices are of eccentricity \(1\).

Proof. By Lemma 2, for any \(k\)-apex tree \(G\), we have \(Ecc(G) \ge Ecc(KX_G+S_G)\). In this case star will be of order \(2\) and therefore for every vertex \(v \in KX_G+S_G\), \(ecc(v)=1\) and hence $$Ecc(G) \ge n .$$

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

Let \(G = (V,E)\) be a graph. We denote the number of vertices of \(G\) by \(n\) and the number of edges by \(m\), i.e., \(|V(G)| = n\) and \(|E(G)| = m\). The degree of a vertex \(v\), denoted by \(d_G(v)\) is the number of edges incident to \(v\). For undefined terminologies, we refer the reader to [1]. A graph invariant is any function on a graph that does not depend on a labeling of its vertices and are called topological indices. Hundreds of different invariants have been employed to date (with unequal success) in QSAR/QSPR studies. Among them two are Zagreb indices. Due to their chemical relevance, they have been subject of numerous papers in literature [2, 3, 4, 5]. There two invariants are called the first Zagreb index and second Zagreb index [6, 7, 8, 9, 10, 11] and are defined as: $$ M_1(G) = \sum_{u \in V(G)} d_G(u)^2 \hspace{10mm} \mbox{and} \hspace{10mm} M_2(G) = \sum_{uv \in E(G)} d_G(u)\,d_G(v), $$ respectively. In fact, one can rewrite the first Zagreb index as: $$ M_1(G) = \sum_{uv \in E(G)} \big[ d_G(u) +d_G(v) \big] . $$ Noticing the contribution of nonadjacent vertex pairs when computing the weighted Winer polynomials of certain composite graphs, the authors in [6] defined the first Zagreb coindex and the second Zagreb coindex as: $$ \overline{M_1}(G) = \sum_{uv \notin E(G)} \big[ d_G(u) +d_G(v) \big] \hspace{10mm} \mbox{and} \hspace{10mm} \overline{M_2}(G) = \sum_{uv \notin E(G)} d_G(u)\,d_G(v), $$ respectively.

1.1. Transformation and total transformation graphs

Transformation graphs receives information from the original graph and converts source information into a new structure. If it is possible to figure out the given graph from the transformed graph in polynomial time, such operation may be used to survey miscellaneous structural properties of the original graph considering the transformation graphs. Therefore it fosters the research of transformation graphs and their structural properties [12].

Sampathkumar [13] introduced the concepts of semitotal-point graph and semitotal-line graph which are stated as follows:

Let \(G=(V,E)\) be a graph. The semitotal-line graph \(T_2(G)\) is a graph with \(V(T_2(G)) = V(G) \cup E(G)\) and any two vertices \(u,v \in T_2(G)\) are adjacent if and only if

1. \(u\) and \(v\) are adjacent edges in \(G\), and
2. one is a vertex of \(G\) and other is an edge of \(G\) incident with it.

Note that the definition of semitotal-line graph and middle graphs [14] are identical. These two concepts have been introduced in the same year.

The semitotal-point graph \(T_1(G)\) is a graph with \(V(T_1(G)) = V(G) \cup E(G)\) and any two vertices \(u,v \in T_1(G)\) are adjacent if and only if

1. \(u\) and \(v\) are adjacent vertices in \(G\) and
2. one is a vertex of \(G\) and other is an edge of \(G\) incident with it.

The total graph \(T(G)\) of a graph \(G\) is the graph whose vertex set is \(V(G)\cup E(G)\), and in which two vertices are adjacent if and only if they are adjacent or incident in \(G\) [15].

Let \(G=(V,E)\) be a graph and \(x,y,z\) be three variables taking values \(+\) or \(-\). The total transformation graph \(G^{xyz}\) is a graph having \(V(G) \cup E(G)\) as a vertex set, and for \(\alpha, \beta \in V(G) \cup E(G)\), \(\alpha\) and \(\beta\) are adjacent in \(G^{xyz}\) if and only if

1. \(\alpha, \beta \in V(G)\), \(\alpha\), \(\beta\) are adjacent in \(G\) if \(x = +\) and \(\alpha\) and \(\beta\) are not adjacent in \(G\) if \(x = -\).
2. \(\alpha, \beta \in E(G)\), \(\alpha\), \(\beta\) are adjacent in \(G\) if \(y = +\) and \(\alpha\) and \(\beta\) are not adjacent in \(G\) if \(y = -\).
3. \(\alpha \in V(G)\) and \(\beta \in E(G)\), \(\alpha\), \(\beta\) are incident in \(G\) if \(z = +\) and \(\alpha\) and \(\beta\) are not incident in \(G\) if \(z = -\).

Since there are eight distinct 3-permutations of \(\{+, -\}\), we obtain eight graphical transformations of \(G\). It is interesting to see that \(G^{+++}\) is exactly the total graph \(T(G)\) of \(G\) and \(G^{---}\) is the complement of \(T(G)\). Also for a given graph \(G\), \(G^{++-}\) and \(G^{--+}\), \(G^{+-+}\) and \(G^{-+-}\), \(G^{-++}\) and \(G^{+--}\) are the other three pairs of complementary graphs.

The basic properties of these total transformation can be seen in [12,16,17, 18].

In this paper, we obtained some new properties of Zagreb indices. We mainly give explicit formulae for the second Zagreb index of semitotal-point graph, semitotal-line graph and eight total transformation graphs.

2. Results

We begin with the following straightforward observations.

Observation 1. For a positive integer \(k\), we have \(\xi_{k}(G) = \sum_{v \in V(G)} (d_G(v))^{k}\). One can see that \(\xi_1(G)\) is just the number of edges in \(G\), and \(\xi_2(G)\) is the first Zagreb index \(M_1(G)\).

Observation 2. For any nonempty graph \(G\), we have $$ \sum_{uv \in E(G)} \big[d_G(u)^2 + d_G(v)^2 \big] = \sum_{w \in V(G)} d_G(w)^3 = \xi_3(G). $$

Theorem 1.[6] Let \(G\) be any nontrivial graph of order \(n\) and size \(m\). Then $$ \overline{M_2}(G) =2m^{2} - M_{2}(G) - \frac{1}{2}M_{1}(G) . $$

In the next theorem, the explicit formulas of first Zagreb index are given [19].

Theorem 2.[19] Let \(G\) be any nontrivial graph of order \(n\) and size \(m\). Then \begin{eqnarray*} M_1(T_1(G)) &=& M_1(G)+2M_2(G)+\xi_3(G) ,\\ M_1(T_2(G)) &=& 4(m + M_1(G)),\\ M_1(G^{+++}) &=& 4M_1(G)+ 2M_2(G) + \xi_3(G),\\ M_1(G^{---}) &=& (m+n)[(m+n)^2+6m -2n +1]+ 8m + 2(m+n-3)M_1(G)+ 2M_2(G) + \xi_3(G),\\ M_1(G^{++-}) &=& mn(m+n-8)+ 16m + 2(n-4)M_1(G) + 2M_2(G) + \xi_3(G), \\ M_1(G^{--+}) &=& n(n-1)^{2} +m(m+3)^{2}-2(m +3)M_1(G) + 2M_2(G) + \xi_3(G),\\ M_1(G^{+-+}) &=& m(m+3)^2 -2(m + 1)M_1(G) + 2M_2(G) + \xi_3(G). \end{eqnarray*} \begin{eqnarray*} M_1(G^{-+-}) &=& (m+n)[n(m+n) -2(n+4m)] + m[(n-4)^2+9]+ 2(n-2)M_1(G) + 2M_2(G) + \xi_3(G) ,\\ M_1(G^{-++}) &=& n(n-1)^2 + 2M_2(G) + \xi_3(G),\\ M_1(G^{+--}) &=& m[(nm +1) +(m+n)(m +n -2)]- 2(m+n-1)M_1(G) + 2M_2(G) + \xi_3(G). \end{eqnarray*}

In the following Lemma, the order and size of transformation graphs are given [19].

Lemma 1. [19] Let \(G\) be a nontrivial graph of order \(n\) and size \(m\). Then \begin{eqnarray*} |V(T_{1}(G))| &=& m+n , |E(T_1(G))| =m= \frac{1}{2}[2m + M_1(G)]. \\ |V(T_{2}(G))| &=& m+n , |E(T_2(G))| =3m.\\ |V(G^{+++})| &=& m +n, |E(G^{+++})| = m = \frac{1}{2}[4m + M_1(G)].\\ |V(G^{---})| &=& m +n, |E(G^{---})| = m =\frac{1}{2}[(m+n-1)(m+n)-4m- M_1(G)]. \\ |V(G^{++-})| &=& m +n, |E(G^{++-})| =m =\frac{1}{2}[2m(n-2)+ M_1(G)].\\ |V(G^{--+})| &=& m +n, |E(G^{--+})| =m =\frac{1}{2}[m(m+n) + n(n+1)- M_1(G)].\\ |V(G^{+-+})| &=& m +n, |E(G^{+-+})| =m =\frac{1}{2}[m(m+7) - M_1(G)].\\ |V(G^{-+-})| &=& m +n, |E(G^{-+-})| =m =\frac{1}{2}[n(m+n-1) + m(n-8) + M_1(G)].\\ |V(G^{-++})| &=& m +n, |E(G^{-++})| =m =\frac{1}{2}[n(n-1) + M_1(G)].\\ |V(G^{+--})| &=& m +n, |E(G^{+--})| =m =\frac{1}{2}[m(m +2n -1) - M_1(G)]. \end{eqnarray*}

In the next Lemma, the edge partition of transformation graphs in terms of \(E(G)\) and \(E(L(G))\) are given.

Lemma 2. Let \(G\) be a nontrivial graph of order \(n\) and size \(m\). Then

1. \(E(G^{+++}) = E(G) \cup E(L(G)) \cup 2E(G) ,\)
2. \(E(G^{---}) = E(\overline{G}) \cup E(\overline{L(G)}) \cup (n-2) \cdot E(G),\)
3. \(E(G^{++-}) = E(G) \cup E(L(G)) \cup (n-2) \cdot E(G) ,\)
4. \(E(G^{--+}) = E(\overline{G}) \cup E(\overline{L(G)}) \cup 2E(G) ,\)
5. \(E(G^{+-+}) = E(G)\cup E(\overline{L(G)}) \cup 2E(G) ,\)
6. \(E(G^{-+-}) = E(\overline{G}) \cup E(L(G)) \cup (n-2) \cdot E(G) ,\)
7. \(E(G^{-++}) = E(\overline{G}) \cup E(L(G)) \cup 2E(G) ,\)
8. \(E(G^{+--}) = E(G) \cup E(\overline{L(G)}) \cup (n-2) \cdot E(G) .\)

Theorem 3. Let \(G\) be a nontrivial graph of order \(n\) and size \(m\). Then \(M_2(T_1(G)) = 4 (4m + M_{1}(G))\).

Proof. Note that for \(u \in V(T_1(G)) \cap V(G) \), \(d_{_{T_1(G)}}(u) =2d_G(u)\) and for \(u \in V(T_1(G) \cap E(G)\), \(d_{_{T_1(G)}}(u) =2\). Therefore by Lemma 2, \begin{eqnarray*} M_2(T_1(G)) &=& \sum_{u \in V(T_1(G))} (2d(u))^2 + 2\sum_{u \in V(T_1(G)) \cap V(G)} 4d(u) = 4 (4m + M_{1}(G)). \end{eqnarray*} as desired.

Theorem 4. Let \(G\) be a nontrivial graph of order \(n\) and size \(m\). Then \(M_2(T_2(G)) = 2M_{1}(G) +4M_{2}(G) + \xi_{3}(G).\)

Proof. Suppose \(e =uv\) is a vertex in \(T_2(G)\). It can be easily seen that \(d_{T_2(G)}(e) = d_G(u) + d_G(v)\) and if \(u \in V(T_2(G)) \cap V(G)\), then \(d_{T_2(G)}(u) =d_G(u)\). Therefore by Lemma 2, \begin{eqnarray*} M_2(T_2(G)) &=& \sum_{_{u \in V(T_2(G)) \cap V(G)}} (d(u) + d(v))^2 + 2d(u)\sum_{_{u \in V(T_2(G) \cap E(G))}} (d(u) + d(v))= 2M_{1}(G) +4M_{2}(G) + \xi_{3}(G) . \end{eqnarray*} This completes the proof.

Theorem 5. Let \(G\) be a nontrivial graph of order \(n\) and size \(m\). Then \( M_2(G^{+++}) = 8M_1(G)+ 6M_2(G) + \xi_3(G) . \)

Proof. Note that \(E(G^{+++}) = E(G) \cup E(L(G) \cup 2E(G)\) and for \(u \in V(G^{+++}) \cap V(G)\), \(d_{G^{+++}}(u) = 2d_G(u)\) and for \(u \in V(G^{+++}) \cap E(G)\), \(d_{G^{+++}}(u) = d_G(u)+d_G(v)\). Therefore by Lemma 2, \begin{eqnarray*} M_2(G^{+++}) &=& \sum \limits_{u \in V(G^{+++}) \cap V(G)} 2d(u))^2 + \sum\limits_{u \in E(G^{+++}) \cap V(G)} (d(u)+d(v))^2 + 4d(u)\sum \limits_{u, v \in E(G) } (d(u)+d(v)) \\ &=& 4M_{1}(G) + 2M_{2}(G) + \xi_{3}(G) +4M_{1}(G) + 4M_{2}(G) \\ &=& 8M_1(G)+ 6M_2(G) + \xi_3(G). \end{eqnarray*} as asserted.

Theorem 6. Let \(G\) be a nontrivial graph of order \(n\) and size \(m\). Then \( M_2(G^{---}) = mn(m +n)^{2} -2m(m+n)(3n+4) +m(n +16)- 3(m+n-1)M_{1}(G)+ 2(n-1)M_{2}(G) +\xi_{3}(G). \)

Proof. Note that \(E(G^{---}) = E(\overline{G}) \cup E(\overline{L(G)}) \cup (n-2) \cdot E(G)\), and for \(u \in V(G^{---}) \cap V(G)\), \(d_{G^{---}}(u) = m + n -1 - 2d_G(u)\) and for \(u \in V(G^{---}) \cap E(G)\), \(d_{G^{---}}(u) = m + n -1 - (d_G(u) +d_G(v))\). Therefore by Lemma 2, \begin{eqnarray*} M_2(G^{---}) &=& \sum_{u \in V(G^{---}) \cap V(G)} (m + n -(1 + 2d_G(u)))^2 + \sum_{u \in V(G^{---})\cap E(G)} (m + n -(1 + (d_G(u) +d_G(v))))^2 \\ &&+ (n-2)(m + n -(1 + 2d_G(u)))\sum_{u ,v \in E(G)} (m + n -(1 + (d_G(u) +d_G(v)))) \\ &=& \big[m(m+n)^{2} +2m(m+n) +17m + 4M_{1}(G) \big] + \big[m(m+n)^{2} +2m(m+n) + m \\&&-2(m +n -1) M_{1}(G) +2M_{2}(G) + \xi_{3}(G) \big]+ (n-2) \big[m(m+n)^{2} -6m(m+n) +5m \\&&- (m +n +3) M_{1}(G) +2M_{2}(G) \big]. \end{eqnarray*} This completes the proof.

In fully analogous manner we arrive also at:

Theorem 7. Let \(G\) be a nontrivial graph of order \(n\) and size \(m\). Then \begin{eqnarray*} M_2(G^{++-}) &=& m^{3} +m(n-4) \big[ n(m+1) -2(m+2) \big] + \big[n(m+2) -2(m+4) \big] M_{1}(G)+ 2M_{2}(G) + \xi_{3}(G), \\ M_2(G^{--+}) &=& m(m+3)^{2} +m(n-1)[2m+n+5] -2(m+n+2) M_{1}(G)+ 2M_{2}(G) + \xi_{3}, \\ M_2(G^{+-+}) &=& m(m+3)(m +11)-2(m +3)M_{1}(G)- 2M_2(G) + \xi_3(G), \\ M_2(G^{-+-}) &=& m\big[(m +n)^{2} + (n-4)^{2} \big] +m(n-2)(n-4)(n-4m) - 10m(m+n)\\&&+ 9m \big[ (n-2)^{2} +mn -11 \big] M_{1}(G) + \xi_{3}(G), \\ M_2(G^{-++}) &=& m(n-1)^2 + 2(n-1)M_{1}(G)+ 2M_2(G) + \xi_3(G), \end{eqnarray*} \begin{eqnarray*} M_2(G^{+--}) &=& m(m^{2}+1) + m(m+n)(m +n -2) +m^{2}(n-2)(m +n+1)- (n(m +2)-2)M_{1}(G) \\&&+2M_{2}(G) + \xi_{3}(G). \end{eqnarray*}

Applying Theorem 1, from the results of Theorems 3-7 and Lemma 1, we can deduce expressions for the second Zagreb coindex of the transformation graphs and total transformation graphs \(G^{xyz}\). These are collected in the following:

Corollary 8. Let \(G\) be a graph of order \(n\) and size \(m.\) Then \begin{eqnarray*} \overline{M_{2}}(T_{1}(G)) &=& 2m^{2} -16m +(M_{1}(G))^{2} +4 (m-1)M_{1}(G) - \frac{1}{2} \big[M_{1}(G) +2M_{2}(G) + \xi_{3}(G) \big], \\ \overline{M_{2}}(T_{2}(G)) &=& 18m^{2} -2m -4M_{1}(G) -4M_{2}(G) - \xi_{3}(G),\\ \overline{M_{2}}(G^{+++}) &=& \frac{1}{2} \big[4m^{2} +(M_{1}(G))^{2} +4(2m-1)M_{1}(G) -2M_{2}(G) - \xi_{3}(G) \big]- 8M_{1}(G) -6M_{2}(G) - \xi_{3}(G),\\ \overline{M_{2}}(G^{---}) &=& \frac{1}{2} \big[(m+n-1)(m+n) -(4m+M_{1}(G)) \big]^{2}- mn(m +n)^{2} + 2m(m+n)(3n+4) -m(n +16) \\ &&+ 3(m+n-1)M_{1}(G)- 2(n-1)M_{2}(G) -\xi_{3}(G) - \frac{1}{2} \big[(m+n)[(m+n)^2+6m -2n +1]\\&&77+ 8m + 2(m+n-3)M_1(G)+ 2M_2(G) + \xi_3(G) \big],\\ \overline{M_{2}}(G^{++-}) &=& \frac{1}{2} \big[(2m(n-2)) +M_{1}(G)^{2} -mn(m+n-8) -16m - 2(n-4)M_{1}(G) -2M_{2}(G) - \xi_{3}(G)\big] \\&&- \big[ m^{3} +m(n-4) \big[ n(m+1) -2(m+2) \big] + \big[n(m+2) -2(m+4) \big] M_{1}(G) + 2M_{2}(G) + \xi_{3} \big], \\ \overline{M_{2}}(G^{--+}) &=& \frac{1}{2} \bigg[(m(m+n) +n(n+1) -M_{1}(G))^{2} - \big[ n(n-1)^{2} +m(m+3)^{2}-2(m +3)M_1(G) + 2M_2(G) \\&&+ \xi_3(G) \big] \bigg] -\big[ m(m+3)^{2} +m(n-1)[2m+n+5] -2(m+n+2) M_{1}(G) + 2M_{2}(G) + \xi_{3}\big], \\ \overline{M_{2}}(G^{+-+}) &=& \frac{1}{2} \bigg[[m(m+7) - M_1(G)]^{2} - \big[ m(m+3)^2 -2(m + 1)M_1(G) + 2M_2(G) + \xi_3(G) \big] \bigg]\\ &&- \big[ m(m+3)(m +11)-2(m +3)M_{1}(G)- 2M_2(G) + \xi_3(G\big],\\ \overline{M_{2}}(G^{-+-}) &=& \frac{1}{2} \bigg[[n(m+n-1) + m(n-8) + M_1(G)]^{2} - \big[(m+n)[n(m+n) -2(n+4m)]\\ &&- m[(n-4)^2+9] + 2(n-2)M_1(G) + 2M_2(G) + \xi_3(G) \big] \bigg] - \bigg[m\big[(m +n)^{2} + (n-4)^{2} \big] \\ &&-m(n-2)(n-4)(n-4m) - 10m(m+n) + 9m \big[ (n-2)^{2} +mn -11 \big] M_{1}(G) + \xi_{3}(G) \bigg], \end{eqnarray*} \begin{eqnarray*} \overline{M_{2}}(G^{-++}) &=& \frac{1}{2} \bigg[[n(n-1) + M_1(G)]^{2} - \big[n(n-1)^2 + 2M_2(G) + \xi_3(G) \big] \bigg] - \bigg[m(n-1)^2 \\ &&+ 2(n-1)M_{1}(G)+ 2M_2(G) + \xi_3(G) \bigg],\\ \overline{M_{2}}(G^{+--}) &=& \frac{1}{2} \bigg[[m(m +2n -1) - M_1(G)]^{2} - \big[m[(nm +1) +(m+n)(m +n -2)] \\ &&- 2(m+n-1)M_1(G) + 2M_2(G) + \xi_3(G) \big] \bigg] -\bigg[m(m^{2}+1) + m(m+n)(m +n -2) \\ &&- m^{2}(n-2)(m +n+1) - (n(m +2)-2)M_{1}(G) +2M_{2}(G) + \xi_{3}(G). \bigg].\end{eqnarray*}

Proof. The proof follows from the Theorems 1 and 3-7.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.