1. Introduction

Let \(G=(V(G), E(G))\) be a finite simple connected graph with \(n=|V(G)|\) vertices and \(m=|E(G)|\) edges. The number of edges in \(G\) that are incident to a vertex \(v\in V(G)\) is called its degree and denoted by \(d_{G}(v)\). A sequence of positive integers \(\pi=(d_{1}, d_{2}, \cdots, d_{n})\) is called the degree sequence of \(G\) if \(d_{i}=d_{G}(v_i)\), \(i=1, 2, \cdots, n\), holds for any \(v_i\in{V(G)}\). In particular, if the vertex degrees is non-decreasing, we use \(\pi=(d_{1}\leq d_{2}\leq\cdots\leq d_{n})\) to denote the degree sequence for simplicity. We denote by \(K_{n}\) and \(\overline{K}_{n}\) the complete graph with \(n\) vertices and its complement graph, respectively.

A cycle (resp. path) passing through each vertex of a graph is said to be a Hamilton cycle (resp. Hamilton path). We call the graph is Hamiltonian (resp. traceable) if there exists a Hamilton cycle (resp. Hamilton path) in it. For some integer \(k\), a connected graph \(G\) is said to be \(k\)-edge-hamiltonian if any collection of vertex-disjoint paths with at most \(k\) edges altogether belong to a hamiltonian cycle in \(G\). A graph is \(k\)-path-coverable if its vertex set can be covered by \(k\) or fewer vertex-disjoint paths, and we call a graph is Hamilton-connected if every two vertices in \(G\) are connected by a Hamiltonian path. For a graph \(G\), a subset \(I\) of \(V(G)\) is said to be an independent set of \(G\) if the induced subgraph \(G[I]\) is a graph with \(|I|\) isolated vertices. The independence number, denoted by \(\alpha(G)\), of \(G\) is the number of vertices in the largest independent set of \(G\), and we call a graph is \(k^{-}\)-independent if its independent number does not exceed to a positive real number \(k\). In what follows, we always omit the subscript \(G\) from the notation if there is no confuse from the context. For standard graph-theoretic notation and terminology the reader is referred to [1].

2. Motivation

In theoretical chemistry molecular structure descriptor, also called topological indices, are used to characterize the properties of the corresponding graph. Up to now, a series of topological indices, such as Wiener index [2] and Harary index [3,4], have been introduced and found a large amount of useful applications. Other nice related results and information could be found in [5,6,7] and therein.

The inverse degree, denoted by \(ID(G)\), of a graph \(G\) was defined as the sum of the inverses of the degrees of the vertices, formally

\begin{equation*} \begin{split} ID(G)=\sum_{u\in V}\frac{1}{d_{G}(u)}, \end{split} \end{equation*} which maybe firstly be investigated in the conjectures of computer program [8]. It was stated that Zhang et al., [9] gave out a counterexample of the Graffiti's conjecture, and obtained the best bound upper and lower bounds on \(ID(T)+\beta(T)\) for any tree \(T\), where \(\beta(T)\) denotes the matching number of \(T\). Two years later, Hu et al., [10] characterized the extremal graphs with respect to the inverse degree among all connected graphs of order \(n\) and with \(m\) edges. In 2008, Dankelmann et al., [11] proved that, if \(G\) is connected and of order \(n\), then the diameter of \(G\) is less than \((3ID(G)+2+o(1))\frac{log n}{log{log n}}\) which improves a bound by Erdös et al., [12]. About one year after, Dankelmann et al., [13] found a relation between the inverse degree and edge-connectivity of graph. In [14], Mukwembi presented a better bound on diameter by the inverse degree than those mentioned in the previous two papers. It is worth mentioning that Li et al., [15] improved the bound on diameter in terms of the inverse degree by Dankelmann et al., [11] for trees and unicyclic graphs. In 2013, Chen and Fujita [16] obtained a nice relation between diameter and the inverse degree of a graph, which settled a conjecture in [14]. In 2016, Xu and co-author determined some upper and lower bounds on the inverse degree for a connected graph in terms of other graph parameters, such as chromatic number, clique number, connectivity, number of cut edges and matching number [17]. We encourage the interested reader to consult [18,19,20,21] and the references therein for more details.

The problem of determining whether a graph keeps certain reasonable property is often difficult and meaningful in graph theory. It is reported in [22] that determining whether a graph is traceable or Hamiltonian is \(NP\)-complete. From then on, exploring such sufficient conditions for graphs attracts a vast number of mathematicians. For example, the authors in [23] studied the traceability of graphs by using a kind of distance-based topological index, the Harary index. In the same year, similar problem was also considered in [24] and a new sufficient condition was found for a graph to be traceable based on the Wiener index. Subsequently, these results mentioned previously were generalized by means of other techniques, we encourage readers to consult [25,26] for more details and information. To the best of our knowledge, there are absolutely few such conditions in terms of the well-known degree-based and distance-based topological indices.

Motivated by the results in [27], in the subsequent sections we attempt to explore sufficient conditions in terms of the inverse degree for graphs to be \(k\)-path-coverable, \(k\)-edge-hamiltonian, Hamilton-connected, traceable and \(k^{-}\)-independent, respectively.

3. \(k\)-path-coverable graphs

In this section, a sufficient condition for a graph to be \(k\)-path-coverable, graphs is presented. To do this, we need the following well-known theorem, which could be found in the book of Bondy and Lesniak, respectively.

Lemma 1. ([28,29]) Let \(\pi=(d_1\leq d_2\leq \cdots \leq d_n)\) be a degree sequence, and also let \(k \geq 1\). If \[d_{i+k}\leq i\Rightarrow d_{n-i}\geq n-i-k~\text{for}~1\leq i< \frac{n-k}{2},\] then \(\pi\) enforces \(k\)-path-coverable.

Now we shall state the main result:

Theorem 1. Let \(G\) be a connected graph of order \(n\geq 8\) and \(k \geq 1\). If \(ID(G)< Q_{1}(n,k)\), where \begin{equation*} Q_{1}(n,k)= \begin{cases} \frac{- k^2 - 3n^2 + 2n + 1}{2(n - 1)(k - n + 1)}& \text{if}~n-k-1~\text{is even};\\[3mm] \frac{- k^3 + (n - 2)k^2 + (- 3n^2 + 4n)k + 3n^3 - 2n^2 - 16n + 16}{2(n - 1)(k^2 - 2kn + 2k + n^2 - 2n)} & \text{if}~n-k-1~\text{is odd},\\ \end{cases} \end{equation*} then \(G\) is \(k\)-path-coverable. Moreover, \(ID(G)=Q_{1}(n,k)\) if and only if \(G\cong K_{\frac{n-k-1}{2}}+(K_1\cup \overline{\frac{n+k-1}{2}})\) if \(n-k-1\) is even; and \(G\cong K_{\frac{n-k-2}{2}}+(K_2\cup \overline{K_\frac{n+k-2}{2}})\) if \(n-k-1\) is odd.

Proof. Suppose that \(G\) is not \(k\)-path-coverable. In view of Lemma 1, there exists an integer \(i\) such that \(d_{i+k}\leq i\) and \(d_{n-i}\leq n-i-k-1\) for \(1\leq i\leq \frac{n-k-1}{2}\). Hence, we have \begin{equation*} \begin{split} ID(G)&=\sum_{i=1}^{n}\frac{1}{d_i}\\ &\geq \frac{i+k}{i}+ \frac{n-2i-k}{n-i-k-1}+ \frac{i}{n-1}. \end{split} \end{equation*} For simplicity, we define the following function on \([1, \frac{n-k-1}{2}]\): \begin{equation*} \begin{split} \mathcal{A}_1(x)=\frac{x+k}{x}+ \frac{n-2x-k}{n-x-k-1}+ \frac{x}{n-1}. \end{split} \end{equation*} It is routine to check that the derivative of \(\mathcal{A}_1(x)\) equals to \begin{equation*} \begin{split} \mathcal{A}_1'(x)=\frac{r_1(x)}{x^2(n - 1)(x + k - n + 1)^2}, \end{split} \end{equation*} where \begin{equation*} r_1(x)=x^4 + (2k - 2n + 2)x^3 + (k^2 + (2 - 2n)k + n - 1)x^2-k(n - 1)(k - n + 1)(2x + k - n + 1). \end{equation*} Similarly, the second derivative of \(\mathcal{A}_1''(x)\) is \begin{equation*} \begin{split} \mathcal{A}_1''(x)=\frac{\eta(x)}{x^3(x + k - n + 1)^3}\,, \end{split} \end{equation*} where \begin{equation*} \begin{split} \eta(x)&=(2n - 4)x^3 + (6k^2 + ( - 6n+ 6)k)x^2 + (6k^3 + ( - 12n+ 12)k^2 + (6n^2 - 12n + 6)k)x \\ & \;\;\;+ 2k^4 + ( - 6n- 6)k^3 + (6n^2 - 12n + 6)k^2 + (- 2n^3 + 6n^2 - 6n + 2)k. \end{split} \end{equation*} By simple calculations, we have \begin{equation*} \begin{split} \eta''(x)=(12n - 24)x + 12k^2 + (12 - 12n)k, \end{split} \end{equation*} and its unique root \(\eta_0=\frac{- k^2 + (n - 1)k}{n - 2}\) satisfies the following property:

Fact 1. \(1< \eta_0\leq \frac{n-k-1}{2}\) if \(k\in [1, \frac{n-2}{2}]\) and \(\eta_0\geq \frac{n-k-1}{2}\) if \(k\in [\frac{n-2}{2}, n-3]\).

In fact, it can be easily seen that \(\eta_0>1\). It remains to prove the last assertion. Let \(g(k)=2(n-2)(\eta_0-\frac{n-k-1}{2})=- 2k^2 + 3kn - 4k - n^2 + 3n - 2\), which has an root, \(\frac{n-2}{2}\) , in the interval \([1, n-3]\). Thus we get \(g(k)\leq \frac{n-k-1}{2}\) if \(k\in [1, \frac{n-2}{2}]\) and \(g(k)\geq \frac{n-k-1}{2}\) if \(k\in [\frac{n-2}{2}, n-3]\), implying the correction of Fact 1.

It is routine to check that \begin{equation*} \begin{cases} &\eta(\eta_0)=\frac{2k(k - n + 1)^3}{(n - 2)^2} (2k^2 + ( - 3n+ 6)k + n^2 - 4n + 4)\\[3mm] &\eta(\frac{n-k-1}{2})=\frac{(k - n + 1)^3(2k - n + 2)}{4}, \end{cases} \end{equation*} and we also can obtain two auxiliary properties for the function \(\eta(x):\)

Fact 2. \(\eta(\eta_0)\leq 0\) if \(k\in [1, \frac{n-2}{2}]\) and \(\eta(\eta_0)\geq 0\) if \(k\in [ \frac{n-2}{2}, n-3]\).

In fact, let \(g_1(k)=2k^2+(-3n+6)k+n^2-4n+4\). It is routine to check that \(g_1(k)\) has an root, \(\frac{n-2}{2}\), in the interval \([1, n-3]\), implying \(g_1(k)\leq 0\) if \(k\in [1, \frac{n-2}{2}]\) and \(g_1(k)\geq 0\) if \(k\in [ \frac{n-2}{2}, n-3]\). This completes the proof of Fact 2.

Fact 3. \(\eta(\frac{n-k-1}{2})\geq 0\) if \(k\in [1, \frac{n-2}{2}]\) and \(\eta(\frac{n-k-1}{2})\leq 0\) if \(k\in [ \frac{n-2}{2}, n-3]\).

In fact, we use \(g_2(k)\) to denote the right-side of \(\eta(\frac{n-k-1}{2})\). It is routine to check that \(g_2(k)\) has an root, \(\frac{n-2}{2}\), in the interval \([1, n-3]\). As desired.

In what follows, we will confirm that \(\mathcal{A}_1'(x)< 0\) in the whole interval \([1, \frac{n-k-1}{2}]\). It is sufficient to show the following three claims.

Claim 1. \(\mathcal{A}_1'(x)< 0\) for \(k \in [1,\frac{n-2}{2}]\) and \(x\in [1,\eta_0]\) .

Direct calculations that

\begin{equation*} \mathcal{A}_1'(\eta_0)=\frac{r_2(k)}{k(n - 1)(k - n + 1)^2(k - n + 2)^2}, \end{equation*} where \begin{equation*} \begin{split} r_2(k)&=k^5 + ( - 4n+6)k^4 + (6n^2 - 18n + 13)k^3 + (- 4n^3 + 18n^2 - 26n + 12)k^2\\ &\;\;\;+ (2n^4 - 13n^3 + 31n^2 - 32n + 12)k - n^5 + 9n^4 - 32n^3 + 56n^2 - 48n + 16. \end{split} \end{equation*} Note that the denominator of \(\mathcal{A}_1'(\eta_0)\) is non-negative, and the third derivative \(r_2^{(3)}(k)=60k^2+(-96n+ 144)k+36n^2-108n+78\) is a convex function in the interval \([1, \frac{n-2}{2}]\). Hence, \(r_2^{(3)}(k)\geq \min\{r_2^{(3)}(1), r_2^{(3)}(\frac{n-2}{2})\}=\min\{36n^2 - 204n + 282, 3n^2 - 6\}>0\), implying that \(r_2'(k)\) is a convex function. Similarly, we have \(r_2'(k)\geq \min\{r_2'(1),r_2'(\frac{n-2}{2})\}=\min\{2n^4-21n^3+85n^2-154n+104, \frac{(n-2)^2(13n^2-44n+32)}{16}\}>0\), which shows that \(r_2(k)\) is monotonously increasing in the accordingly interval. It is routine to check that \(r_{2}(k)\leq r_2(\frac{n-2}{2})=\frac{-(n - 2)^3(15n^2 - 48n + 32)}{32}< 0\), as desired we prove that \(\mathcal{A}_1'(\eta_0)< 0\).

It follows from Fact 1 that \(1\leq \eta_0 \leq \frac{n-k-1}{2}\), implying that \(\eta''(x)=(12n - 24)x + 12k^2 + (12 - 12n)k\leq 0 \). Hence, \(\eta'(x)\) is a decreasing function in the interval \([1, \eta_0]\). Consequently,

\begin{equation*} \eta'(x)\geq \eta'(\eta_0)=\frac{-6k(k - n + 1)^2(k - n + 2)}{n - 2}>0. \end{equation*} Hence, \(\eta(x)\) is monotonously increasing in the accordingly interval.

To accomplish the proof of Claim 1, it remains to prove that \(\mathcal{A}_1''(x)>0\), which is equivalent to that fact \(\eta(x)< 0.\) It follows from Fact 2, together with \(\eta(x)\) is increasing, that \(\eta(x)\leq \eta(\eta_0)< 0\). This implies that \(\mathcal{A}_1'(x)\) is an increasing function in the interval \([1, \eta_0]\). It then yields that \(\mathcal{A}_1'(x)\leq \mathcal{A}_1'(\eta_0)< 0,\) which completes the proof of Claim 1.

Hence, \(ID(G)\geq \mathcal{A}_1(x)\geq \mathcal{A}_1(\eta_0)\).

Claim 2. \(\mathcal{A}_1'(x)< 0\) for \(k \in [1,\frac{n-2}{2}]\) and \(x\in [\eta_0, \frac{n-k-1}{2}]\) .

We begin with such an optimization problem:

\begin{align*} \max \quad & \mathcal{A}_1'(x, k, n)\\ \mbox{s.t.}\quad &\eta_0\leq x\leq \frac{n-k-1}{2}\\[3mm] &1\leq k \leq \frac{n-2}{2}\\[3mm] &n\geq 8. \end{align*} Throughout this paper, we always assume that the order \(n\) of the graph does not exceed \(10^{10}\). It follows that the global optimal solution of \(\mathcal{A}_1'(x, k, n)\) is \(x=2, k=1, n=8\) through Lingo software after iterating 209 times, and the corresponding optimal value is \(-0.4196429\). This implies that \(\mathcal{A}_1'(x)< 0\).

It yields from Claim 2 that \(\mathcal{A}_1(x)\) is decreasing in the interval \([\eta_0, \frac{n-k-1}{2}]\). Therefore, \(ID(G)\geq \mathcal{A}_1(\frac{n-k-1}{2})\).

Claim 3. \(\mathcal{A}_1'(x)< 0\) for \(k \in [\frac{n-2}{2},n-3]\) and \(x\in [1, \frac{n-k-1}{2}]\).

It directly follows from Fact 1 that \(\eta''(x)\leq 0\), which implies that \(\eta'(x)\) is a decreasing function in the interval \(x\in [1, \frac{n-k-1}{2}]\). Hence, \(\eta'(x)\geq \eta'(\frac{n-k-1}{2})=\frac{3(n - 2)(k - n + 1)^2}{2}>0\), implying that \(\eta(x)\) is monotonously increasing in the accordingly interval. It follows from Fact 3 that \(\eta(x)\leq\eta(\frac{n-k-1}{2})\leq 0\), and therefore we have \(\mathcal{A}_1''(x)>0\). Hence, \(\mathcal{A}_1'(x)\) is an increasing function in \([1,\frac{n-k-1}{2}]\).

Direct calculations show that

\begin{equation*} \mathcal{A}_1'(\frac{n-k-1}{2})=\frac{r_3(k)}{(n - 1)(k - n + 1)^2}, \end{equation*} where \(r_3(k)=k^2 + (2 - 2n)k - 3n^2 + 10n - 7.\) It is obvious to find that \(r_3(k)\) is a convex function in the interval \([\frac{n-2}{2},n-3]\). Note that \(r_3(\frac{n-2}{2})=-\frac{15n^2}{4}+12n-8< 0\) and \(r_3(k)=-4n^2+12n-4< 0\), we get \(\mathcal{A}_1'(x)\leq\mathcal{A}_1'(\frac{n-k-1}{2})< 0\). Thus, we have \(ID(G)\geq \mathcal{A}_1(x)\geq \mathcal{A}_1(\frac{n-k-1}{2})\).

Combining Claims 1, 2 and 3, we get \(\mathcal{A}_1(x)\) is decreasing in the whole interval \([1, \frac{n-k-1}{2}]\), which achieves its minimum value at the right end-point of this interval.

Recall that \(x\) is an integer, we need consider the following two cases:

Case 1. \(n-k-1\) is even.

It immediately yields that \(\mathcal{A}_1(x)\geq \mathcal{A}_1(\frac{n-k-1}{2})\), and therefrore

\begin{equation*} \begin{aligned} ID(G) &\geq \frac{- k^2 - 3n^2 + 2n + 1}{2(n - 1)(k - n + 1)}\doteq \widehat{Q}_{1}(n,k), \end{aligned} \end{equation*} contradicting the hypothesis. Hence, the conclusion follows.

Furthermore, the condition in Theorem 1 cannot be dropped. If \(G\cong K_{\frac{n-k-1}{2}}+(K_1\cup \overline{K_{\frac{n+k-1}{2}}})\), then direct computations yields that \(ID(G)=\widehat{Q}_{1}(n,k)\). Conversely, let \(ID(G)=\widehat{Q}_{1}(n,k)\), then all inequalities in the proof should be equalities. Hence, \(i=\frac{n-k-1}{2}\) and therefore \(d_1=\cdots=d_{\frac{n+k-1}{2}}=\frac{n-k-1}{2}\), \(d_{\frac{n+k+1}{2}}=\frac{n-k-1}{2}\) and \(d_{\frac{n+k+3}{2}}=\cdots=d_n=n-1\). This implies that \(G\cong K_{\frac{n-k-1}{2}}+(K_1\cup \overline{K_{\frac{n+k-1}{2}}})\).

Case 2. \(n-k-1\) is odd.

According to previous analysis, we have \(\mathcal{A}_1(x)\geq \mathcal{A}_1(\frac{n-k-2}{2})\). It follows by simple computations that

\begin{equation*} \begin{aligned} ID(G) &\geq \frac{- k^3 + (n - 2)k^2 + (- 3n^2 + 4n)k + 3n^3 - 2n^2 - 16n + 16}{2(n - 1)(k^2 - 2kn + 2k + n^2 - 2n)}\doteq \widetilde{Q}_{1}(n,k), \end{aligned} \end{equation*} again a contradiction, and the conclusion follows.

Furthermore, the condition in Theorem 1 cannot be dropped. If \(G\cong K_{\frac{n-k-2}{2}}+(K_2\cup \overline{K_\frac{n+k-2}{2}})\), then direct computations yields that \(ID(G)=\widetilde{Q}_{1}(n,k)\). Conversely, let \(ID(G)=\widetilde{Q}_{1}(n,k)\), then all inequalities in the proof should be equalities. Hence, \(i=\frac{n-k-2}{2}\) and therefore \(d_1=\cdots=d_{\frac{n+k-2}{2}}=\frac{n-k-2}{2}\), \(d_{\frac{n+k}{2}}=d_{\frac{n+k+2}{2}}=\frac{n-k}{2}\) and \(d_{\frac{n+k+4}{2}}=\cdots=d_n=n-1\). This implies that \(G\cong K_{\frac{n-k-2}{2}}+(K_2\cup \overline{K_\frac{n+k-2}{2}})\).

4. \(k\)-edge-hamiltonian graphs

We begin by presenting an elementary result for \(k\)-edge-hamiltonian graphs.

Lemma 2. ([30]) Let \(\pi=(d_1\leq d_2\leq \cdots \leq d_n)\) be a degree sequence with \(0\leq k\leq n-3\). If \[d_{i-k}\leq i\Rightarrow d_{n-i}\geq n-i+k~\text{for}~k+1\leq i< \frac{n+k}{2},\] then \(\pi\) enforces \(k\)-edge-hamiltonian.

Let \(k_0, k_1, k_2\) be three non-negative real numbers in terms of \(n\):

\begin{equation*} \begin{split} \left\{ \begin{aligned} &k_0=\frac{- n^2+n + \sqrt{n(n^3 + 2n^2 - 15n + 16)}}{2n}\\ &k_1=\frac{n^2-4n+2-\sqrt{n^4-12n^3+32n^2-24n+4}}{2n}\\ &k_2=\frac{n^2-4n+2+\sqrt{n^4-12n^3+32n^2-24n+4}}{2n}. \end{aligned} \right. \end{split} \end{equation*} The main result is the following:

Theorem 2. Let \(G\) be a connected graph of order \(n\geq 9\) and \(0\leq k\leq n-3\). If \(ID(G)< Q_{2}(n,k) \) , where \begin{equation*} Q_{2}(n, k)= \begin{cases} \frac{- k^2+(n^2 - 2n - 1)k + 2n^2 - 5n + 2}{(k + 1)(n^2 - 3n + 2)}&\text{if}~k\in[k_1, k_2], n+k-1=2p\\[3mm] \frac{- k^2+(n^2 - 2n - 1)k + 2n^2 - 5n + 2}{(k + 1)(n^2 - 3n + 2)}&\text{if}~k\in[k_0, n-4], n+k-1=2p+1\\[3mm] \frac{k^3 + (n - 2)k^2 + (3n^2 - 4n)k + 3n^3 - 2n^2 - 16n + 16}{2(n - 1)(k^2 + 2kn - 2k + n^2 - 2n)}&\text{if}~k\in[0, k_0], n+k-1=2p+1\\[3mm] \frac{k^2+3n^2-2n-1}{2(n-1)(k+n-1)}&\text{if}~k\in[0, k_1]\cup [k_2, n-3], n+k-1=2p, \end{cases} \end{equation*} then \(G\) is \(k\)-edge-hamiltonian. Moreover, \(ID(G)=Q_{2}(n,k)\) if and only if \(G\cong K_{k+1}+(K_1 \cup K_{n-k-2} )\) if \(k\in[k_1, k_2]\) and \(n+k-1\) is even or \(k\in[k_0, n-4]\) and \(n+k-1\) is odd; \(G\cong K_{\frac{n+k-2}{2}}+(K_2 \cup \overline{K_\frac{n-k-2}{2}} )\) if \(k\in[0, k_0]\) and \(n+k-1\) is odd; and \(G\cong K_{\frac{n+k-1}{2}}+\overline{K_\frac{n-k+1}{2}}\) if \(k\in[0, k_1]\cup [k_2, n-3]\) and \(n+k-1\) is even.

Proof. Suppose that \(G\) is not \(k\)-edge-hamiltonian. By Lemma 2, we know that there exists integer \(i\) such that \(d_{i-k}\leq i\) and \(d_{n-i}\leq n-i+k-1\) for \(k+1\leq i\leq \frac{n+k-1}{2}\). Then we have \begin{equation*} \begin{split} ID(G)&=\sum_{i=1}^{n}\frac{1}{d_i}\\ &\geq \frac{i-k}{i}+\frac{n-2i+k}{n-i+k-1}+\frac{i}{n-1}. \end{split} \end{equation*} For simplicity, we define the following function on \([k+1, \frac{n+k-1}{2}]\): \begin{equation*} \begin{split} \mathcal{A}_2(x)=\frac{x-k}{x}+\frac{n-2x+k}{n-x+k-1}+\frac{x}{n-1}, \end{split} \end{equation*} and the corresponding second derivative is \begin{equation*} \begin{split} \mathcal{A}_2''(x)=\frac{\zeta(x)}{x^3(x - k - n + 1)^3}, \end{split} \end{equation*} where \begin{equation*} \begin{split} \zeta(x)=(2n - 4)x^3 + 6k(k + n - 1)x^2 - 6k(k + n - 1)^2x + 2k(k + n - 1)^3. \end{split} \end{equation*} Let \(z=k+n-1\), then \(\zeta(x)=(2n-4)x^3+2kz(3x^2-3zx+z^2)\), and consequently we have \(\zeta(x)\geq (2n-4)x^3+2kz(2\sqrt{3}-3)xz>0\). Hence, \(\mathcal{A}_2(x)\) is a concave function, since \(x^3(x - k - n + 1)^3< 0\) for \([k+1, \frac{n+k-1}{2}].\)

To accomplish the proof, in what follows we need consider whether \(n+k-1\) is odd or even.

Case 1. \(n+k-1\) is odd.

In this case, it is not difficult to find that \(k+1\leq x\leq \frac{n+k-2}{2}\) and \(0\leq k \leq n-4\). Hence, \(\mathcal{A}_2(x)\geq \min\{\mathcal{A}_2(k+1), \mathcal{A}_2(\frac{n+k-2}{2})\}\). Direct calculations yields that

\begin{equation*} \begin{cases} \mathcal{A}_2(k+1)=\frac{- k^2+(n^2 - 2n - 1)k + 2n^2 - 5n + 2}{(k + 1)(n^2 - 3n + 2)}\\[3mm] \mathcal{A}_2(\frac{n+k-2}{2})=\frac{k^3 + (n - 2)k^2 + (3n^2 - 4n)k + 3n^3 - 2n^2 - 16n + 16}{2(n - 1)(k^2 + 2kn - 2k + n^2 - 2n)}, \end{cases} \end{equation*} and consequently we get \begin{equation*} \begin{split} \mathcal{A}_2(k+1)-\mathcal{A}_2\left(\frac{n+k-2}{2}\right)&=\frac{\sigma(k)} {2(k + 1)(n^2 - 3n + 2)(k^2 + 2kn - 2k + n^2 - 2n)}, \end{split} \end{equation*} where \begin{equation*} \begin{split} \sigma(k)=&-nk^4 + (n^2 - 5n)k^3 + (n^3 - n^2 - 6n + 4)k^2\\ +& (- n^4 + 5n^3 - 24n + 24)k + n^4 - 10n^3 + 36n^2 - 56n + 32. \end{split} \end{equation*} It is routine to check that \(\sigma(k)\) has two distinct roots in the interval \([0, n-4]\), say \(k_0\) and \(k_0'\) respectively. Formally \begin{equation*} \begin{cases} k_0=\frac{- n^2+n + \sqrt{n(n^3 + 2n^2 - 15n + 16)}}{2n}\\[3mm] k_0'=n-4. \end{cases} \end{equation*} In the following, we shall confirm that \(0< k_0< n-5\). In fact, the left-side of the inequality always holds under our initial conditions. It is sufficient to verify the last part. Simple calculations show that \begin{equation*} \begin{split} 2n(k_0-(n-5))=- 3n^2+11n + \sqrt{n^4 + 2n^3 - 15n^2 + 16n} , \end{split} \end{equation*} which is non-positive since \(\left(\sqrt{n^4 + 2n^3 - 15n^2 + 16n}\right)^{2}-(3n^2 -11n)^2=- 8n^4 + 68n^3 - 136n^2 + 16n< 0\), as desired.

It then follows from direct calculations that

\begin{equation*} \begin{cases} \sigma(0)=n^4 - 10n^3 + 36n^2 - 56n + 32>0\\[3mm] \sigma(n-5)=- 6n^3 + 51n^2 - 102n + 12< 0. \end{cases} \end{equation*} Applying Rolle's Theorem for the function \(\sigma(k)\), we obtain that \(\sigma(k)\geq 0\) if \(k\in [0, k_0]\), and \(\sigma(k)\leq 0\) if \(k\in [k_0, n-4].\)

To continue to the proof, we need consider the following possibilities.

Case 1.1. \(k\in [0, k_0]\).

Considering that \(\sigma(k)\geq 0\) and applying the hypothesis, we obtain \(\mathcal{A}_2\left(\frac{n+k-2}{2}\right)\leq \mathcal{A}_2(k+1)\). It immediately yields that

\begin{equation*} \begin{split} ID(G)&\geq \mathcal{A}_2\left(\frac{n+k-2}{2}\right)\\ &=\frac{k^3 + (n - 2)k^2 + (3n^2 - 4n)k + 3n^3 - 2n^2 - 16n + 16}{2(n - 1)(k^2 + 2kn - 2k + n^2 - 2n)}\doteq \widetilde{Q}_{2}(n,k). \end{split} \end{equation*} Thus we obtain a contradiction, completing the proof.

Furthermore, the corresponding condition in Theorem 2 cannot be dropped. If \(G\cong K_{\frac{n+k-2}{2}}+(K_2 \cup \overline{K_\frac{n-k-2}{2}}) \), then directly computations yields that \(ID(G)=\widetilde{Q}_{2}(n,k)\). Conversely, let \(ID(G)=\widetilde{Q}_{2}(n,k)\), then all inequalities in the proof should be equalities. Hence, \(i=\frac{n+k-2}{2}\) and therefore \(d_1=\cdots=d_{\frac{n-k-2}{2}}=\frac{n+k-2}{2}\), \(d_{\frac{n-k}{2}}=d_{\frac{n-k+2}{2}}=\frac{n+k}{2}\) and \(d_{\frac{n-k+4}{2}}=\cdots=d_n=n-1\). This implies that \(G\cong K_{\frac{n+k-2}{2}}+(K_2 \cup \overline{K_\frac{n-k-2}{2}})\).

Case 1.2. \(k\in [k_0, n-4]\).

Note that \(\sigma(k)\leq0\), which implies that \(\mathcal{A}_2\left(\frac{n+k-2}{2}\right)\geq\mathcal{A}_2(k+1)\). It immediately yields that

\begin{equation*} \begin{split} ID(G)&\geq \mathcal{A}_2(k+1)\\ &=\frac{- k^2+(n^2 - 2n - 1)k + 2n^2 - 5n + 2}{(k + 1)(n^2 - 3n + 2)}\doteq \widehat{Q}_{2}(n,k), \end{split} \end{equation*} again a contradiction. Hence, \(G\) is \(k\)-edge-hamiltonian.

Furthermore, the corresponding condition in Theorem 2 cannot be dropped. If \(G\cong K_{k+1}+(K_1 \cup K_{n-k-2} )\), then directly computations yields that \(ID(G)=\widehat{Q}_{2}(n,k)\). Conversely, let \(ID(G)=\widehat{Q}_{2}(n,k)\), then all inequalities in the proof should be equalities. Hence, \(i=k+1\) and therefore \(d_1=k+1\), \(d_{2}=\cdots=d_{n-k-1}=n-2\) and \(d_{n-k}=\cdots=d_n=n-1\). This implies that \(G\cong K_{k+1}+(K_1 \cup K_{n-k-2} )\).

Case 2. \(n+k-1\) is even.

In this case, it is routine to check that \(1\leq x\leq\frac{n+k-1}{2}\) and \(k\in [0, n-3]\). Hence, \(\mathcal{A}_2(x)\geq \min\{\mathcal{A}_2(1), \mathcal{A}_2(\frac{n+k-1}{2})\}\). Direct calculations yields that

\begin{equation*} \mathcal{A}_2\left(\frac{n+k-1}{2}\right)=\frac{k^2 + 3n^2 - 2n - 1}{2(n - 1)(k + n - 1)}, \end{equation*} and consequently we have \begin{equation*} \begin{split} \mathcal{A}_2(k+1)-\mathcal{A}_2\left(\frac{n+k-1}{2}\right) =\frac{\varsigma(k)}{2(k + 1)(k + n - 1)(n^2 - 3n + 2)}, \end{split} \end{equation*} where \(\varsigma(k)=-nk^3 + (2n^2 - 7n + 2)k^2 + (- n^3 + 6n^2 - 11n + 4)k + n^3 - 6n^2 + 11n - 6\). It is routine to check that \(\varsigma(k)\) has three distinct roots, says \(k_1, k_2\) and \(k_3\), in the interval \([0, n-3]\). Formally \begin{equation*} \begin{cases} k_1=\frac{n^2 - 4n + 2 -\Delta}{2n}\\[3mm] k_2=\frac{n^2 - 4n + 2 +\Delta}{2n}\\[3mm] k_3=n-3, \end{cases} \end{equation*} where \(\Delta=\sqrt{n^4 - 12n^3 + 32n^2 - 24n + 4}\). It is not difficult to verify that \(0< k_1< k_2< n-4\), and \begin{equation*} \begin{cases} \varsigma(0)=n^3 - 6n^2 + 11n - 6>0\\[3mm] \varsigma\left(\frac{k_1+k_2}{2}\right)=\frac{- n^6 + 14n^5 - 54n^4 + 64n^3 + 12n^2 - 40n + 8}{8n^2}< 0\\[3mm] \varsigma(n-4)=n^2 - 5n + 10>0. \end{cases} \end{equation*} Again applying Rolle's Theorem for the function \(\varsigma(k)\), we obtain that \(\varsigma(k)\geq 0\) if \(k\in [0, k_1] \cup [k_2, n-3]\), and \(\varsigma(k)\leq 0\) if \(k\in[k_1, k_2].\)

To continue to the proof, we need consider the following possibilities.

Case 2.1. \(k\in [0, k_1] \cup [k_2, n-3]\).

Recall that \(\varsigma(k)\geq0\), then we have \(\mathcal{A}_2(k+1)-\mathcal{A}_2\left(\frac{n+k-1}{2}\right)\geq 0\). It yields that

\begin{equation*} \begin{split} ID(G)\geq \mathcal{A}_2\left(\frac{n+k-1}{2}\right)=\frac{k^2 + 3n^2 - 2n - 1}{2(n - 1)(k + n - 1)}\doteq \widehat{\widehat{Q}}_{2}(n,k), \end{split} \end{equation*} which contradicts with our assumption. Hence, \(G\) is \(k\)-edge-hamiltonian.

Furthermore, the corresponding condition in Theorem 2 cannot be dropped. If \(G\cong K_{\frac{n+k-1}{2}}+ \overline{K_\frac{n-k+1}{2}} \), then direct computations yields that \(ID(G)=\widehat{\widehat{Q}}_{2}(n,k)\). Conversely, let \(ID(G)=\widehat{\widehat{Q}}_{2}(n,k)\), then all inequalities in the proof should be equalities. Hence, \(i=\frac{n+k-1}{2}\) and therefore \(d_1=\cdots=d_{\frac{n-k-1}{2}}=\frac{n+k-1}{2}\), \(d_{\frac{n-k+1}{2}}=\frac{n+k-1}{2}\) and \(d_{\frac{n-k+3}{2}}=\cdots=d_n=n-1\). This implies that \(G\cong K_{\frac{n+k-1}{2}}+\overline{K_\frac{n-k+1}{2}} \).

Case 2.2. \(k\in[k_1,k_2]\).

Recall that \(\varsigma(k)\leq0\), then we have \(\mathcal{A}_2(k+1)-\mathcal{A}_2\left(\frac{n+k-1}{2}\right)\leq 0\). It immediately yields that

\begin{equation*} \begin{split} ID(G)\geq \mathcal{A}_2(k+1)=\frac{- k^2+(n^2 - 2n - 1)k + 2n^2 - 5n + 2}{(k + 1)(n^2 - 3n + 2)}\doteq \widetilde{\widetilde{Q}}_{2}(n,k), \end{split} \end{equation*} contradicting the hypothesis. The assertion is proved.

Furthermore, the corresponding condition in Theorem 2 cannot be dropped. If \(K_{k+1}+(K_1 \cup K_{n-k-2} )\), then simple calculations yield that \(ID(G)=\widetilde{\widetilde{Q}}_{2}(n,k)\). Conversely, let \(ID(G)=\widetilde{\widetilde{Q}}_{2}(n,k)\), then all inequalities in the proof should be equalities. Hence, \(i=k+1\) and therefore \(d_1=k+1\), \(d_{2}=\cdots=d_{n-k-1}=n-2\) and \(d_{n-k}=\cdots=d_n=n-1\). This implies that \(K_{k+1}+(K_1 \cup K_{n-k-2} )\).

5. Hamilton-connected graphs

Lemma 3.([31]) Let \(\pi=(d_1\leq d_2\leq \cdots \leq d_n)\) be a degree sequence with \(n\geq 3\). If \[d_{k-1}\leq k\Rightarrow d_{n-k}\geq n-k+1~\text{for}~2\leq k\leq \frac{n}{2},\] then \(\pi\) enforces Hamilton-connected.

Now we shall state the main result:

Theorem 3. Let \(G\) be a connected graph of order \(n.\) If \begin{equation*} \begin{split} ID(G)< \frac{k^3 + (2n - 3)k^2 + (2 - 2n^2)k + n^2 - n}{k(k - n)(n - 1)} \doteq Q_{3}(n,k), \end{split} \end{equation*} then \(G\) is Hamilton-connected. Moreover, \(ID(G)=Q_{3}(n,k)\) if and only if \(G\cong {K_{k}}+(\overline{K_{k-1}} \cup K_{n-2k+1})\).

Proof. Suppose that \(G\) is not Hamilton-connected. Accordingly to Lemma 3, there must exist an integer \(k\) such that \(d_{k-1}\leq k\) and \(d_{n-k}\leq n-k\) for \(2\leq k\leq \frac{n}{2}\). It then follows that \begin{equation*} \begin{split} ID(G)&\geq \frac{k-1}{k}+\frac{n-2k+1}{n-k}+\frac{k}{n-1}\\ &= \frac{k^3 + (2n - 3)k^2 + (2 - 2n^2)k + n^2 - n}{k(k - n)(n - 1)}, \end{split} \end{equation*} which contradicts to our initial assumption. Hence the result follows.

Furthermore, the condition in Theorem 3 cannot be dropped. If \(G\cong {K_{k}}+(\overline{K_{k-1}} \cup K_{n-2k+1})\), then one can easily see that \(ID(G)=Q_{3}(n,k)\). Conversely, let \(ID(G)=\widehat{Q}_{3}(n,k)\), then all inequalities in the proof should be equalities. Therefore, \(d_1=\cdots=d_{k-1}=k\), \(d_{k}=\cdots=d_{n-k}=n-k\) and \(d_{n-k+1}=\cdots=d_n=n-1\). Hence, \(G\cong {K_{k}}+(\overline{K_{k-1}} \cup K_{n-2k+1})\).

6. Traceable graphs

Lemma 4.([1]) Let \(G\) be a nontrivial graph of order \(n\geq 4\), with degree sequence \(\pi=(d_1\leq d_2\leq \cdots \leq d_n)\). Suppose that there is no integer \(k< \frac{n+1}{2}\) such that \(d_k \leq k-1\) and \(d_{n-k+1}\leq n-k-1 \). Then \(G\) is traceable.

Now we shall state the main result:

Theorem 4. Let \(G\) be a connected graph of order \(n.\) If \begin{equation*} \begin{split} ID(G)< \frac{k^3 + (2n - 4)k^2 + (- 2n^2 + 2n + 1)k + n^2 - n}{(k - 1)(n - 1)(k - n + 1)} \doteq Q_{4}(n,k), \end{split} \end{equation*} then \(G\) is traceable. Moreover, \(ID(G)=Q_{4}(n,k)\) if and only if \(G\cong {K_{k-1}}+(\overline{K_{k}} \cup K_{n-2k+1})\).

Proof. Suppose that \(G\) is not traceable. Then it follows from Lemma 4 that \(d_{k}\leq k-1\) and \(d_{n-k+1}\leq n-k-1\). Hence, from the definition of the inverse degree, we have \begin{equation*} \begin{split} ID(G)&\geq \frac{k}{k-1}+\frac{n-2k+1}{n-k-1}+\frac{k-1}{n-1}\\ &= \frac{k^3 + (2n - 4)k^2 + (- 2n^2 + 2n + 1)k + n^2 - n}{(k - 1)(n - 1)(k - n + 1)} \end{split} \end{equation*} which is a contradiction. This completes the proof.

Furthermore, the condition in Theorem 4 cannot be dropped. If \(G\cong {K_{k-1}}+(\overline{K_{k}} \cup K_{n-2k+1})\), then one can easily see that \(ID(G)=Q_{4}(n,k)\). Conversely, let \(ID(G)=\widehat{Q}_{4}(n,k)\), then all inequalities in the proof should be equalities. Hence, therefore \(d_1=\cdots=d_{k}=k-1\), \(d_{k+1}=\cdots=d_{n-k+1}=n-k-1\) and \(d_{n-k+2}=\cdots=d_n=n-1\). Hence, \(G\cong {K_{k-1}}+(\overline{K_{k}} \cup K_{n-2k+1})\).

7. \(k^{-}\)-independent graphs

More recently, An et al., [27] considered the property of \(k^{-}\)-independent graphs by using the first Zagreb index for a graph to be \(k^{-}\)-independent. In this section, we continue this program to explore sufficient conditions for a graph to be \(k^{-}\)-independent in terms of the inverse degree.

The following result is due to a survey by Bauer et al., [32].

Lemma 5. ([32]) Let \(\pi=(d_1\leq d_2\leq \cdots \leq d_n)\) be a graphical sequence and \(k\geq 1\). If \(d_{k+1}\geq n-k\), then \(\pi\) enforces \(k^{-}\)-independent.

We conclude this paper with the following structural result.

Theorem 5. Let \(G\) be a connected graph of order \(n.\) If \begin{equation*} \begin{split} ID(G)< \frac{- k^2 + kn - k - n^2 + n}{(n - 1)(k - n + 1)} \doteq Q_{5}(n,k), \end{split} \end{equation*} then \(G\) is \(k^{-}\)-independent. Moreover, \(F(G)=Q_{5}(n,k)\) if and only if \(G\cong \overline{K_{k+1}}+K_{n-k-1}\).

Proof. Suppose that \(G\) is not \(k^{-}\)-independent. Then it follows from Lemma 5 that \(d_{k+1}\leq n-k-1\). From the definition of the inverse degree, we have \begin{equation*} \begin{split} ID(G)&\geq \frac{k+1}{n-k-1}+\frac{n-k-1}{n-1}\\ &= \frac{- k^2 + kn - k - n^2 + n}{(n - 1)(k - n + 1)} \end{split} \end{equation*} which is a contradiction. Hence the result follows.

Furthermore, the condition in Theorem 5 cannot be dropped. If \(G\cong \overline{K_{k+1}}+K_{n-k-1}\), then one can easily see that \(ID(G)=Q_{5}(n,k)\). Conversely, let \(ID(G)=\widehat{Q}_{5}(n,k)\), then all inequalities in the proof should be equalities. Hence, therefore \(d_1=\cdots=d_{k+1}=n-k-1\), and \(d_{k+2}=\cdots=d_n=n-1\). Hence, \(G\cong \overline{K_{k+1}}+K_{n-k-1}\).

Acknowledgments

This research was supported by the Defense Industrial Technology Development Program (JCKY2018205C003) and National Key Research and Development Project (2019YEB2006602).

Author Contributions

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

Conflict of Interests

The authors declare no conflict of interest.

1. Introduction

Let \(C_m\), \(K_m\) and \(K_m -I\) denote cycle of length \(m\), complete graph on \(m\) vertices and complete graph on \(m\) vertices minus a \(1\)-factor respectively. By an \(m\)-cycle we mean a cycle of length \(m\). All graphs considered in this paper are simple and finite. A graph is said to be decomposed into subgraphs \(H_1\) and \(H_2\) which is denoted by \(G= H_1 \oplus H_2\), if \(G\) is the edge disjoint union of \(H_1\) and \(H_2\). If \(G= H_1 \oplus H_2 \oplus \cdots \oplus H_k\), where \(H_1\), \(H_2\), ..., \(H_k\) are all isomorphic to \(H\), then \(G\) is said to be \(H\)-decomposable. Furthermore, if \(H\) is a cycle of length \(m\) then we say that \(G\) is \(C_m\)-decomposable and this can be written as \(C_m|G\). A \(k\)-factor of \(G\) is a \(k\)-regular spanning subgraph. A \(k\)-factorization of a graph \(G\) is a partition of the edge set of \(G\) into \(k\)-factors. A \(C_k\)-factor of a graph is a \(2\)-factor in which each component is a cycle of length \(k\). A resolvable \(k\)-cycle decomposition (for short \(k\)-RCD) of \(G\) denoted by \(C_k ||G\), is a \(2\)-factorization of \(G\) in which each \(2\)-factor is a \(C_k\)-factor.

For two graphs \(G\) and \(H\) their tensor product \(G \times H\) has vertex set \(V(G) \times V(H)\) in which two vertices \((g_1,h_1)\) and \((g_2,h_2)\) are adjacent whenever \(g_1g_2 \in E(G)\) and \(h_1h_2 \in E(H)\). From this, note that the tensor product of graphs is distributive over edge disjoint union of graphs, that is if \(G= H_1 \oplus H_2 \oplus \cdots \oplus H_k\), then \(G \times H = (H_1 \times H)\oplus (H_2 \times H) \oplus \cdots \oplus(H_k \times H)\). Now, for \(h \in V(H)\), \(V(G) \times h = \{(v,h)|v \in V(G)\}\) is called a column of vertices of \(G \times H\) corresponding to \(h\). Further, for \(y \in V(G)\), \(y \times V(H) = \{(y,v)|v \in V(H)\}\) is called a layer of vertices of \(G \times H\) corresponding to \(y\).

In [1], Oyewumi et al., obtained an interesting result on the decompositions of certain graphs. The problem of finding \(C_k\)-decomposition of \(K_{2n+1}\) or \(K_{2n}-I\) where \(I\) is a \(1\)-factor of \(K_{2n}\), is completely settled by Alspach, Gavlas and Sajna in two different papers (see [2,3]). A generalization to the above complete graph decomposition problem is to find a \(C_k\)-decomposition of \(K_m \ast \overline K_n\), which is the complete \(m\)-partite graph in which each partite set has \(n\) vertices. The study of cycle decompositions of \(K_m \ast \overline K_n\) was initiated by Hoffman et al., [4]. In the case when \(p\) is a prime, the necessary and sufficient conditions for the existence of \(C_p\)-decomposition of \(K_m \ast \overline K_n\), \(p\geq 5\) is obtained by Manikandan and Paulraja in [5,6,7]. Billington [8] studied the decomposition of complete tripartite graphs into cycles of length 3 and 4. Furthermore, Cavenagh and Billington [9] studied \(4\)-cycle, \(6\)-cycle and \(8\)-cycle decomposition of complete multipartite graphs.

Billington et al., [10] solved the problem of decomposing \((K_m \ast \overline K_n)\) into \(5\)-cycles. Similarly, when \(p \geq 3\) is a prime, the necessary and sufficient conditions for the existence of \(C_{2p}\)-decomposition of \(K_m \ast \overline K_n\) was obtained by Smith (see [11]). For a prime \(p \geq 3\), it was proved in [12] that \(C_{3p}\)-decomposition of \(K_m \ast \overline K_n\) exists if the obvious necessary conditions are satisfied. As the graph \(K_m \times K_n \cong K_m \ast \overline K_n - E(nK_m)\) is a proper regular spanning subgraph of \(K_m \ast \overline K_n\). It is natural to think about the cycle decomposition of \(K_m \times K_n\).

The results in [5,6,7] also give necessary and sufficient conditions for the existence of a \(p\)-cycle decomposition, (where \(p \geq 5\) is a prime number) of the graph \(K_m \times K_n\). In [13] it was shown that the tensor product of two regular complete multipartite graph is Hamilton cycle decomposable. Muthusamy and Paulraja in [14] proved the existence of \(C_{kn}\)-factorization of the graph \(C_k \times K_{mn}\), where \(mn \neq 2 (\textrm{mod}\ 4)\) and \(k\) is odd. While Paulraja and Kumar [15] showed that the necessary conditions for the existence of a resolvable \(k\)-cycle decomposition of tensor product of complete graphs are sufficient when \(k\) is even. Oyewumi and Akwu [16] proved that \(C_4\) decomposes the product \(K_m \times K_n\), if and only if either (1) \(n \equiv 0 \ (\textrm{mod}\ 4)\) and \(m\) is odd, (2) \(m \equiv 0 \ (\textrm{mod}\ 4)\) and \(n\) is odd or (3) \(m \ or \ n \equiv 1 \ (\textrm{mod}\ 4)\).

As a companion to the work in [16], i.e., to consider the decomposition of the tensor product of complete graphs into cycles of even length. This paper proves that the obvious necessary conditions for \(K_m \times K_n\), \(2 \leq m,n\), to have a \(C_6\)-decomposition are also sufficient. Among other results, here we prove the following main results.

It is not surprising that the conditions in Theorem 1 are "symmetric" with respect to \(m\) and \(n\) since \(K_m \times K_n \cong K_n \times K_m\).

Theorem 1. Let \(2 \leq m,n, \) then \(C_6| K_m \times K_n\) if and only if \(m \equiv 1 \ or \ 3\ (\textrm{mod}\ 6)\) or \(n \equiv 1 \ or \ 3\ (\textrm{mod}\ 6)\).

Theorem 2. Let \(m\) be an even integer and \(m \geq 6\), then \(C_6| K_m - I \times K_n\) if and only if \(m \equiv 0 \ \text{or} \ 2\ (\textrm{mod}\ 6)\).

2. \(C_6\) decomposition of \(C_3 \times K_n\)

Theorem 3. Let \(n \in N\), then \(C_6|C_3 \times K_n\).

Proof. Following from the definition of tensor product of graphs, let \(U^1= \{u_1,v_1,w_1\}\), \(U^2= \{u_2,v_2,w_2\} \),\(...\), \(U^n=\{u_n,v_n,w_n\}\) form the partite set of vertices in \(C_3 \times K_n\). Also, \(U^i\) and \(U^j\) has an edge in \(C_3 \times K_n\) for \(1\leq i,j\leq n\) and \(i\neq j\) if the subgraph induce \(K_{3,3}-I\), where \(I\) is a \(1\)-factor of \(K_{3,3}\). Now, each subgraph \(U^i \cup U^j\) is isomorphic to \(K_{3,3}-I\). But \(K_{3,3}-I\) is a cycle of length six. Hence the proof.

Example 1. The graph \(C_3 \times K_7\) can be decomposed into cycles of length \(6\).

Proof. Let the partite sets (layers) of the tripartite graph \(C_3 \times K_7\) be \(U=\{u_1, u_2, . . . , u_7\}\), \(V=\{v_1, v_2, . . . , v_7\}\) and \(W=\{w_1,w_2, . . . ,w_7\}\). We assume that the vertices of \(U,V\) and \(W\) having same subscripts are the corresponding vertices of the partite sets. A \(6\)-cycle decomposition of \(C_3 \times K_7\) is given below:

\(\{u_1,v_2,w_1,u_2,v_1,w_2\}\),\(\{u_1,v_3,w_1,u_3,v_1,w_3\}\),\(\{u_2,v_3,w_2,u_3,v_2,w_3\}\),

\(\{u_1,v_4,w_1,u_4,v_1,w_4\}\),\(\{u_2,v_4,w_2,u_4,v_2,w_4\}\),\(\{u_3,v_4,w_3,u_4,v_3,w_4\}\),

\(\{u_1,v_5,w_1,u_5,v_1,w_5\}\),\(\{u_2,v_5,w_2,u_5,v_2,w_5\}\),\(\{u_3,v_5,w_3,u_5,v_3,w_5\}\),

\(\{u_4,v_5,w_4,u_5,v_4,w_5\}\),\(\{u_1,v_6,w_1,u_6,v_1,w_6\}\),\(\{u_2,v_6,w_2,u_6,v_2,w_6\}\),

\(\{u_3,v_6,w_3,u_6,v_3,w_6\}\),\(\{u_4,v_6,w_4,u_6,v_4,w_6\}\),\(\{u_5,v_6,w_5,u_6,v_5,w_6\}\),

\(\{u_1,v_7,w_1,u_7,v_1,w_7\}\),\(\{u_2,v_7,w_2,u_7,v_2,w_7\}\),\(\{u_3,v_7,w_3,u_7,v_3,w_7\}\),

\(\{u_4,v_7,w_4,u_7,v_4,w_7\}\),\(\{u_5,v_7,w_5,u_7,v_5,w_7\}\),\(\{u_6,v_7,w_6,u_7,v_6,w_7\}\).

Theorem 4. [17] Let \(m\) be an odd integer and \(m \geq 3\). If \(m \equiv 1 \ or \ 3\ (\textrm{mod}\ 6)\) then \(C_3|K_m\).

Theorem 5. [3] Let \(n\) be an even integer and \(m\) be an odd integer with \(3 \leq m \leq n\). The graph \(K_n- I\) can be decomposed into cycles of length \(m\) whenever \(m\) divides the number of edges in \(K_n - I\).

3. \(C_6\) decomposition of \(C_6 \times K_n\)

Theorem 6. [3] Let \(n\) be an odd integer and \(m\) be an even integer with \(3 \leq m \leq n\). The graph \(K_n\) can be decomposed into cycles of length \(m\) whenever \(m\) divides the number of edges in \(K_n\).

Lemma 1. \(C_6|C_6 \times K_2\).

Proof. Let the partite set of the bipartite graph \(C_6 \times K_2\) be \(\{u_1,u_2,...,u_6\}\), \(\{v_1,v_2,...,\) \(v_6\}\). We assume that the vertices having the same subscripts are the corresponding vertices of the partite sets. Now \(C_6 \times K_2\) can be decomposed into \(6\)-cycles which are \(\{u_1,v_2,u_3,v_4,u_5,v_6\}\) and \(\{v_1,u_2,v_3,u_4,v_5,u_6\}\).

Theorem 7. For all \(n\), \(C_6|C_6 \times K_n\).

Proof. Let the partite set of the \(6\)-partite graph \(C_6 \times K_n\) be \(U=\{u_1,u_2,...,u_n\}\), \(V=\{v_1,v_2,...,v_n\}\), \(W=\{w_1,w_2,...,w_n\}\), \(X=\{x_1,x_2,...,x_n\}\), \(Y=\{y_1,y_2,...,y_n\}\) and \(Z=\{z_1,z_2,...,z_n\}\), we assume that the vertices of \(U,V,W,X,Y\) and \(Z\) having the same subscripts are the corresponding vertices of the partite sets. Let \(U^1=\{u_1,v_1,w_1,x_1,y_1,z_1\}\), \(U^2=\{u_2,v_2,w_2,x_2,y_2,z_2\}\), ..., \(U^n=\{u_n,v_n,w_n,x_n,y_n,z_n\}\) be the sets of these vertices having the same subscripts. By the definition of the tensor product, each \(U^i\), \(1 \leq i \leq n\) is an independent set and the subgraph induced by each \(U^i \cup U^j\), \(1\leq i,j\leq n\) and \(i\neq j\) is isomorphic to \(C_6 \times K_2\). Now by Lemma 1 the graph \(C_6 \times K_2\) admits a \(6\)-cycle decomposition. This completes the proof.

4. \(C_6\) decomposition of \(K_m \times K_n\)[Proofs of main Theorems]

Proof of Theorem 1. Assume that \(C_6|K_m \times K_n\) for some \(m\) and \(n\) with \(2 \leq m,n\). Then every vertex of \(K_m \times K_n\) has even degree and \(6\) divides in the number of edges of \(K_m \times K_n\). These two conditions translate to \((m-1)(n-1)\) being even and \(6|m(m-1)n(n-1)\) respectively. Hence, by the first fact \(m\) or \(n\) has to be odd, i.e., has to be congruent to \(1 \ or \ 3 \ or \ 5\ (\textrm{mod}\ 6)\). The second fact can now be used to show that they cannot both be congruent to \(5\ (\textrm{mod}\ 6)\). It now follows that \(m \equiv 1 \ or \ 3\ (\textrm{mod}\ 6)\) or \(n \equiv 1 \ or \ 3\ (\textrm{mod}\ 6)\).

Conversely, let \(m \equiv 1 \ or \ 3\ (\textrm{mod}\ 6)\). By Theorem 4, \(C_3|K_m\) and hence \(K_m \times K_n =((C_3 \times K_n)\oplus \cdots \oplus(C_3 \times K_n))\). Since \(C_6|C_3 \times K_n\) by Theorem 3.

Finally, if \(n \equiv 1 \ or \ 3\ (\textrm{mod}\ 6)\), the above argument can be repeated with the roles of \(m\) and \(n\) interchanged to show again that \(C_6|K_m \times K_n\). This completes the proof.

Proof of Theorem 2. Assume that \(C_6|K_m-I \times K_n, m\geq 6\). Certainly, \(6|mn(m-2)(n-1)\). But we know that if \(6|m(m-2)\) then \(6|mn(m-2)(n-1)\). But \(m\) is even therefore \(m \equiv 0 \ or \ 2\ (\textrm{mod}\ 6)\).

Conversely, let \(m \equiv 0 \ or \ 2 \ (\textrm{mod}\ 6)\). Notice that for each \(m\), \(\frac{m(m-2)}{2}\) is a multiple of \(3\). Thus by Theorem 5 \(C_3|K_m -I\) and hence \(K_m - I \times K_n =((C_3 \times K_n)\oplus \cdots \oplus(C_3 \times K_n))\). From Theorem 3, \(C_6|C_3 \times K_n\). The proof is complete.

5. Conclusion

In view of the results obtained in this paper we draw our conclusion by the following corollary.

Corollary 1. For any simple graph \(G\). If

• 1. \(C_3 |G\) then \(C_6 |G \times K_n\), whenever \(n \geq 2\).
• 2. \(C_6 |G\) then \(C_6 |G \times K_n\), whenever \(n \geq 2\).

Proof. We only need to show that \(C_3|G\). Applying Theorem 3 gives the result.

Acknowledgments

The authors would like to thank the referees for helpful suggestions which has improved the present form of this work.

Author Contributions

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

Conflict of Interests

The authors declare no conflict of interest.

1. Introduction

Theory of chemical graphs is the branch of mathematical chemistry that applies theory of graphs to mathematical modeling of chemical phenomena. In chemical graph theory a molecular graph is a simple graph in which the vertices and edges represent atoms and chemical bonds between them. In this paper, \(G\) be a simple connected graph with vertex set \(V(G)\) and edge set \(E(G)\). The number of elements in \(V(G)\) and \(E(G)\) is represented as \(|V (G)|\) and \(|E(G)|\), respectively. For a vertex \(u \in V (G)\), the number of vertices adjacent to the vertex \(u\) is called the degree of \(u\), denoted by \(\delta_G (u)\). The complement of \(G\), denoted by \(\overline{G}\) , is a simple graph on the same set of vertices \(V(G)\) in which two vertices \(u\) and \(v\) are adjacent, i.e., connected by an edge \(uv\), if and only if they are not adjacent in \(G\). Hence, \(uv\in E(\overline {G})\), if and only if \(uv\notin E(G)\). Obviously \(E(G)\cup E(\overline {G})=E(K_n)\), and \(\overline {m} =|E(\overline{G})|=\binom n{2}-m\), the degree of a vertex \(u\) in \(\overline {G}\), is the number of edges incident to \(u\), denoted by \(\delta_{\overline {G}}(u)=n-1-\delta_{G}(u)\) [1]. The well-known Zagreb indices introduced in [2] are among the most important topological indices. The first and second Zagreb indices \(M _1\) and \(M _2\), respectively, are defined for a molecular graph G as: \[{M_1}(G) = \sum\limits_{v \in V(G)} {\delta _G}^{2}(v)=\sum\limits_{uv \in E(G)} [{\delta _G}(u) + {\delta _G}(v)], \quad \quad {M_2}(G) = \sum\limits_{uv \in E(G)} {{\delta _G}(u)\,{\delta _G}(v)}. \] The first and second Zagreb coindices have been introduced by Ashrafi et al., in 2010 [3], they are respectively defined as: \[{\overline{M}_1}(G) = \sum\limits_{uv \notin E(G)} [{\delta _G}(u) + {\delta _G}(v)], \quad \quad {\overline{M}_2}(G) = \sum\limits_{uv \notin E(G)} {{\delta _G}(u)\,{\delta _G}(v)}. \] Furtula and Gutman in 2015 introduced forgotten index (F-index) [4] which is defined as: \[F(G) = \sum\limits_{v \in V(G)} {{ {\delta _G}^3(v)} }= \sum\limits_{uv \in E(G)} {\left( {{\delta _G}^2(u)\, + {\delta _G}^2(v)} \right)}. \] De et al., in 2016 defined forgotten coindex (F-coindex)[5] which is defined as: \[\overline {F}(G) = \sum\limits_{uv \notin E(G)} {\left( {{\delta _G}^2(u)\, + {\delta _G}^2(v)} \right)}. \] Then, De et al., in 2016 [6] computed the forgotten index of join \(G_1+G_2\), tensor product \(G_1 \otimes G_2\), Cartesian product \(G_1\times G_2\), composition \(G_1\circ G_2\), strong product \(G_1\ast G_2\), disjunction \(G_1\vee G_2\) and symmetric difference \(G_1\oplus G_2\) of two graphs. Here we continue this line of research by exploring the behavior of the forgotten index under the same operations of complement graphs. The results are applied to molecular graph of nanotorus and titania nanotubes. In recent years, there has been considerable interest in general problems of determining topological indices and their operations [1,7,8,9].

2. Preliminaries

In this section we give some basic and preliminary concepts which we shall use later.

Lemma 1.[10,11] Let \(G_1\) and \(G_2\) be two connected graphs with \(|V (G_1 )|=n_1\), \(|V (G_2 )|=n_2\), \( |E (G_1 )|=m_1\), and \(|E (G_2 )|=m_2\). Then

• 1. \(|V (G_1 \times G_2 )|= |V (G_1\vee G_2 )|= |V (G_1 \circ G_2 )|= |V (G_1 \otimes G_2 )|= |V (G_1 \ast G_2 )|=|V (G_1 \oplus G_2 )|=n_1 n_2,\)

  \(|V (G_1+ G_2 )|= n_1+ n_2,\)

• 2. \(|E(G_1 \times G_2)| = m_1 n_2 + n_1 m_2,\)

  \(|E(G_1 \ast G_2 )|= m_1 n_2+n_1 m_2+2m_1 m_2,\)

  \( |E(G_1+G_2)| = m_1+m_2+n_1 n_2,\)

  \(|E(G_1 \circ G_2)| = m_1 {n_2}^2 +m_2 n_1,\)

  \(|E(G_1\vee G_2 )|=m_1 {n_2}^2 +m_2 {n_1}^2 - 2m_1 m_2,\)

  \( | E(G_1 \otimes G_2)| = 2m_1 m_2,\)

  \(|E(G_1 \oplus G_2)| = m_1 {n_2}^2 + m_2 {n_1}^2 - 4m_1 m_2.\)

Proposition 1. [12] Let \(G\) be a simple graph on \(n\) vertices and \(m\) edges. Then. \[ F (\overline{G} )= n(n - 1)^3 - 6m(n - 1)^2 + 3(n - 1)M_1(G) - F(G).\]

3. Discussion and main results

In this section, we study the forgotten index of various complement graph binary operations such as join, tensor product, Cartesian product, composition, strong product, disjunction and symmetric difference of two simple connected graphs. We use the notation \(V( G_i)\) for the vertex set, \(E( G_i)\), \(E(\overline G_i)\) for the edge sets, \(n_i\) for the number of vertices and \(m_i\), \(\overline m_i\) for the number of edges of the graph \(G_i\), \(\overline G_i\) respectively. All graphs here offer are simple graphs.

Definition 1(Join). The join \(G_{1}+ G_{2}\) of two graphs \(G_{1}\) and \(G_{2}\) is a graph with vertex set \(V (G_{1}+G_{2} )=V(G_{1}) \cup V(G_{2})\) and edge set \(E(G_{1})\cup E(G_{1}) \cup \{{uv| u \in V(G_{1}), v\in V (G_{2})}\}\).

Theorem 1. The \(F-index\) of the complement of \(G_1+ G_2 \) is given by \begin{eqnarray*} F(\overline{G_1+ G_2}) &=&(n_1+n_2)(n_1+n_2 - 1)^3 - 6(m_1+m_2+n_1 n_2)(n_1+n_2- 1)^2 \\ &&+ 3(n_1+n_2- 1)[M_1 (G_1 )+M_1 (G_2 )+n_1 n_2^2+n_2 n_1^2+4m_1 n_2+4m_2 n_1]\\ && -[F(G_1 )+F( G_2 )+3n_2 M_1 (G_1 )+3n_1 M_1 (G_2 )+6n_2^2 m_1+6n_1^2 m_2+n_1 n_2^3+n_2 n_1^3]. \end{eqnarray*}

Proof. From Proposition 1, we have \(F (\overline{G} )= n(n - 1)^3 - 6m(n - 1)^2 + 3(n - 1)M_1(G) - F(G) \), and since \(M_1 (G_1+ G_2 )=M_1 (G_1 )+M_1 (G_2 )+n_1 n_2^2+n_2 n_1^2+4m_1 n_2+4m_2 n_1\), given in [10], \(F(G_1+ G_2 )=F(G_1 )+F( G_2 )+3n_2 M_1 (G_1 )+3n_1 M_1 (G_2 )+6n_2^2 m_1+6n_1^2 m_2+n_1 n_2^3+n_2 n_1^3\), given in [6]. and \( |E(G_1+ G_2 )|=m_1+m_2+n_1 n_2,\quad |V(G_1+ G_2)|=n_1+n_2\) given in Lemma 1. Then \begin{eqnarray*} F(\overline{G_1+ G_2}) &=&|V(G_1+ G_2)|(|V(G_1+ G_2)| - 1)^3 - 6|E(G_1+ G_2 )|(|V(G_1+ G_2)| - 1)^2 \\ && +3(|V(G_1+ G_2)| - 1)M_1(G_1+ G_2 ) - F(G_1+ G_2 )\\ &=&(n_1+n_2)(n_1+n_2 - 1)^3 - 6(m_1+m_2+n_1 n_2)(n_1+n_2- 1)^2 \\ && +3(n_1+n_2- 1)[M_1 (G_1 )+M_1 (G_2 )+n_1 n_2^2+n_2 n_1^2+4m_1 n_2+4m_2 n_1]\\ && -[F(G_1 )+F( G_2 )+3n_2 M_1 (G_1 )+3n_1 M_1 (G_2 )+6n_2^2 m_1+6n_1^2 m_2+n_1 n_2^3+n_2 n_1^3]. \end{eqnarray*}

Definition 2(Tensor product). The tensor product \(G_{1}\otimes G_{2}\) of two graphs \(G_{1}\) and \(G_{2}\) is the graph with vertex set \(V(G_{1})\times V(G_{2})\) and any two vertices \((u_{1},v_{1})\) and \((u_{2},v_{2})\) are adjacent if and only if \(u_{1}u_{2}\in E(G_1)\) and \(v_{1}v_{2}\in E(G_2)\).

Theorem 2. The \(F-index\) of the complement of \(G_1\otimes G_2 \) is given by \begin{eqnarray*} F(\overline{G_1\otimes G_2}) &=&n_1n_2(n_1n_2- 1)^3 - 12m_1 m_2(n_1n_2- 1)^2 + 3(n_1n_2- 1)M_1 (G_1 ) M_1 ( G_2 )- F(G_1 )F( G_2 ). \end{eqnarray*}

Proof. From Proposition 1 we have \(F (\overline{G} )= n(n - 1)^3 - 6m(n - 1)^2 + 3(n - 1)M_1(G) - F(G) \), and since \(M_1 (G_1\otimes G_2 )=M_1 (G_1 ) M_1 ( G_2 )\) given in [13], \(F(G_1\otimes G_2 )=F(G_1 )F( G_2 )\) given in [6]. and \( |E(G_1\otimes G_2 )|=2m_1 m_2,\quad |V(G_1\otimes G_2)|=n_1n_2\) given in Lemma 2.1. Then. \begin{eqnarray*} F(\overline{G_1\otimes G_2}) &=&|V(G_1\otimes G_2)|(|V(G_1\otimes G_2)| - 1)^3 - 6|E(G_1\otimes G_2 )|(|V(G_1\otimes G_2)| - 1)^2 \\ && +3(|V(G_1\otimes G_2)| - 1)M_1(G_1\otimes G_2 ) - F(G_1\otimes G_2 )\\ &=&n_1n_2(n_1n_2- 1)^3 - 12m_1 m_2(n_1n_2- 1)^2 +3(n_1n_2- 1)M_1 (G_1 ) M_1 ( G_2 )- F(G_1 )F( G_2 ). \end{eqnarray*}

Definition 3(Cartesian product). The Cartesian product \(G_{1}\times G_{2}\), of two simple and connected graphs \(G_{1}\) and \(G_{2}\) has the vertex set\(V(G_{1}\times G_{2})=V(G_{1})\times V(G_{2})\) and \((a,x)(b,y)\) is an edge of \(G_{1}\times G_{2}\) if \(a = b\) and \(xy\in E(G_{2})\), or \(ab\in E(G_{1})\) and \(x = y\).

Theorem 3. The \(F-index\) of the complement of \(G_1\times G_2 \) is given by \begin{eqnarray*} F(\overline{G_1\times G_2}) &=&n_1n_2(n_1n_2- 1)^3 - 6(m_1 n_2 + n_1 m_2)(n_1n_2- 1)^2+ 3(n_1n_2- 1)[n_2 M_1 (G_1 )+n_1 M_1 (G_2 )+8m_1 m_2]\\ && -[n_2 F(G_1 )+n_1 F( G_2 )+6m_2 M_1 (G_1 )+6m_1 M_1 (G_2 )]. \end{eqnarray*}

Proof. From Proposition 1, we have \(F (\overline{G} )= n(n - 1)^3 - 6m(n - 1)^2 + 3(n - 1)M_1(G) - F(G) \), and since \(M_1 (G_1\times G_2 )=n_2 M_1 (G_1 )+n_1 M_1 (G_2 )+8m_1 m_2\), given in [14], \(F(G_1\times G_2 )=n_2 F(G_1 )+n_1 F( G_2 )+6m_2 M_1 (G_1 )+6m_1 M_1 (G_2 ),\) given in [6]. and \( |E(G_1\times G_2 )|=m_1 n_2 + n_1 m_2,\quad |V(G_1\times G_2)|=n_1n_2\) given in Lemma 1. Then \begin{eqnarray*} F(\overline{G_1\times G_2}) &=&|V(G_1\ast G_2)|(|V(G_1\ast G_2)| - 1)^3 - 6|E(G_1\ast G_2 )|(|V(G_1\ast G_2)| - 1)^2 \\ && +3(|V(G_1\ast G_2)| - 1)M_1(G_1\ast G_2 ) - F(G_1\ast G_2 )\\ &=&n_1n_2(n_1n_2- 1)^3 - 6(m_1 n_2 + n_1 m_2)(n_1n_2- 1)^2 \\ && +3(n_1n_2- 1)[n_2 M_1 (G_1 )+n_1 M_1 (G_2 )+8m_1 m_2]\\ && -[n_2 F(G_1 )+n_1 F( G_2 )+6m_2 M_1 (G_1 )+6m_1 M_1 (G_2 )]. \end{eqnarray*}

Definition 4(Composition). The composition \(G_{1}\circ G_{2}\), of two simple and connected graphs \(G_{1}\) and \(G_{2}\) with disjoint vertex sets \( V(G_{1})\) and \( V(G_{2})\) and edge sets \( E(G_{1})\) and \( E(G_{2})\) is the graph with vertex set \(V(G_{1})\times V(G_{2})\) and \(u = (u_{1},v_{1})\) is adjacent with \(v=(u_{2},v_{2})\) whenever (\(u_{1}\) is adjacent with \(u_{2}\)) or {\(u_{1}=u_{2}\) and \(v_{1}\) is adjacent with \(v_{2}\)}.

Theorem 4. The \(F-index\) of the complement of \(G_1\circ G_2 \) is given by \begin{eqnarray*} F(\overline{G_1\circ G_2}) &=&n_1n_2(n_1n_2- 1)^3 - 6(m_1 {n_2}^2 +m_2 n_1)(n_1n_2- 1)^2 \\ && +3(n_1n_2- 1)[n_2^3 M_1 (G_1 )+ n_1 M_1 (G_2 )+ 8n_2 m_2 m_1]\\ && -[n_2^4 F(G_1 )+n_1 F( G_2 )+6n_2^2 m_2 M_1 (G_1 )+6n_2 m_1 M_1 (G_2 )]. \end{eqnarray*}

Proof. From Proposition 1, we have \(F (\overline{G} )= n(n - 1)^3 - 6m(n - 1)^2 + 3(n - 1)M_1(G) - F(G) \), and since \(M_1 (G_1\circ G_2 )= n_2^3 M_1 (G_1 )+ n_1 M_1 (G_2 )+ 8n_2 m_2 m_1\), given in [14]. \(F(G_1 \circ G_2)=n_2^4 F(G_1 )+n_1 F( G_2 )+6n_2^2 m_2 M_1 (G_1 )+6n_2 m_1 M_1 (G_2 ),\) given in [6]. and \( |E(G_1\circ G_2 )|=m_1 {n_2}^2 +m_2 n_1,\quad |V(G_1\circ G_2)|=n_1n_2\) given in Lemma 1. Then \begin{eqnarray*} F(\overline{G_1\circ G_2}) &=&|V(G_1\circ G_2)|(|V(G_1\circ G_2)| - 1)^3 - 6|E(G_1\circ G_2 )|(|V(G_1\circ G_2)| - 1)^2 \\ && +3(|V(G_1\circ G_2)| - 1)M_1(G_1\circ G_2 ) - F(G_1\circ G_2 )\\ &=&n_1n_2(n_1n_2- 1)^3 - 6(m_1 {n_2}^2 +m_2 n_1)(n_1n_2- 1)^2 \\ && +3(n_1n_2- 1)[n_2^3 M_1 (G_1 )+ n_1 M_1 (G_2 )+ 8n_2 m_2 m_1]\\ && -[n_2^4 F(G_1 )+n_1 F( G_2 )+6n_2^2 m_2 M_1 (G_1 )+6n_2 m_1 M_1 (G_2 )]. \end{eqnarray*}

Definition 5 (Strong product). The strong product \(G_{1}\ast G_{2}\), of two simple and connected graphs \(G_{1}\) and \(G_{2}\) is a graph with vertex set \(V (G_{1}\ast G_{2})=V(G_{1})\times V(G_{2})\) and any two vertices \((u_{1},v_{1})\) and \((u_{2},v_{2})\) are adjacent if and only if {\(u_{1}= u_{2}\in V(G_{1})\) and \(v_{1} v_{2}\in E(G_{2})\)} or {\(v_{1}= v_{2}\in V(G_{2})\) and \(u_{1} u_{2}\in E(G_{1})\)}.

Proposition 2. [15] Let \(G_1 ,G_2 \) be two graphs with \(n_1,n_2\) vertices and \(m_1,m_2\) edges, respectively. Then \[M_1 (G_1 \ast G_2 )=(n_2+6m_2)M_1 (G_1 )+8m_2 m_1+(6m_1+n_1)M_1 (G_2 )+2M_1 (G_1 ) M_1 (G_2 ).\]

Theorem 5. The \(F-index\) of the complement of \(G_1\ast G_2 \) is given by \begin{eqnarray*} F(\overline{G_1\ast G_2}) &=&n_1n_2(n_1n_2- 1)^3 - 6(m_1 n_2+n_1 m_2+2m_1 m_2)(n_1n_2- 1)^2 \\ &&+ 3(n_1n_2- 1)[(n_2+6m_2)M_1 (G_1 )+8m_2 m_1+(6m_1+n_1)M_1 (G_2 )+2M_1 (G_1 ) M_1 (G_2 )]\\ &&-[n_2 F(G_1 )+n_1 F(G_2 )+F(G_1 )F( G_2 )+6m_2 M_1 (G_1 )+6m_1 M_1 (G_2 )\\ &&+6m_2 F(G_1 )+6m_1 F(G_2 )+3F(G_2 ) M_1 (G_1 )+3F(G_1 ) M_1 (G_2 )+6M_1 (G_1 ) M_1 (G_2 )]. \end{eqnarray*}

Proof. From Proposition 1, we have \(F (\overline{G} )= n(n - 1)^3 - 6m(n - 1)^2 + 3(n - 1)M_1(G) - F(G) \), and since \(M_1 (G_1\ast G_2 )=(n_2+6m_2)M_1 (G_1 )+8m_2 m_1+(6m_1+n_1)M_1 (G_2 )+2M_1 (G_1 ) M_1 (G_2 ),\) given in Proposition 2, and by [6] we have \begin{eqnarray*} F(G_1\ast G_2 ) &=&n_2 F(G_1 )+n_1 F(G_2 )+F(G_1 )F( G_2 )+6m_2 M_1 (G_1 )+6M_1 (G_1 ) M_1 (G_2 )\\ &&+6m_1 M_1 (G_2 )+6m_2 F(G_1 )+6m_1 F(G_2 )+3F(G_2 ) M_1 (G_1 )+3F(G_1 ) M_1 (G_2 ). \end{eqnarray*} And since \( |E(G_1\ast G_2 )|=m_1 n_2+n_1 m_2+2m_1 m_2,\quad |V(G_1\ast G_2)|=n_1n_2\), given in Lemma 1. Then \begin{eqnarray*} F(\overline{G_1\ast G_2}) &=&|V(G_1\ast G_2)|(|V(G_1\ast G_2)| - 1)^3 - 6|E(G_1\ast G_2 )|(|V(G_1\ast G_2)| - 1)^2 \\ &&+ 3(|V(G_1\ast G_2)| - 1)M_1(G_1\ast G_2 ) - F(G_1\ast G_2 )\end{eqnarray*} \begin{eqnarray*} &=&n_1n_2(n_1n_2- 1)^3 - 6(m_1 n_2+n_1 m_2+2m_1 m_2)(n_1n_2- 1)^2 \\ &&+ 3(n_1n_2- 1)[(n_2+6m_2)M_1 (G_1 )+8m_2 m_1+(6m_1+n_1)M_1 (G_2 )+2M_1 (G_1 ) M_1 (G_2 )]\\ &&-[n_2 F(G_1 )+n_1 F(G_2 )+F(G_1 )F( G_2 )+6m_2 M_1 (G_1 )+6m_1 M_1 (G_2 )\\ &&+6m_2 F(G_1 )+6m_1 F(G_2 )+3F(G_2 ) M_1 (G_1 )+3F(G_1 ) M_1 (G_2 )+6M_1 (G_1 ) M_1 (G_2 )]. \end{eqnarray*}

Definition 6 (Disjunction). The disjunction \(G_{1}\vee G_{2}\) of graphs \(G_{1}\) and \(G_{2}\) is the graph with vertex set \(V(G_{1})\times V(G_{2})\) and \((u_{1},v_{1})\) is adjacent with \((u_{2},v_{2})\), whenever \((u_{1},u_{2})\in E(G_{1})\) or \((v_{1},v_{2})\in E(G_{2})\).

Theorem 6. The first Zagreb index of \(G_1\vee G_2 \) is given by \begin{eqnarray*} M_1 (G_1 \vee G_2 ) &=&[n^3_2-4n_2m_2]M_1(G_1)+[n^3_1-4n_1m_1]M_1(G_2)+ 8n_1n_2 m_1m_2+M_1(G_1)M_1(G_2). \end{eqnarray*}

Theorem 7. The \(F-index\) of the complement of \(G_1\vee G_2 \) is given by \begin{eqnarray*} F(\overline{G_1\vee G_2}) &=&n_1n_2(n_1n_2- 1)^3 - 6[m_1 {n_2}^2 +m_2 {n_1}^2 - 2m_1 m_2](n_1n_2- 1)^2 \\ &&+ 3(n_1n_2- 1)\big{[}[n^3_2-4n_2m_2]M_1(G_1)+[n^3_1-4n_1m_1]M_1(G_2)\\ &&+ 8n_1n_2 m_1m_2+M_1(G_1)M_1(G_2)\big{]}- [n_2^4 F(G_1 )+n_1^4 F(G_2 )-F(G_1 )F(G_2 )\\ &&+6n_1 n_2^2 m_2 M_1 (G_1 )+6n_2 n_1^2 m_1 M_1 (G_2 )+3n_2 F(G_1 ) M_1 (G_2 )+3n_1 F(G_2 ) M_1 (G_1 )\\ &&-6n_2^2 m_2 F(G_1 )-6n_1^2 m_1 F(G_2 )-6n_1 n_2 M_1 (G_1 ) M_1 (G_2 )]. \end{eqnarray*}

Proof. From Proposition 1, we have \(F (\overline{G} )= n(n - 1)^3 - 6m(n - 1)^2 + 3(n - 1)M_1(G) - F(G)\), and by Theorem 6 and [6], respectively, we have \begin{eqnarray*} M_1 (G_1 \vee G_2 ) &=&[n^3_2-4n_2m_2]M_1(G_1)+[n^3_1-4n_1m_1]M_1(G_2)+ 8n_1n_2 m_1m_2+M_1(G_1)M_1(G_2). \end{eqnarray*} \begin{eqnarray*} F(G_1 \vee G_2 ) &=&n_2^4 F(G_1 )+n_1^4 F(G_2 )-F(G_1 )F(G_2 )+6n_1 n_2^2 m_2 M_1 (G_1 )\\ &&+6n_2 n_1^2 m_1 M_1 (G_2 )+3n_2 F(G_1 ) M_1 (G_2 )+3n_1 F(G_2 ) M_1 (G_1 )\\ &&-6n_2^2 m_2 F(G_1 )-6n_1^2 m_1 F(G_2 )-6n_1 n_2 M_1 (G_1 ) M_1 (G_2 ). \end{eqnarray*} And since \(|E(G_1\vee G_2)|=m_1 {n_2}^2 +m_2 {n_1}^2 - 2m_1 m_2,\quad |V(G_1\vee G_2)|=n_1n_2\) given in Lemma 1. Then \begin{eqnarray*} F(\overline{G_1\vee G_2}) &=&|V(G_1\vee G_2)|(|V(G_1\vee G_2)| - 1)^3 - 6|E(G_1\vee G_2 )|(|V(G_1\vee G_2)| - 1)^2 \\ &&+ 3(|V(G_1\vee G_2)| - 1)M_1(G_1\vee G_2 ) - F(G_1\vee G_2 )\\ &=&n_1n_2(n_1n_2- 1)^3 - 6[m_1 {n_2}^2 +m_2 {n_1}^2 - 2m_1 m_2](n_1n_2- 1)^2 \\ &&+ 3(n_1n_2- 1)\big{[}[n^3_2-4n_2m_2]M_1(G_1)+[n^3_1-4n_1m_1]M_1(G_2)\\ &&+ 8n_1n_2 m_1m_2+M_1(G_1)M_1(G_2)\big{]}- [n_2^4 F(G_1 )+n_1^4 F(G_2 )-F(G_1 )F(G_2 )\\ &&+6n_1 n_2^2 m_2 M_1 (G_1 )+6n_2 n_1^2 m_1 M_1 (G_2 )+3n_2 F(G_1 ) M_1 (G_2 )+3n_1 F(G_2 ) M_1 (G_1 )\\ &&-6n_2^2 m_2 F(G_1 )-6n_1^2 m_1 F(G_2 )-6n_1 n_2 M_1 (G_1 ) M_1 (G_2 )]. \end{eqnarray*}

Definition 7 (Symmetric difference). The symmetric difference \(G_{1}\oplus G_{2}\), of two simple and connected graphs \(G_{1}\) and \(G_{2}\) is the graph with vertex set \(V(G_{1})\times V(G_{2})\) and \(E(G_{1}\oplus G_{2})=(u_{1},u_{2})(v_{1},v_{2} )| u_{1} v_{1}\in E(G_{1})\) or \(u_{2} v_{2}\in E(G_{2})\) but not both.

Theorem 8. The first Zagreb index of \(G_1\oplus G_2 \) is given by \begin{eqnarray*} M_1 (G_1 \oplus G_2 ) &=&[n^3_2-8n_2m_2]M_1(G_1)+[n^3_1-8n_1m_1]M_1(G_2)+ 8n_1n_2 m_1m_2+4M_1(G_1)M_1(G_2). \end{eqnarray*}

Theorem 9. The \(F-index\) of the complement of \(G_1\oplus G_2 \) is given by \begin{eqnarray*} F(\overline{G_1\oplus G_2}) &=&n_1n_2(n_1n_2- 1)^3-6[m_1 {n_2}^2 + m_2 {n_1}^2 - 4m_1 m_2](n_1n_2- 1)^2 \\ &&+ 3(n_1n_2- 1)\big{[}[n^3_2-8n_2m_2]M_1(G_1)+[n^3_1-8n_1m_1]M_1(G_2)\\ &&+ 8n_1n_2 m_1m_2+4M_1(G_1)M_1(G_2)\big{]}- \big{[}n_2^4 F(G_1 )+n_1^4 F(G_2 )\\ &&-8F(G_1 )F(G_2 )+6n_1 n_2^2 m_2 M_1 (G_1 )+6n_2 n_1^2 m_1 M_1 (G_2 )\\ &&+12n_2 F(G_1 ) M_1 (G_2 )+12n_1 F(G_2 ) M_1 (G_1 )-12n_2^2 m_2 F(G_1 )\\ &&-12n_1^2 m_1 F(G_2 )-12n_1 n_2 M_1 (G_1 ) M_1 (G_2)\big{]}. \end{eqnarray*}

Proof. From Proposition 1, we have \(F (\overline{G} )= n(n - 1)^3 - 6m(n - 1)^2 + 3(n - 1)M_1(G) - F(G)\), and by Theorem 8 and [6], respectively, we have \begin{eqnarray*} M_1 (G_1 \oplus G_2 ) &=&[n^3_2-8n_2m_2]M_1(G_1)+[n^3_1-8n_1m_1]M_1(G_2)+ 8n_1n_2 m_1m_2+4M_1(G_1)M_1(G_2). \end{eqnarray*} \begin{eqnarray*} F(G_1 \oplus G_2 ) &=&n_2^4 F(G_1 )+n_1^4 F(G_2 )-8F(G_1 )F(G_2 )+6n_1 n_2^2 m_2 M_1 (G_1 )\\ &&+6n_2 n_1^2 m_1 M_1 (G_2 )+12n_2 F(G_1 ) M_1 (G_2 )+12n_1 F(G_2 ) M_1 (G_1 )\\ &&-12n_2^2 m_2 F(G_1 )-12n_1^2 m_1 F(G_2 )-12n_1 n_2 M_1 (G_1 ) M_1 (G_2 ). \end{eqnarray*} And since \( |E(G_1\oplus G_2 )|=m_1 {n_2}^2 + m_2 {n_1}^2 - 4m_1 m_2,\quad |V(G_1\oplus G_2)|=n_1n_2\) given in Lemma 1. Then \begin{eqnarray*} F(\overline{G_1\oplus G_2}) &=&|V(G_1\oplus G_2)|(|V(G_1\oplus G_2)| - 1)^3 - 6|E(G_1\oplus G_2 )|(|V(G_1\oplus G_2)| - 1)^2 \\ &&+ 3(|V(G_1\oplus G_2)| - 1)M_1(G_1\oplus G_2 ) - F(G_1\oplus G_2 )\\ &=&n_1n_2(n_1n_2- 1)^3-6[m_1 {n_2}^2 + m_2 {n_1}^2 - 4m_1 m_2](n_1n_2- 1)^2 \\ &&+ 3(n_1n_2- 1)\big{[}[n^3_2-8n_2m_2]M_1(G_1)+[n^3_1-8n_1m_1]M_1(G_2)\\ &&+ 8n_1n_2 m_1m_2+4M_1(G_1)M_1(G_2)\big{]}- \big{[}n_2^4 F(G_1 )+n_1^4 F(G_2 )\\ &&-8F(G_1 )F(G_2 )+6n_1 n_2^2 m_2 M_1 (G_1 )+6n_2 n_1^2 m_1 M_1 (G_2 )\\ &&+12n_2 F(G_1 ) M_1 (G_2 )+12n_1 F(G_2 ) M_1 (G_1 )-12n_2^2 m_2 F(G_1 )\\ &&-12n_1^2 m_1 F(G_2 )-12n_1 n_2 M_1 (G_1 ) M_1 (G_2)\big{]}. \end{eqnarray*}

4. Application

In this section, the forgotten index have been investigated for complement titania \(TiO_2\) nanotubes and molecular graph of nanotorus.

Corollary 1. The forgotten index of complement \(TiO_2 [n,m]\) nanotube Figure 1 is given by \begin{eqnarray*} F (\overline{TiO_2[n,m]}) &=&(6mn+ 6n)(6mn+ 6n- 1)^3 - 4(10mn+8n)(6mn+ 6n- 1)^2 \\ &&+ 3(6mn+ 6n- 1)(76mn + 48n) - 320mn - 160n . \end{eqnarray*}

Proof. By Proposition 1, we have \(F (\overline{G} )= n(n - 1)^3 - 4m(n - 1)^2 + 3(n - 1)M_1(G) - F(G)\), and since \(F(TiO_2 [n,m] )=320mn + 160n\), and \(M_1 (TiO_2 [n,m] )=76mn + 48n\) given in [16]. In [17] the partitions of the vertex set and edge set \(\sum| V(TiO_2[n,m])|=6mn+ 6n\), \(\sum| E(TiO_2[n,m])|=10mn+8n\) of titania nanotubes. Then \begin{eqnarray*} F (\overline{TiO_2[n,m]}) &=&\sum| V(TiO_2[n,m])|(\sum| V(TiO_2[n,m])| - 1)^3- 4\sum| E(TiO_2[n,m])|(\sum| V(TiO_2[n,m])| - 1)^2 \\ &&+ 3(\sum| V(TiO_2[n,m])| - 1)M_1(TiO_2[n,m]) - F(TiO_2[n,m])\\ &=&(6mn+ 6n)(6mn+ 6n- 1)^3 - 4(10mn+8n)(6mn+ 6n- 1)^2 \\ &&+ 3(6mn+ 6n- 1)(76mn + 48n) - 320mn - 160n .

Corollary 2. Let \(T = T [p, q]\) be the molecular graph of a nanotorus such that \(|V (T )|=pq\), \(|E (T )|=\frac{3}{2} pq\), Figure 2. Then

• a. \(F (\overline{T [p, q]} )=pq[(pq- 1)^2( pq- 7) + 27(pq- 1)- 27] .\)
• b. \( F (P_n\times T)= pq [125n-122] .\)
• c. \(F( \overline {P_n\times T})=pq[(npq- 1)^2(n^2pq-11n+4)+ 3(npq- 1)(25n-18) -125n+122].\)

Proof.

• a. By Proposition 1, we have \(F (\overline{G} )= n(n - 1)^3 - 4m(n - 1)^2 + 3(n - 1)M_1(G) - F(G)\), and since \(M_1 (T ) =9 pq \) given in [14]. and \(F (T ) =27 pq \) given in [18]. Then \begin{eqnarray*} F (\overline{T [p, q]} ) &=& |V (T )|(|V (T )| - 1)^3 - 4|E(T )|(|V (T )| - 1)^2 + 3(|V (T )| - 1)M_1(T) - F(T)\\ &=& pq(pq- 1)^3 - 6 pq(pq- 1)^2 + 27 pq(pq- 1)- 27 pq \\ &=& pq[(pq- 1)^2( pq- 7) + 27(pq- 1)- 27] . \end{eqnarray*}
• b. By \(F(G_1\times G_2 )=n_2 F(G_1 )+n_1 F( G_2 )+6m_2 M_1 (G_1 )+6m_1 M_1 (G_2 ),\) given in [6]. and since \(M_1 (P_n ) =4n-6 \), \(M_1 (T ) =9 pq \) given in [14]. and \(F (P_n ) =8n-14 \), \(F (T ) =27 pq \) given in [18]. Then \begin{eqnarray*} F (P_n \times T ) &=& |V (T )| F(P_n )+|V (P_n )| F( T )+6|E (T )| M_1 (P_n )+6|E (P_n )| M_1 (T )\\ &=& 2pq (4n-7)+27npq+18pq (2n-3)+54pq(n-1)\\ &=& pq [125n-122]. \end{eqnarray*}
• c. By [19], \(M_1 (P_n\times T ) =pq(25n-18),\) and by using Lemma 1, \(|E(P_n \times T)| =pq(\frac{5}{2}n-1)\), \(|V(P_n \times T)| =npq \), and applying Proposition 1 and item (b) we get \begin{eqnarray*} F (\overline{P_n \times T} ) &=& |V (P_n \times T )|(|V (P_n \times T )| - 1)^3 - 4|E(P_n \times T)|(|V (P_n \times T )| - 1)^2\\ &&+ 3(|V (P_n \times T )| - 1)M_1(P_n \times T) - F(P_n \times T)\\ &=& npq(npq- 1)^3 - 4pq(\frac{5}{2}n-1)(npq- 1)^2 \\ &&+ 3pq(npq- 1)(25n-18) -pq [125n-122]\\ &=& pq[(npq- 1)^2(n^2pq-11n+4)+ 3(npq- 1)(25n-18) -125n+122]. \end{eqnarray*}

5. Conclusion

The forgotten index one of the most important topological indices which preserve the symmetry of molecular structures and provide a mathematical formulation to predict their physical and chemical properties. In this article, we computed the forgotten index of some basic mathematical operations and obtained explicit formula for their values under complement graph operations, and we computed the forgotten index of molecular complement graph of nanotorus and titania nanotubes \(TiO_2 [n,m]\).

Authorcontributions

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

Conflictofinterests

The authors declare no conflict of interest.

1. Preliminaries

Let \(G\) be a simple graph with vertex set \(\mathit{V}({G})=\{1,2,\ldots,n\}\), having neither loops nor multiple edges. The adjacency matrix \(\mathbf{A}\) is the \(n\times n\) matrix whose \(ij^{th}\) entry is \(1\) if vertices \(i\) and \(j\) are connected by an edge and is \(0\) otherwise.

Let \(\mathbf{I}\) denote the identity matrix. The \(k^{th}\) column of \(\mathbf{I}\) is \(\mathbf{e}_k\) for all \(k\). The characteristic polynomial \(|x\mathbf{I}-\mathbf{A}|\) of \(\mathbf{A}\), denoted by \(\phi(G,x)\), is a monic polynomial with integer coefficients. Since the edges of \(G\) are undirected, \(\mathbf{A}\) is symmetric and the roots of \(\phi(G,x)\), which are the eigenvalues of \(\mathbf{A}\) (or of \(G\)), are real numbers. Thus, the eigenvalues of \(G\) are \(n\) totally real algebraic integers, with possible repetitions. A simple eigenvalue is a simple root of \(\phi(G,x)\). As we shall see in the next sections, we shall be focusing on the \(s\) distinct eigenvalues \(\lambda_1>\ldots>\lambda_s\) of \(G\), largely ignoring their multiplicities. To this end, if \(\phi(G,x)=\prod_{k=1}^s (x-\lambda_k)^{q_k}\), where \(q_k\) is the multiplicity of \(\lambda_k\) for all \(k\), then we denote the product of distinct factors \(\prod_{k=1}^s (x-\lambda_k)\) by \(m(G,x)\). Since \(\mathbf{A}\) is symmetric, it is diagonalizable; thus, \(m(G,x)\) is the matrix minimal polynomial of \(\mathbf{A}\).

By differentiating \(\prod_{k=1}^s (x-\lambda_k)^{q_k}\) with respect to \(x\) to obtain \(\phi^\prime(G,x)\), we discover that \(\lambda\) is a multiple root of \(\phi(G,x)\) if and only if it is also a root of \(\phi^\prime(G,x)\). Thus, we have the following theorem.

Theorem 1. \(m(G,x)=\phi(G,x)\) if and only if \(\phi(G,x)\) and \(\phi^\prime(G,x)\) have no common polynomial factors.

In this paper, we propose to consider the factorization of the minimal polynomial \(m(G,x)\) over \(\mathbb{Q}\), rather than that over \(\mathbb{R}\), the latter of which yields the distinct eigenvalues of \(G\). This will reveal a restriction on the possible values of the rank of any walk matrix of \(G\).

The rest of the paper is structured as follows. The next section shall present elementary results on minimal polynomials of roots of \(m(G,x)\). Section 3 reminds the reader of the basic results on walks of graphs, then presents the concepts of pseudo walk matrices and their walk vectors, originally introduced in [1]. The following section mentions the important results from that paper that relate to pseudo walk matrices, which leads to the significant result in Theorem 12 that restricts the matrix rank of any pseudo walk matrix to a few possible values depending on the factorization of \(m(G,x)\).

Section 5 provides a brief preliminary on the control theoretic aspects relevant to this paper, leading to the justification for generalizing controllable pairs \((\mathbf{A},\mathbf{v})\) to the case where \(\mathbf{v}\) is a walk vector. The next section focuses on graphs having an irreducible characteristic polynomial, which are shown to occur relatively frequently. For such graphs, it is proved that \((\mathbf{A},\mathbf{v})\) is controllable for all walk vectors \(\mathbf{v}\). Further interesting results on controllable pairs are provided in Section 7. The final section introduces recalcitrant pairs, which, in a sense, are the direct opposite of controllable pairs because they occur when the rank of a pseudo walk matrix attains the lower bound set by Theorem 12. Several related results on recalcitrant pairs are revealed.

2. Minimal polynomials of eigenvalues

A polynomial \(p(x)\) with integer coefficients (of positive degree) is called primitive if its coefficients have no common integer factors other than \(\pm 1\). Since both \(m(G,x)\) and \(\phi(G,x)\) are monic, they are clearly both primitive. The following lemma tells us that monic polynomials with coefficients in \(\mathbb{Q}\) must have integer coefficients whenever their product has integer coefficients.

Lemma 1. Let \(p_1(x)\) and \(p_2(x)\) be two monic polynomials with rational coefficients, both having degree at least one. Suppose \(p(x)=p_1(x)\,p_2(x)\) is a monic polynomial with integer coefficients. Then the coefficients of both \(p_1(x)\) and \(p_2(x)\) are integers.

Proof. Rewrite \(p_1(x)\) and \(p_2(x)\) as \(\frac1{a}(a\,p_1(x))\) and \(\frac1{b}(b\,p_1(x))\) respectively, where the integers \(a\) and \(b\) are chosen such that \(a\,p_1(x)\) and \(b\,p_2(x)\) are primitive polynomials with integer coefficients. By Gauss' Lemma [2], the product \((a\,p_1(x))(b\,p_2(x))\), or \((ab)\,p(x)\), is also primitive. But \(p(x)\) is a monic polynomial with integer coefficients; consequently, it is primitive itself. This is only possible if \(ab=\pm 1\). Hence \(p_1(x)\) and \(p_2(x)\) actually have integer coefficients, as required.

Theorem 2. If \(f(x)\), a polynomial with rational coefficients, is a factor of a monic polynomial with integer coefficients, then \(f(x)\) is monic and has integer coefficients.

Even though Theorem 2 appears to be basic, it will have important implications later on in this paper.

Let us consider the minimal polynomial (in field theory) of each eigenvalue \(\lambda_k\) of \(\mathbf{A}^{1}\). This paper requires the reader to distinguish between the minimal polynomial of \(\mathbf{A}\), denoted by \(m(G,x)\), and the minimal polynomial of each eigenvalue \(\lambda_k\). The fact that mathematics uses the same terminology for both is unfortunate. For all eigenvalues \(\lambda_k\), this is the (monic) polynomial \(p_k(x)\) with integer coefficients having the smallest possible degree that satisfies \(p_k(\lambda_k)=0\). The algebraic conjugates of \(\lambda_k\) are all the roots of its minimal polynomial, including \(\lambda_k\) itself. Thus, the conjugates of \(\lambda_k\) are also eigenvalues of \(\mathbf{A}\), otherwise \(m(G,x)\) would not have integer coefficients.

Thus, we decompose the set of distinct eigenvalues \(\Lambda=\{\lambda_1,\ldots,\lambda_s\}\) into \(r\) mutually disjoint subsets \(\Lambda_1,\ldots,\Lambda_r\). Two eigenvalues are in the same set \(\Lambda_i\) if and only if they are algebraic conjugates. We note that, for all \(i\), \(|\Lambda_i|\), the cardinality of set \(\Lambda_i\), is the degree of the minimal polynomial shared by each of its elements; in particular, \(\Lambda_i\) has only one element if and only if its element is an integer. We define the minimal polynomial of \(\Lambda_i\) (for some \(i\)) to be that of any of its elements; we denote this by \(m(\Lambda_i)\). Thus, at worst, \(r=s\) and \(\Lambda\) would be split into \(s\) singletons \(\Lambda_1=\{\lambda_1\},\ldots,\Lambda_s=\{\lambda_s\}\). This happens when all the distinct eigenvalues of the graph \(G\) are integers, in which case \(G\) is an integral graph [3, Section 8.4]. At the other end of the spectrum, \(r\) could be equal to \(1\) and \(m(G,x)\) would then be irreducible over \(\mathbb{Q}\). In this case, all the \(s\) eigenvalues are conjugates, and we only have one set \(\Lambda_1=\{\lambda_1,\ldots,\lambda_s\}\). We shall focus on this case in Section 6, as it seems to be quite common.

Example 1. The eigenvalues of the adjacency matrix of the cycle \(C_7\) on seven vertices are \(\Lambda=\{2,1.247,-0.445,-1.802\}\), all of which are repeated twice except for the eigenvalue \(2\). The characteristic polynomial of \(C_7\) is \(x^7-7x^5+14x^3-7x-2\), which factorizes into \((x-2)(x^3+x^2-2x-1)^2\). Thus \(m(C_7,x)=(x-2)(x^3+x^2-2x-1)=x^4-x^3-4x^2+3x+2\). Hence \(m(\Lambda_1)=x-2\) and \(m(\Lambda_2)=x^3+x^2-2x-1\). Moreover, the set \(\Lambda\) decomposes into \(\Lambda_1=\{2\}\) and \(\Lambda_2=\{1.247,-0.445,-1.802\}\).

3. Walks and Pseudo walk matrices

A walk of length \(\ell\) on \(G\), starting from vertex \(j\) and ending at vertex \(k\), is a sequence of \(\ell+1\) vertices \(v_1,v_2,\ldots,v_{\ell+1}\) (not necessarily distinct) such that \(v_1=j\), \(v_{\ell+1}=k\) and every two consecutive vertices in the sequence are connected by an edge in \(G\).

Let \(S\) be any subset of the Cartesian product \(\mathcal{V}^2=\mathcal{V}(G)\times\mathcal{V}(G)\). If \(u=(j,k)\in S\), then \(w_\ell(\{u\})\) is the number of walks of length \(\ell\) in \(G\) that start at \(j\) and end at \(k\). It is well-known that

the entry of \(\mathbf{A}^\ell\) in the \(j^{th}\) row and \(k^{th}\) column [3, Proposition 1.3.4]. We shall consider the total number of such walks for all \(u\in S\), which we denote by \(w_\ell(S)\). In other words, we define If \(S\) is deducible from the context, then we occasionally shorten the notation \(w_\ell(S)\) to \(w_\ell\).

Since the matrix inverse \((\mathbf{I}-x\mathbf{A})^{-1}\) may be expanded into the formal power series \(\sum_{j=0}^\infty \mathbf{A}^jx^j\), the entry \(\left[(\mathbf{I}-x\mathbf{A})^{-1}\right]_{jk}\), or \(\mathbf{e}_k^{T}(\mathbf{I}-x\mathbf{A})^{-1}\mathbf{e}_j\), can be written as the formal power series \(w_0+w_1x+w_2x^2+w_3x^3+\cdots\), where \(w_0,w_1,w_2,\ldots\) are, respectively, the number of walks of length \(0,1,2,\ldots\) starting from vertex \(j\) and ending at vertex \(k\). Thus, for any \(S\subseteq\mathcal{V}^2\), \[\sum_{(j,k)\in S}\!\!\left[(\mathbf{I}-x\mathbf{A})^{-1}\right]_{jk}=w_0(S)+(w_1(S))x+(w_2(S))x^2+\cdots.\]

In particular, if \(S=V_1\times V_2\), where \(V_1\) and \(V_2\) are both subsets of \(\mathcal{V}(G)\), then the left hand side of the above relation would be simply \(\mathbf{b}_1^{T}(\mathbf{I}-x\mathbf{A})^{-1}\mathbf{b}_2\), where \(\mathbf{b}_1\) is the \(0\)-\(1\) vector where, for all \(k\), its \(k^{th}\) entry is equal to \(1\) if and only if \(k\in V_1\), and \(\mathbf{b}_2\) is defined analogously for \(V_2\). We say that each of \(\mathbf{b}_1\) and \(\mathbf{b}_2\) is an indicator vector for \(V_1\) and \(V_2\) respectively.

In the literature, the walk matrix \(\mathbf{W}\) of \(G\) is generally taken to be the \(n\times n\) matrix where, for all \(k\), its \(k^{th}\) column is the vector \(\mathbf{A}^{k-1}\mathbf{j}\), where \(\mathbf{j}\) is the \(n\times 1\) vector of all ones [4,5,6]. By (1) and (2), \(\left[\mathbf{W}\right]_{jk}=w_{k-1}(S_j)\), where \(S_j=\{j\}\times\mathcal{V}(G)\). Other authors additionally consider walk matrices of the form \(\mathbf{W}_\mathbf{b}=\begin{pmatrix}\mathbf{b} & \mathbf{A}\mathbf{b} & \mathbf{A}^2\mathbf{b} & \cdots & \mathbf{A}^{n-1}\mathbf{b}\end{pmatrix}\), where \(\mathbf{b}\) is any \(0\)-\(1\) vector [1,7,8]. The \(jk^{th}\) entry of \(\mathbf{W}_\mathbf{b}\) is equal to \(w_{k-1}(S^\prime_j)\), where \(S^\prime_j=\{j\}\times B\) and \(B\) is the subset of \(\mathcal{V}(G)\) such that \(\mathbf{b}\) is its indicator vector.

It is clear that

and for all \(j\) and \(k\), \(\left[{\mathbf{W}_\mathbf{b}}^{T}\mathbf{W}_\mathbf{b}\right]_{jk}=w_{j+k-2}(B\times B)\). Since \({\mathbf{W}_\mathbf{b}}^{T}\mathbf{W}_\mathbf{b}\) has constant skew diagonals, it is a so-called Hankel matrix, so we denote it by \(\mathbf{H}_\mathbf{b}\). The matrix \(\mathbf{H}_\mathbf{b}\) is the Gram matrix of the columns of \(\mathbf{W}_\mathbf{b}\). It is well-known that \(\mathbf{H}_\mathbf{b}\) and \(\mathbf{W}_\mathbf{b}\) have the same matrix rank [9,Section 0.4.6(d)].

In this paper, we go even further, considering Hankel matrices whose \(jk^{th}\) entry is \(w_{j+k-2}(S)\), where \(S\) is any subset of \(\mathcal{V}^2\). The author of this paper had considered such matrices in [1]. In that paper, it was proved that for any \(S\), there exists a vector \(\mathbf{v}\) (indeed, often more than one) such that the Gram matrix of the columns \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\ldots,\mathbf{A}^{n-1}\mathbf{v}\) is the Hankel matrix \(\mathbf{H}_\mathbf{v}\) whose constant skew diagonal entries are \(w_0(S),w_1(S),\ldots,w_{2n-2}(S)\). The matrix \(\mathbf{W}_\mathbf{v}=\begin{pmatrix}\mathbf{v} & \mathbf{A}\mathbf{v} & \mathbf{A}^2\mathbf{v} & \cdots & \mathbf{A}^{n-1}\mathbf{v}\end{pmatrix}\) is called a pseudo walk matrix, because its entries are usually not walk enumerations, even though the matrix \(\mathbf{H}_\mathbf{v}={\mathbf{W}_\mathbf{v}}^{T}\mathbf{W}_\mathbf{v}\), as already mentioned, contains the walk enumerations \(w_0(S),w_1(S),\ldots,w_{2n-2}(S)\) on its constant skew diagonals.

Definition 3 ([1]). A pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) associated with \(S\subseteq\mathcal{V}^2\) is the \(n\times n\) matrix with columns \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\cdots,\mathbf{A}^{n-1}\mathbf{v}\) such that the entries of the Hankel matrix \(\mathbf{H}_\mathbf{v}={\mathbf{W}_\mathbf{v}}^{T}\mathbf{W}_\mathbf{v}\) satisfy \(\left[\mathbf{H}_\mathbf{v}\right]_{ij}=w_{i+j-2}(S)\) for all \(i\) and \(j\). The vector \(\mathbf{v}\) in the previous sentence is a walk vector associated with \(S\). If \(\mathbf{v}\) is a \(0\)-\(1\) vector, then \(\mathbf{W}_\mathbf{v}\) may be simply called a walk matrix associated with \(S\).

It is proved in [1, Theorem 2.1] that any two distinct walk vectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\) associated with a fixed \(S\subseteq\mathcal{V}^2\) produce two different pseudo walk matrices \(\mathbf{W}_{\mathbf{v}_1}\) and \(\mathbf{W}_{\mathbf{v}_2}\) having the same matrix rank. This fact ensures that any walk vector associated with \(S\), no matter how unusual (some walk vectors may even contain imaginary entries) is fine, keeping in mind that the rank of \(\mathbf{W}_\mathbf{v}\) is significant to the controllability aspects of the graph \(G\), as we shall see later in Section 5.

We note that if the rank of \(\mathbf{W}_\mathbf{v}\) is \(R\), then the \(n\times c\) matrix \(\begin{pmatrix}\mathbf{v} & \mathbf{A}\mathbf{v} & \cdots & \mathbf{A}^{c-1}\mathbf{v}\end{pmatrix}\) has rank \(R\) for all \(c\geq R\). This is because, for any vector \(\mathbf{v}\), we may write down the list of vectors \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\ldots,\mathbf{A}^k\mathbf{v}\), stopping this list when the last vector is linearly dependent on the previous ones. When this happens, \(\mathbf{A}^k\mathbf{v}=\sum_{j=0}^{k-1} a_j\mathbf{A}^j\mathbf{v}\) for some scalars \(a_j\). But by premultiplying both sides of this equation by \(\mathbf{A}^i\), for \(i\) a positive integer, we realize that \(\mathbf{A}^{i+k}\mathbf{v}\) may also be written as a linear combination of the vectors \(\mathbf{v},\mathbf{A}\mathbf{v},\ldots,\mathbf{A}^{k-1}\mathbf{v}\). Consequently, we may say that the rank of \(\mathbf{W}_\mathbf{v}\) is the number \(R\) such that the vectors \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\ldots,\mathbf{A}^{R-1}\mathbf{v}\) are linearly independent but the vectors \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\ldots,\mathbf{A}^R\mathbf{v}\) are linearly dependent.

Because of this, in certain papers such as [1,10], walk matrices are not \(n\times n\) matrices but \(n\times R\) matrices, where \(R\) is as defined in the previous paragraph. We distinguish between the \(n\times n\) pseudo walk matrices defined in Definition 3 from these \(n\times R\) pseudo walk matrices by denoting the latter by \(\overline{W}_\mathbf{v}\).

Earlier, we mentioned that, for a pseudo walk matrix of rank \(R\), there must exist scalars \(a_j\) such that \(\mathbf{A}^R\mathbf{v}=\sum_{j=0}^{R-1} a_j\mathbf{A}^j\mathbf{v}\). This may be rewritten as \(\mathbf{A}^R\mathbf{v}=\overline{W}_\mathbf{v}\mathbf{c}\), where \(\mathbf{c}=\begin{pmatrix}a_0 & a_1 & \cdots & a_{R-1}\end{pmatrix}^{T}\). Clearly the matrix \(\begin{pmatrix}\mathbf{A}\mathbf{v} & \mathbf{A}^2\mathbf{v} & \cdots & \mathbf{A}^R\mathbf{v}\end{pmatrix}\) is equal to \(\overline{W}_\mathbf{v}\begin{pmatrix}\mathbf{e}_2 & \mathbf{e}_3 & \cdots & \mathbf{c}\end{pmatrix}\). Alternatively, \(\mathbf{A}\overline{W}_\mathbf{v}=\overline{W}_\mathbf{v}\mathbf{C}_\mathbf{v}\), where the \(R\times R\) matrix \(\mathbf{C}_\mathbf{v}\) is the companion matrix \(\begin{pmatrix}\mathbf{e}_2 & \mathbf{e}_3 & \cdots & \mathbf{c}\end{pmatrix}\) of \(\overline{W}_\mathbf{v}\) [9,10]. By expanding the determinant \(|x\mathbf{I}-\mathbf{C}_\mathbf{v}|\) along its last column using the Laplace determinant expansion, the characteristic polynomial of \(\mathbf{C}_\mathbf{v}\) is seen to be \(x^R-a_{R-1}x^{R-1}-a_{R-2}x^{R-2}-\cdots-a_0\). We denote this polynomial by \(\phi_\mathbf{v}(x)\) and call it the companion polynomial of \(\mathbf{W}_\mathbf{v}\) (or of \(\overline{W}_\mathbf{v}\)). Note that we started this paragraph with the equation \(\left(\mathbf{A}^R-a_{R-1}\mathbf{A}^{R-1}-a_{R-2}\mathbf{A}^{R-2}-\cdots-a_0\mathbf{I}\right)\mathbf{v}=\mathbf{0}\) --- thus, we have the Cayley-Hamiltonian-like result \((\phi_\mathbf{v}(\mathbf{A}))\mathbf{v}=\mathbf{0}\) for any walk vector \(\mathbf{v}\).

In the literature, an eigenvalue is termed main if it has an associated eigenvector that is not orthogonal to \(\mathbf{j}\). Here, we generalize this definition to \(\mathbf{v}\)--main eigenvalues:

Definition 4. An eigenvalue of \(\mathbf{A}\) is \(\mathbf{v}\)-main if it has an associated eigenvector that is not orthogonal to the walk vector \(\mathbf{v}\).

4. Results on the rank of Pseudo walk matrices

The significance of the companion polynomial is the following result. It is only mentioned in [1], but not proved. The proof below follows that of the similar result in [10], there considering only the particular case \(\mathbf{v}=\mathbf{j}\).

Theorem 5 ([1]). For any walk vector \(\mathbf{v}\), \(\phi_\mathbf{v}(x)=\prod(x-\lambda_i)\), with the product running over all the \(\mathbf{v}\)-main distinct eigenvalues of \(G\).

Proof. Suppose \(\lambda\) is an eigenvalue of \(G\) having an associated eigenvector \(\mathbf{x}\) that is not orthogonal to \(\mathbf{v}\). Since \(\mathbf{A}\overline{W}_\mathbf{v}=\overline{W}_\mathbf{v}\mathbf{C}_\mathbf{v}\), we have \(\mathbf{x}^{T}\mathbf{A}\overline{W}_\mathbf{v}=\mathbf{x}^{T}\overline{W}_\mathbf{v}\mathbf{C}_\mathbf{v}\), or \(\lambda (\mathbf{x}^{T}\overline{W}_\mathbf{v}) = (\mathbf{x}^{T}\overline{W}_\mathbf{v})\mathbf{C}_\mathbf{v}\). Thus, the vector \((\overline{W}_\mathbf{v})^{T}\mathbf{x}\) is a left eigenvector associated with the eigenvalue \(\lambda\) of \(\mathbf{C}_\mathbf{v}\). Note that \((\overline{W}_\mathbf{v})^{T}\mathbf{x}\ne\mathbf{0}^{T}\), since it is equal to \(\begin{pmatrix}\mathbf{v}^{T}\mathbf{x} & \mathbf{v}^{T}\mathbf{A}\mathbf{x} & \mathbf{v}^{T}\mathbf{A}^2\mathbf{x} & \cdots & \mathbf{v}^{T}\mathbf{A}^{R-1}\mathbf{x}\end{pmatrix}^{T}\), or \(\left(\mathbf{v}^{T}\mathbf{x}\right)\begin{pmatrix}1 & \lambda & \lambda^2 & \cdots & \lambda^{R-1}\end{pmatrix}^{T}\).

Conversely, suppose \(\lambda\) is an eigenvalue of \(\mathbf{C}_\mathbf{v}\) with associated eigenvector \(\mathbf{y}\). We prove that \(\lambda\) is an eigenvalue of \(G\) having an associated eigenvector not orthogonal to \(\mathbf{v}\). Restarting again from \(\mathbf{A}\overline{W}_\mathbf{v}=\overline{W}_\mathbf{v}\mathbf{C}_\mathbf{v}\), \((x\mathbf{I}_n-\mathbf{A})\overline{W}_\mathbf{v}=\overline{W}_\mathbf{v}(x\mathbf{I}_R-\mathbf{C}_\mathbf{v})\), where \(\mathbf{I}_k\) denotes the \(k\times k\) identity matrix. Hence

Substituting \(x=\lambda\) in (4), we obtain \((\lambda\mathbf{I}_n-\mathbf{A})\overline{W}_\mathbf{v}\mathbf{y}=\mathbf{0}\), or \(\mathbf{A}\left(\overline{W}_\mathbf{v}\mathbf{y}\right)=\lambda\left(\overline{W}_\mathbf{v}\mathbf{y}\right)\). We now show that \(\mathbf{v}^{T}\left(\overline{W}_\mathbf{v}\mathbf{y}\right)\ne 0\).

On the one hand,

On the other hand Combining (5) and (6) for \(k=0,1,2,\ldots,R-1\), we obtain or \(\left((\overline{W}_\mathbf{v})^{T}\overline{W}_\mathbf{v}\right)\mathbf{y}=\left(\mathbf{v}^{T}\overline{W}_\mathbf{v}\mathbf{y}\right)\begin{pmatrix}1 & \lambda & \lambda^2 & \cdots & \lambda^{R-1}\end{pmatrix}^{T}\). Since \((\overline{W}_\mathbf{v})^{T}\overline{W}_\mathbf{v}\) is the Gram matrix of the \(R\) linearly independent columns of \(\overline{W}_\mathbf{v}\), it is invertible. If we suppose that \(\mathbf{v}^{T}\overline{W}_\mathbf{v}\mathbf{y}=0\), then (7) would become \(\left((\overline{W}_\mathbf{v})^{T}\overline{W}_\mathbf{v}\right)\mathbf{y}=\mathbf{0}\) and consequently \(\mathbf{y}\), an eigenvector, would absurdly be the zero vector. Thus \(\mathbf{v}^{T}\overline{W}_\mathbf{v}\mathbf{y}\ne 0\), proving the result.

We immediately have the following corollary.

Corollary 1 ([1]). For any walk vector \(\mathbf{v}\) associated with the pseudo walk matrix \(\mathbf{W}_\mathbf{v}\), the following four quantities are equal:

• The rank of \(\mathbf{W}_\mathbf{v}\);
• The number of columns of \(\overline{W}_\mathbf{v}\);
• The number of \(\mathbf{v}\)-main eigenvalues of \(G\);
• The degree of \(\phi_\mathbf{v}(x)\).

Moreover, we have the following result, the first part of which is also a consequence of Theorem 5.

Theorem 6([1]). The rank of any pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) is at most equal to \(s\), the number of distinct eigenvalues of \(G\). This upper bound is attained by the walk vector \(\mathbf{p}=\sum_{i=1}^n \mathbf{x}_i\), the sum of any set of \(n\) mutually orthonormal eigenvectors associated with the \(n\) (not necessarily distinct) eigenvalues of \(G\). This walk vector \(\mathbf{p}\) is associated with the set \(\{(v,v)\mid v\in\mathcal{V}(G)\}\) counting the number of all closed walks of \(G\).

A nice result in [1, Theorem 2.3] expresses each coefficient of \(\phi_\mathbf{v}(x)\) in terms of the ratio of two determinants.

Theorem 7([1, Theorem 2.3]). The coefficient of \(x^k\) of the polynomial \(\phi_\mathbf{v}(x)\) of degree \(R\) is \((-1)^{R-k}\frac{|\mathbf{N}|}{|\mathbf{H}|}\), where \(\mathbf{H}=\overline{W}_\mathbf{v}^{T}\overline{W}_\mathbf{v}\) and \(\mathbf{N}\) is the matrix whose first \(k\) columns are those of \(\mathbf{H}\) and whose last \(r-k\) columns are those of \(\mathbf{H}^\prime=\overline{W}_\mathbf{v}^{T}\mathbf{A}\overline{W}_\mathbf{v}\).

Clearly the determinants \(|\mathbf{H}|\) and \(|\mathbf{N}|\) in the above theorem are integers, because their entries are walk enumerations on the graph. Hence \(\phi_\mathbf{v}(x)\) has rational coefficients. However, by Theorem 5, \(\phi_\mathbf{v}(x)\) divides \(m(G,x)\) for any walk vector \(\mathbf{v}\). We thus apply Theorem 2 to obtain the following important result.

Theorem 8. For any walk vector \(\mathbf{v}\), \(\phi_\mathbf{v}(x)\) is a monic polynomial with integer coefficients.

Theorem 8 extends the result in [11], there proved only for \(\mathbf{v}=\mathbf{j}\), to any walk vector of any pseudo walk matrix with starting and ending vertices in any subset of \(\mathcal{V}^2\).

The following is a rather noteworthy corollary of Theorem 8.

Corollary 2. In the statement of Theorem 7, \(|\mathbf{N}|\) is an integer multiple of \(|\mathbf{H}|\) for any coefficient of \(\phi_\mathbf{v}(x)\).

Recall that we had split the set of distinct eigenvalues \(\Lambda\) into \(r\) mutually disjoint subsets \(\Lambda_1,\ldots,\Lambda_r\), where each subset \(\Lambda_i\) contains all the conjugates of some eigenvalue \(\lambda_i\). Because of Theorem 8, the set of \(\mathbf{v}\)-main eigenvalues of \(G\) must be the union of some or all of these \(\Lambda_i\) subsets.

Theorem 9. If \(\lambda\in\Lambda_i\) is a \(\mathbf{v}\)-main eigenvalue of \(G\), then any other \(\lambda_k\) in \(\Lambda_i\) is also a \(\mathbf{v}\)-main eigenvalue of \(G\).

Proof. By Theorem 8, \(\phi_\mathbf{v}(x)\) is monic with integer coefficients. Let us write \(\phi_\mathbf{v}(x)\) as the product \(p_1(x)\ldots p_t(x)\) of all polynomial factors over \(\mathbb{Q}\), all of which must also be monic with integer coefficients thanks to Theorem 2. Since \(\lambda\) is a \(\mathbf{v}\)-main eigenvalue, its minimal polynomial \(\displaystyle\!\!\prod_{\lambda_k\in\Lambda_i}\!\!(x-\lambda_k)\) must be one of these \(p_j(x)\). This proves that all eigenvalues in \(\Lambda_i\) are \(\mathbf{v}\)-main.

By the Perron-Frobenius theorem for irreducible non-negative matrices, the largest eigenvalue \(\lambda_1\) of the adjacency matrix of any connected graph is simple and the components of its associated eigenvector are all positive [3, Theorem 1.3.6]. Thus, if the walk vector \(\mathbf{v}\) is a \(0\)-\(1\) vector corresponding to a walk matrix (rather than a pseudo walk matrix), then \(\lambda_1\) will necessarily be \(\mathbf{v}\)-main. It turns out that this is also true for walk vectors associated with pseudo walk matrices.

In order to prove this result, we first recall the main result of [1] in Theorem 10 below that explicitly obtains a walk vector \(\mathbf{v}\) for any subset \(S\) of \(\mathcal{V}^2\). Recall that what we mean by this is that \(\mathbf{v}\) can always be obtained from \(S\) so that the Gram matrix of the columns of \(\mathbf{W}_\mathbf{v}\) is a Hankel matrix with constant skew diagonals containing the walk enumerations \(w_0(S),\ldots,w_{2n-2}(S)\).

Theorem 10 ([1, Theorem 3.1]). Let \(S\) be any subset of \(\mathcal{V}^2\). Then \(\mathbf{v}\) is a walk vector associated with \(S\) if \(\mathbf{v}=\mathbf{X}\mathbf{d}\), where \(\mathbf{X}\) is an orthogonal matrix whose columns are \(n\) orthonormal eigenvectors of \(G\) associated with its \(n\) (possibly not distinct) eigenvalues and \(\mathbf{d}\) is any column vector where, for \(k=1,2,\ldots,n\), its \(k^{th}\) entry \(\left[\mathbf{d}\right]_k\) is \(\pm\sqrt{\sum_{(u,v)\in S}\left[\mathbf{X}\right]_{uk}\left[\mathbf{X}\right]_{vk}}\).

Theorem 11. \(\lambda_1\) is a \(\mathbf{v}\)-main eigenvalue for any walk vector \(\mathbf{v}\).

Proof. Let \(\mathbf{X}\) be as in the statement of Theorem 10. Without loss of generality, we assume that the first column of \(\mathbf{X}\) is \(\mathbf{x}_1\), the eigenvector associated with \(\lambda_1\). Then, by Theorem 10, \(\mathbf{v}^{T}\mathbf{x}_1=\mathbf{d}^{T}\mathbf{X}^{T}\mathbf{x}_1=\mathbf{d}^{T}\mathbf{e}_1=\left[\mathbf{d}\right]_1\). By the same theorem, this number is equal to \(\pm\sqrt{\sum_{(u,v)\in S}\left[\mathbf{X}\right]_{u1}\left[\mathbf{X}\right]_{v1}}\), or \(\pm\sqrt{\sum_{(u,v)\in S}\left[\mathbf{x}_1\right]_u\left[\mathbf{x}_1\right]_v}\). Since the entries of \(\mathbf{x}_1\) are all positive, this quantity is nonzero, as required.

Combining the results in this section, we reveal the following important theorem, upon which the remainder of this paper is based.

Theorem 12. The rank of \(\mathbf{W}_\mathbf{v}\), for any walk vector \(\mathbf{v}\), is equal to \(|\Lambda_1|+g_2|\Lambda_2|+g_3|\Lambda_3|+\cdots+g_r|\Lambda_r|\), where each of \(g_2,g_3,\ldots,g_r\) is either \(0\) or \(1\) (and these \(g_i\)'s may possibly be different for different walk vectors \(\mathbf{v}\)).

Theorem 12 limits the possible values for the rank of \(\mathbf{W}_\mathbf{v}\) quite significantly, especially if \(r\) is small. Indeed, if \(r=1\), we have the following interesting result.

Theorem 13. If \(\phi(G,x)\) is irreducible over \(\mathbb{Q}\), then \(\mathbf{W}_\mathbf{v}\) has rank \(n\) for all walk vectors \(\mathbf{v}\).

If \(r=2\), that is, if \(m(G,x)\) factors into just two monic polynomials over \(\mathbb{Q}\) (both with integer coefficients), then each pseudo walk matrix associated with \(G\) has just two possible values for its rank.

Theorem 14. Suppose \(m(G,x)=p_1(x)\,p_2(x)\) where both \(p_1(x)\) and \(p_2(x)\) are irreducible polynomials (over \(\mathbb{Q}\)) with integer coefficients. If \(p_1(x)\) is a polynomial of degree \(d\) having \(\lambda_1\) as a root, then for any walk vector \(\mathbf{v}\), the rank of \(\mathbf{W}_\mathbf{v}\) is either \(d\) or \(s\), the number of distinct eigenvalues of \(G\).

In Example 1, we mentioned that the cycle \(C_7\) has minimal polynomial \((x-2)(x^3+x^2-2x-1)\). By Theorem 14, we immediately infer that the rank of any pseudo walk matrix of \(C_7\) is either \(1\) or \(4\).

In the case of Example 1, the rank of \(\mathbf{W}_\mathbf{j}\) is clearly one, since \(C_7\) is a regular graph. The following result tells us that, indeed, such pseudo walk matrices of rank one only occur for regular graphs.

Theorem 15. If a graph \(G\) is not regular, then none of its pseudo walk matrices has rank one.

Proof. Suppose the pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) has rank one. Then \(\mathbf{A}\mathbf{v}=\mu\mathbf{v}\) where \(\mu\) is the largest eigenvalue of \(\mathbf{A}\), which must be an integer by Theorem 8. The number of walks \(w_0,w_1,w_2,w_3,\ldots\) of length \(0,1,2,3,\ldots\) within the set \(S\) for which \(\mathbf{v}\) is a walk vector are thus \(\mathbf{v}^{T}\mathbf{v},\mu(\mathbf{v}^{T}\mathbf{v}),\mu^2(\mathbf{v}^{T}\mathbf{v}),\mu^3(\mathbf{v}^{T}\mathbf{v}),\ldots\). In other words, \(w_k=\mu\,w_{k-1}\) for all walk lengths \(k\geq 1\). This is only possible if \(G\) is a regular graph of degree \(\mu\).

Since \(\mathbf{W}_\mathbf{j}\) is always a pseudo walk matrix of rank one for regular graphs, we have:

Corollary 3. A pseudo walk matrix associated with some subset \(S\) of \(\mathcal{V}^2\) of a graph \(G\) has rank one if and only if \(G\) is a regular graph.

In Section 8, we shall reveal more results on pseudo walk matrices of regular graphs.

Because of Theorem 12, we propose that rather than finding the set of eigenvalues of \(G\) to determine the rank of a (pseudo) walk matrix, we first perform a polynomial factorization of \(m(G,x)\) over \(\mathbb{Q}\). The number of such factors and their degrees sheds a very important light on the rank of any pseudo walk matrix. Indeed, in the case when such a factorization is not possible, the rank is immediately inferred to be \(n\).

5. Control theory

A graph is said to be controllable if \(\mathbf{W}_\mathbf{j}\) is invertible; equivalently, if all the eigenvalues of \(G\) are simple and (j)-main [4,5,12]. Other papers such as [7,8,13] additionally consider pairs \((\mathbf{A},\mathbf{b})\) for \(0\)-\(1\) vectors \(\mathbf{b}\). In the same way as for controllable graphs, such pairs are controllable if \(\mathbf{W}_\mathbf{b}\) is invertible, or, equivalently, if all eigenvalues of \(G\) are simple and \(\mathbf{b}\)-main.

The terminology `controllable' arises from the following class of problems in control theory. Consider the following discrete system with state \(\mathbf{x}[k]\) at time \(k\) \((k=0,1,2,3,\ldots)\) satisfying the recurrence relation

Here, \(\mathbf{M}\) is a fixed \(n\times n\) matrix, \(\mathbf{q}\) is a fixed \(n\times 1\) vector and \(u[k]\) is any sequence serving as an input to the system. The output sequence is \(\mathbf{r}^{T}\mathbf{x}[0],\mathbf{r}^{T}\mathbf{x}[1],\mathbf{r}^{T}\mathbf{x}[2],\ldots\) for a fixed vector \(\mathbf{r}\).

The system governed by (8) is controllable if, given an initial state \(\mathbf{x}[0]\) and a future state \(\mathbf{s}\), there exists a time \(K\geq 0\) and an input \(u[k]\) such that \(\mathbf{x}[K]=\mathbf{s}\) [14, Definition 6.1]. The Kalman controllability criterion, a well-known criterion for system controllability, states that (8) is controllable if and only if the rank of the controllability matrix \(\begin{pmatrix}\mathbf{q} & \mathbf{M}\mathbf{q} & \mathbf{M}^2\mathbf{q} & \ldots & \mathbf{M}^{n-1}\mathbf{q}\end{pmatrix}\) is full. We immediately note that the controllability matrix is precisely the pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) whenever \(\mathbf{M}=\mathbf{A}\) and \(\mathbf{q}=\mathbf{v}\). This explains why we were so keen on obtaining the rank of \(\mathbf{W}_\mathbf{v}\), and why we say that, for a \(0\)-\(1\) vector \(\mathbf{b}\), the pair \((\mathbf{A},\mathbf{b})\) is controllable whenever the rank of \(\mathbf{W}_\mathbf{b}\) is full.

In the control theory literature, there is also the Popov-Belevitch-Hautus (PBH) test, another necessary and sufficient test for the controllability of (8). It says that the system is controllable whenever none of the eigenvectors of \(\mathbf{M}\) is orthogonal to \(\mathbf{q}\) [15, Theorem 6.2-5 (1.)]. This agrees with the more general results presented in Theorem 5 and Corollary 1 of this paper.

We shall henceforth consider the system

where the state \(\mathbf{x}[k]\) is being affected by the adjacencies of the graph \(G\) as per the adjacency matrix \(\mathbf{A}\).

When \(\mathbf{v}\) is a \(0\)-\(1\) vector \(\mathbf{b}\), the input \(u[k]\) will affect the vertices in the set \(V\) indicated by \(\mathbf{b}\). As mentioned in Section 3, the formal power series produced from the expression \(\mathbf{b}^{T}(\mathbf{I}-x\mathbf{A})^{-1}\mathbf{b}\) is \(\sum_{k=0}^\infty (w_k(V\times V))x^k\). We may, however, rewrite the right hand side as \(\displaystyle\sum_{k=1}^s \dfrac{(\mathbf{b}^{T}\mathbf{x}_k)^2}{1-\lambda_kx}\), where, for all \(k\), \(\mathbf{x}_k\) is an appropriate\(^{2}\) choice of eigenvector associated with \(\lambda_k\). Clearly this function will have \(n\) distinct poles when \(n=s\) (that is, if all eigenvalues are distinct) and when no eigenvector \(\mathbf{x}_k\) is orthogonal to \(\mathbf{b}\). Hence the pair \((\mathbf{A},\mathbf{b})\) is controllable if and only if the rational function \(\mathbf{b}^{T}(\mathbf{I}-x\mathbf{A})^{-1}\mathbf{b}\) has \(n\) distinct poles [8, Lemma 2.1].

In the same way, we have defined a walk vector \(\mathbf{v}\) associated with any subset \(S\) of \(\mathcal{V}^2\) such that

\[\mathbf{v}^{T}(\mathbf{I}-x\mathbf{A})^{-1}\mathbf{v}=\sum_{k=0}^\infty (w_k(S))x^k=\sum_{k=1}^s \dfrac{(\mathbf{v}^{T}\mathbf{x}_k)^2}{1-\lambda_kx}.\] The system (9) where \(\mathbf{v}\) is a walk vector associated with \(S\) would have its input \(u[k]\) affecting all the pairs of vertices in \(S\).

Thus, it is natural to extend the consideration of controllable pairs \((\mathbf{A},\mathbf{b})\) pertaining to pairs of vertices in a subset \(V\times V\) of \(\mathcal{V}^2\), where \(\mathbf{b}\) is an indicator vector of \(V\), to the more general consideration of controllable pairs \((\mathbf{A},\mathbf{v})\) pertaining to pairs of vertices in any subset \(S\) of \(\mathcal{V}^2\), where \(\mathbf{v}\) is the walk vector of \(S\). We thus define the pair \((\mathbf{A},\mathbf{v})\) to be controllable if all eigenvalues of \(G\) are simple and \(\mathbf{v}\)-main, or, equivalently, if the pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) is invertible.

6. Graphs with an irreducible characteristic polynomial

Theorem 13 tells us that graphs having an adjacency matrix with an irreducible characteristic polynomial are quite special.

Theorem 16. If \(\phi(G,x)\) is irreducible over \(\mathbb{Q}\), then the pair \((\mathbf{A},\mathbf{v})\) is controllable for any walk vector \(\mathbf{v}\).

Theorem 16 above generalizes the result of [8, Theorem 5.3], which only mentions the walk vectors in the set \(\{\mathbf{j},\mathbf{e}_1,\ldots,\mathbf{e}_n\}\).

It was proved in [16] that the ratio of controllable graphs on \(n\) vertices to all non-isomorphic graphs on \(n\) vertices tends to \(1\) as \(n\) increases. By Theorem 16, any graph with an irreducible characteristic polynomial over \(\mathbb{Q}\) is controllable. The converse is false; the smallest example of a controllable graph with a factorizable polynomial is that of Figure 1, having characteristic polynomial \(x(x^5-8x^3-6x^2+8x+6)\). Observe that, by Theorem 14, the rank of every pseudo walk matrix associated with this graph must be either \(5\) or \(6\).

Table 1 below enumerates the number of non-isomorphic connected graphs \(G(n)\), the number of connected controllable graphs \(C(n)\) and the number of graphs with an irreducible characteristic polynomial \(I(n)\) on up to 10 vertices. The data in the second row was obtained from [17]. The numbers in the third row were obtained from [6]. The enumerations in the fourth row were produced using a mathematics software package that checked the irreducibility, or otherwise, of the characteristic polynomials of all non-isomorphic connected graphs on up to 10 vertices. The collection of these 11989764 non-isomorphic connected graphs were produced beforehand by the algorithm explained in [18].

Table 1. The number of connected graphs \(G(n)\), connected controllable graphs \(C(n)\) and connected graphs with an irreducible characteristic polynomial \(I(n)\) on \(n\) vertices.

\(n\)	1	2	3	4	5	6	7	8	9	10
\(G(n)\)	1	1	2	6	21	112	853	11117	261080	11716571
\(C(n)\)	1	0	0	0	0	8	85	2275	83034	5512362
\(I(n)\)	1	0	0	0	0	7	54	1943	62620	4697820

The characteristic polynomial of a disconnected graph is the product of the characteristic polynomials of each of its components. Thus, no disconnected graph has an irreducible characteristic polynomial, justifying the consideration of only connected graphs in Table 1.

We remark that roughly six out of every seven controllable graphs on up to ten vertices have an irreducible characteristic polynomial. This seems to suggest that the quantity \(\frac{I(n)}{G(n)}\), like \(\frac{C(n)}{G(n)}\), may also approach \(1\) as \(n\) tends to infinity. The empirical evidence warrants the formulation of the following conjecture.

Conjecture 1. \(\displaystyle\lim_{n\to\infty}\frac{I(n)}{G(n)}=1.\)

Among the 31 controllable graphs on seven vertices having a factorizable characteristic polynomial, 28 of them have zero as one of their eigenvalues [6]; indeed, each of their characteristic polynomial factorizes into \(x\) and some irreducible polynomial of degree six. The other three, depicted in Figure 2, factor into \((x+2)\) and an irreducible polynomial of degree six.

When analysing these graphs, it was observed that the characteristic polynomial of the majority of controllable graphs on at most nine vertices is either irreducible or factors into just two irreducible polynomials, one of which is of the form \((x-m)\) for some integer \(m\). Indeed, this is true for all controllable graphs on six and seven vertices. However, there are some exceptional cases when the number of vertices is higher. The characteristic polynomial of the controllable graph in Figure 3 factors into two irreducible polynomials, both of which having degree four. We also note the controllable graph in Figure 4 whose characteristic polynomial has four factors. Similar examples of such exceptional controllable graphs having more than eight vertices also exist.

We conclude this section by posing the following question. Is there a graph such that \((\mathbf{A},\mathbf{v})\) is controllable for all walk vectors \(\mathbf{v}\) and whose characteristic polynomial is not irreducible? In other words, is there a graph that is a counterexample to the converse of Theorem 16? No such graphs have been found so far. We conjecture that there aren't any.

Conjecture 2. Let a graph \(G\) have adjacency matrix \(\mathbf{A}\). The pair \((\mathbf{A},\mathbf{v})\) is controllable for any walk vector \(\mathbf{v}\) if and only if \(\phi(G,x)\) is irreducible over \(\mathbb{Q}\).

7. Results on controllable pairs

Let \(\mathbf{b}_1\) and \(\mathbf{b}_2\) be any two \(0\)-\(1\) vectors. The walk matrix \(\mathbf{W}_{\mathbf{b}_1}\) contains walk enumerations of walks starting and ending within the subset \(V_1\) of \(\mathcal{V}(G)\) such that \(\mathbf{b}_1\) is the indicator vector of \(V_1\); we can say a similar statement for \(\mathbf{W}_{\mathbf{b}_2}\). Moreover, similarly to (3), We remark that the \(ij^{th}\) entry of the matrix in (10) is the number of walks of length \(i+j-2\) starting from vertices in \(V_1\) and ending at vertices in \(V_2\). In other words, \(\left[{\mathbf{W}_{\mathbf{b}_1}}^{T}\mathbf{W}_{\mathbf{b}_2}\right]_{ij}=w_{i+j-2}(V_1\times V_2)\) for all \(i\) and \(j\).

However, by Theorem 10, there exists a walk vector \(\mathbf{v}\) associated with \(V_1\times V_2\) such that the Gram matrix of the columns of the pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) is equal to the matrix in (10). Moreover, this Gram matrix must have the same rank as \(\mathbf{W}_\mathbf{v}\). Thus, we have the following lemma.

Lemma 2. The rank of the pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) associated with the subset \(V_1\times V_2\) of \(\mathcal{V}^2\) is equal to the rank of \({\mathbf{W}_{\mathbf{b}_1}}^{T}\mathbf{W}_{\mathbf{b}_2}\), where \(\mathbf{b}_1\) and \(\mathbf{b}_2\) are the \(0\)-\(1\) indicator vectors of \(V_1\) and \(V_2\) respectively.

Clearly if both \({\mathbf{W}_{\mathbf{b}_1}}\) and \(\mathbf{W}_{\mathbf{b}_2}\) are invertible, then so is \({\mathbf{W}_{\mathbf{b}_1}}^{T}\mathbf{W}_{\mathbf{b}_2}\). Thus we have the following result.

Theorem 17. Let \(\mathbf{b}_1\) and \(\mathbf{b}_2\) be indicator vectors of two subsets \(V_1\) and \(V_2\) of \(\mathcal{V}(G)\). Moreover, let \(\mathbf{v}\) be the walk vector of \(V_1\times V_2\). If \((\mathbf{A},\mathbf{b}_1)\) and \((\mathbf{A},\mathbf{b}_2)\) are both controllable pairs, then so is the pair \((\mathbf{A},\mathbf{v})\).

A graph is omnicontrollable if \((\mathbf{A},\mathbf{e}_i)\) is a controllable pair for all \(i\in\{1,2,\ldots,n\}\) [7,13]. Because of Theorem 17, omnicontrollable graphs have many more controllable pairs apart from \((\mathbf{A},\mathbf{e}_1),\ldots,(\mathbf{A},\mathbf{e}_n)\).

Corollary 4. Let \(\mathbf{v}_{(i,j)}\) be a walk vector associated with the set containing the single pair \(\{(i,j)\}\) whose Gram matrix of the columns of its pseudo walk matrix counts walks starting from vertex \(i\) and ending at vertex \(j\) in an omnicontrollable graph \(G\). The pair \((\mathbf{A},\mathbf{v}_{(i,j)})\) is controllable for any two vertices \(i\) and \(j\) in \(G\).

Proof. When \(i=j\), the result is immediate by definition of an omnicontrollable graph. Thus suppose \(i\ne j\). Then \((\mathbf{A},\mathbf{e}_i)\) and \((\mathbf{A},\mathbf{e}_j)\) are controllable pairs for any \(i\) and \(j\). Since \(\{i\}\times\{j\}=\{(i,j)\}\), the result follows by applying Theorem 17.

8. Recalcitrant pairs

If a pair \((\mathbf{A},\mathbf{v})\) is not controllable, then the pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) would not have full rank. For instance, graphs having at least one repeating eigenvalue can have no controllable pair \((\mathbf{A},\mathbf{v})\) for any walk vector \(\mathbf{v}\); this is a consequence of Theorem 6. Indeed, by using Theorem 1, we can present the following result.

Theorem 18. If \(\phi(G,x)\) and \(\phi^\prime(G,x)\) have some polynomial common factor, then no walk vector \(\mathbf{v}\) exists such that the pair \((\mathbf{A},\mathbf{v})\) is controllable.

By Theorem 12, the rank of any pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) is at least \(|\Lambda_1|\), the degree of the minimal polynomial of the largest eigenvalue \(\lambda_1\). As already mentioned in Section 6, if \(|\Lambda_1|=n\), then \(\phi(G,x)\) would be irreducible and all pairs \((\mathbf{A},\mathbf{v})\) would be controllable. Thus we assume that \(|\Lambda_1|< n\) in this section.

We introduce the following terminology for when the rank of \(\mathbf{W}_\mathbf{v}\) attains this lower bound \(|\Lambda_1|\):

Definition 19. For a graph with adjacency matrix \(\mathbf{A}\) that does not have an irreducible characteristic polynomial, the pair \((\mathbf{A},\mathbf{v})\) is called recalcitrant if the rank of the pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) associated with the walk vector \(\mathbf{v}\) is equal to the degree of the minimal polynomial of the largest eigenvalue of \(\mathbf{A}\).

We make use once again of Lemma 2. It is well-known (see, for example, [9, Section 0.4.5(c)]) that the rank of the product of two matrices is at most equal to the rank of the one having the smaller rank. This means that if, in Lemma 2, one of \(\mathbf{W}_{\mathbf{b}_1}\) or \(\mathbf{W}_{\mathbf{b}_2}\) has the smallest rank possible, then so will the rank of \({\mathbf{W}_{\mathbf{b}_1}}^{T}\mathbf{W}_{\mathbf{b}_2}\).

Theorem 20. Let \(\mathbf{b}_1\) and \(\mathbf{b}_2\) be indicator vectors of two subsets \(V_1\) and \(V_2\) of \(\mathcal{V}(G)\). Moreover, let \(\mathbf{v}\) be the walk vector of \(V_1\times V_2\). If one of \((\mathbf{A},\mathbf{b}_1)\) or \((\mathbf{A},\mathbf{b}_2)\) (or both) is a recalcitrant pair, then the pair \((\mathbf{A},\mathbf{v})\) is also recalcitrant.

Note that if a graph has \(n\) distinct eigenvalues and its characteristic polynomial factors into exactly two monic polynomials with integer coefficients, then \((\mathbf{A},\mathbf{v})\) is either controllable or recalcitrant for all walk vectors \(\mathbf{v}\). Thus, for such graphs, if the rank of all walk matrices (that is, those having a walk vector with \(0\)-\(1\) entries) is known, then the rank of the pseudo walk matrices whose walk vectors are associated with a subset of \(\mathcal{V}^2\) of the form \(V_1\times V_2\) (\(V_1,V_2\subseteq\mathcal{V}(G)\)) may be inferred by simply applying Theorem 17 or Theorem 20.

We can apply Theorem 20 to regular graphs.

Corollary 5. If \(G\) is a regular graph, then the pair \((\mathbf{A},\mathbf{v})\) is recalcitrant for any walk vector \(\mathbf{v}\) associated with the set \(V\times\mathcal{V}(G)\) for all \(V\subseteq\mathcal{V}(G)\). Moreover, the pseudo walk matrices of all such walk vectors have rank one.

Proof. The rank of the walk matrix \(\mathbf{W}_\mathbf{j}\) associated with a regular graph \(G\) is one; indeed, \(\mathbf{A}\mathbf{j}=\Delta\mathbf{j}\) where \(\Delta\) is the common degree (and the largest eigenvalue) of \(G\). Hence \((\mathbf{A},\mathbf{j})\) is a recalcitrant pair. By Theorem 20, any walk vector \(\mathbf{v}\) associated with \(V\times\mathcal{V}(G)\), where \(V\) is any subset of \(\mathcal{V}(G)\), must also be recalcitrant. Moreover, any such pseudo walk matrices \(\mathbf{W}_\mathbf{v}\) must have the same rank as that of \(\mathbf{W}_\mathbf{j}\), that is, rank one.

From the proof of Corollary 5, we remark that a pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) has rank one if and only if \(\mathbf{v}\) is an eigenvector associated with the (integer) eigenvalue \(\lambda_1\) of \(\mathbf{A}\). Because of this, for a connected regular graph, the pseudo walk matrices of rank one must have walk vectors \(\mathbf{v}\) of the form \(\sqrt{\frac{m}{n}}\mathbf{j}\) for some integer \(m\) between \(1\) and \(n\), so that \(\mathbf{v}^{T}\mathbf{v}\), or \(w_0(S)\), would be an integer between \(1\) and \(n\).

Since non-regular graphs cannot have pseudo walk matrices of rank one by Theorem 15, we have the following corollary by combining this theorem with Corollary 5.

Corollary 6. If a non-regular graph has its largest eigenvalue equal to an integer, then \((\mathbf{A},\mathbf{v})\) is not recalcitrant for any pseudo walk vector \(\mathbf{v}\).

The smallest example of a graph having the properties described in Corollary 6 is the star \(K_{1,4}\) on five vertices, whose characteristic polynomial is \(x^3(x-2)(x+2)\). Using Corollary 6 and Theorem 12, we can infer that the rank of any pseudo walk matrix for \(K_{1,4}\) must be either two or three. Thus, in the case of \(K_{1,4}\), any pair \((\mathbf{A},\mathbf{v})\) for any walk vector \(\mathbf{v}\) is neither controllable nor recalcitrant.

Conflict of interests

The author declares no conflict of interest.

1. Introduction

We follow [1,2] and [3] for the definitions and notations in graph theory and number theory respectively. We denote to the greatest common divisor of two positive integers \(m\) and \(n\) by \((m, n)\). The notion of prime labeling introduced by Entringer and appeared in [4] by Tout, Dabboucy and Howalla in \(1982\). A graph \(G\) with vertex set \(V(G)\) is said to have a prime labeling if there exists a bijective function \(f: V(G) \to\{1,2, \ldots,|V(G)|\}\) such that for every edge \(x y \in E(G), f(x)\) and \(f(y)\) are relatively prime. A graph \(G\) that admits a prime labeling is called a prime graph. The prime labeling of some families of graphs were studied by many authors. Entringer conjectured that all trees have prime labeling. Fu and Huang [5] proved that all complete binary trees are prime. Among the other classes of trees known to have prime labeling are: paths, stars, spiders, olive trees, all trees of order up to \(50,\) palm trees, and banana trees (see [2]).

Seoud and Youssef [6] defined \(R_{n}\) as the maximal prime graph of order \(n\) and denoted to the number of edges in \(R_{n}\) by \(\rho(n)\). They gave an exact formula for \(\rho(n)\) and gave some necessary conditions for a graph to be prime and gave a relation between prime labeling and vertex coloring of graphs. They also conjectured that all unicyclic graphs are prime.

In \(2011,\) Vaidya and Prajapati [7] gave a variation of the definition of prime labeling. They called a graph \(G\) of order \(n\) is \(k-\)prime for some positive integer \(k\) if its vertices can be labeled bijectively by the labels \(k, k+1, \ldots, k+n-1\) such that adjacent vertices receive relatively prime labels. For more details about the known results on prime labelings see [2,8,9,10,11,12,13].

In the following we introduce a new variation of the prime labeling.

Definition 1. A graph \(G\) with vertex set \(V(G)\) is said to have an odd prime labeling if there exists an injective function \(f: V(G) \to\{1,3,5, \ldots, \mid V(G) \mid\}\) such that for every edge \(x y \in E(G)\) \(f(x)\) and \(f(y)\) are relatively prime. A graph \(G\) that admits an odd prime labeling is called an odd prime graph.

We give a generalization of the concept of the prime labeling as follows:

Definition 2. Let \(k\) and \(d\) be positive integers. A graph \(G\) with vertex set \(V(G)\) and of order n is said to have a \((k ,d )-\)prime labeling if there exists an injective function \(f: V(G) \to\{k, k+d, k+2 d,\ldots, k+(n-1) d\} \quad\) such \(\quad\) that \(\quad\) for \(\quad\) every \(\quad\) edge \(x y \in E(G)\), \((f(x), f(y))=1.\) A graph \(G\) that admits a \((k, d)-\)prime labeling is called a \((k, d)-\)prime graph.

With this notation, the \((1,1) -\)prime labeling and the prime labeling coincides; the \((1,2) -\)prime labeling is the same as odd prime labeling and the \((k, 1)-\)prime labeling is the same as the \(k-\)prime labeling. In this paper we deal with the odd prime labeling. We give some necessary conditions for a graph to be odd prime and classify some particular families of graphs according to their prime labeling.

2. Some properties of odd prime graphs

In this section we give some necessary conditions for a graph to be an odd prime.

Theorem 1.

• (i) Every spanning subgraph of an odd prime graph is odd prime graph.
• (ii) Every graph is an induced subgraph of an odd prime graph.

Proof.

• (i) Follows directly from the definition.
• (ii) Let \(G\) be a graph of order \(n \geq 2\) and let \(1< p_{1}< p_{2}< \cdots< p_{n-1}\) be the first \(n-1\) odd prime numbers. As the number of odd integers which are greater than \(1\) and less than \(p_{n-1}\) is equal to \(\dfrac{p_{n-1}-3}{2}\) and the number of odd primes less than \(p_{n-1}\) is equal to \(n-2,\) then the number of composite odd positive integers less than \(p_{n-1}\) is equal to \(\dfrac{p_{n-1}-3}{2}-(n-2)=\dfrac{p_{n-1}+1}{2}-n\). Now label the vertices of \(G\) distinctly by \(1, p_{1}, p_{2}, \cdots, p_{n-1}\). Then add \(m\) isolated vertices to \(G,\) where \(m=\dfrac{p_{n-1}+1}{2}-n\). Finally label these isolated vertices distinctly from \(\left\{2 k+1: 1< 2 k+1< p_{n-1}\right\} \backslash\left\{1, p_{1}, p_{2}, \cdots, p_{n-1}\right\}.\)

Definition 3. Let \(M_{n}\) be the graph whose vertex set \(V\left(M_{n}\right)=\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}\) and whose edge set \(\quad E\left(M_{n}\right)\) is defined as \(v_{i} v_{j} \in E\left(M_{n}\right)\) if and only if \((2 i-1,2 j-1)=1,\) for every \(v_{i}, v_{j} \in V\left(M_{n}\right)\) and \(i< j .\) We call \(M_{n},\) the maximal odd prime graph of order \(n .\) We denote the size of \(M_{n}\) by \(\gamma(n)\). It is clear that \(\gamma(n)\) represents the maximum number of edges in the odd prime graph of order \(n\) and any graph of order \(n\) is isomorphic to a spanning subgraph of \(M_{n}\).

Remark 1. One way to obtain the graph \(M_{n}\) is to label the vertices of the complete graph \(K_{n}\) distinctly by the odd integers \(1,3,5, \ldots, 2 n-1\) and then successively delete any edge whose the labels of its end vertices have a common divisor greater than \(1 .\) Then we can get: \(\gamma(2)=1, \gamma(3)=3, \gamma(4)=6, \gamma(9)=1,\) and \(\gamma(6)=14\). Another way to obtain \(M_{n}\) is from \(M_{n-1}\) by adding a new vertex with label \(2 n-1\) to the graph \(M_{n-1}\) such that this vertex is adjacent to a vertex in \(M_{n-1}\) labeled \(2 i-1,1 \leq i \leq n-1\) if \((2 i-1,2 n-1)=1\). Figure 1 shows the maximal prime graph \(R_8\) and the maximal odd prime graph \(M_{8}\).

For a given positive integer \(n\), let \(X_{n}=\{1 \leq k \leq n:(k, n)=1\}\). This gives that \(\phi(n)=\left|X_{n}\right|,\) where \(\phi(n)\) is the Euler's phi function. If \(n\) is odd integer greater than \(1,\) we may decompose \(X_{n}\) into two disjoint subsets as \(X_{n}=O_{n} \cup E_{n},\) where \(O_{n}\) (resp. \(\left.E_{n}\right)\) is the set of all odd (resp. even ) positive integers through 1 to \(n\) which are relatively prime to \(n\). We denote to \(\left|O_{n}\right|\) by \(\phi_{0}(n)\).

In the following lemma we show that both \(O_{n}\) and \(E_{n}\) have the same number of elements if \(n\) is odd integer greater than \(1\).

Lemma 1. If \(n\) is odd integer such that \(n \geq 3,\) then \(\left|O_{n}\right|=\left|E_{n}\right|=\dfrac{\phi(n)}{2}\).

Proof. Since we have \(k \in O_{n}\) if and only if \(n-k \in E_{n},\) we get the result.

The following result gives an exact formula for \(\gamma(n)\).

Theorem 2. If \(n \geq 2,\) then \(\gamma(n)=\dfrac{1}{2} \displaystyle\sum_{i=2}^{n} \phi(2 i-1)\).

Proof. From Remark 1 above, we have for \(n \geq 3, \gamma(n)=\gamma(n-1)+\phi_{0}(2 n-1),\) with \(\gamma(2)=\phi_{o}(3)=1 .\) Solving this recurrence relation we get \(\gamma(n)=\sum^{n} \phi_{0}(2 i-1)\) and the results follows from Lemma 1.

According to the above theorem we get that all graphs of order not exceeding \(7\) are odd prime except \(K_{5}, K_{6},\) and \(K_{7}\).

Seoud and Youssef [6] showed that \(\beta\left(C_{n}^{2}\right)=\left\lfloor {\dfrac{n}{3}} \right\rfloor, n \geq 4 .\) We shall show that if an edge deleted from \(C_{n}^{2}, n \geq 5,,\) then the vertex independence number of the resulting graph is the same as \(C_{n}^{2}\) except the case when \(n \equiv 2(\bmod 3) .\) In the following theorems we study some more properties for the graph \(M_{n}\).

Theorem 3. For \(n \geq 2, \beta\left(M_{n}\right)=\left\lfloor {\dfrac{n+1}{3}} \right\rfloor,\) where \(\beta(G)\) is the vertex independence number of a graph \(G\).

Proof. If \(2 \leq n \leq 4,\) we have \(M_{n}=K_{n}\) and the results follow. For \(n \geq 5,\) consider the graph \(G=C_{n}^{2}-e\) with vertex set \(\mathrm{V}(G)=\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}\) and edge set \(E(G)=\left\{v_{i} v_{j}: i-j \equiv \pm 1(\bmod n)\right\} \cup\left\{v_{i} v_{i+2}: 1 \leq i \leq n-2\right\} \cup\left\{v_{n} v_{2}\right\} .\) We claim that \(G\) is odd prime graph with \(\beta(G)=\left\lfloor {\dfrac{n+1}{3}} \right\rfloor.\) Define a function \(f: \Gamma(G) \to\{1,3,5, \ldots, 2 n-1\}\) in three cases according to \(n\) modulo 3 as follows:

If \(n \equiv 0(\bmod 3)\)

\[\begin{array}{ll} f\left(v_{3{i-2}}\right)=3(2 i-1), & 1 \leq i \leq \dfrac{n}{3}, \\[2mm] f\left(v_{3{i-1}}\right)=3(2 i-1)-2, & 1 \leq i \leq \dfrac{n}{3}, \\[2mm] f\left(v_{3{i}}\right)=3(2 i-1)+2, & 1 \leq i \leq \dfrac{n}{3}. \end{array} \] If \(n \equiv 1(\bmod 3)\) \[\begin{array}{ll} f\left(v_{3{i-2}}\right)=3(2 i-1), & 1 \leq i \leq \dfrac{n-1}{3}, \\[2mm] f\left(v_{3{i-1}}\right)=3(2 i-1)-2, & 1 \leq i \leq \dfrac{n-1}{3},\\[2mm] f\left(v_{3{i}}\right)=3(2 i-1)+2, & 1 \leq i \leq \dfrac{n-1}{3},\\[2mm] f\left(v_{3{n}}\right)=2n-1. \end{array} \] If \(n \equiv 2(\bmod 3)\) \[\begin{array}{ll} f\left(v_{3{i-2}}\right)=3(2 i-1), & 1 \leq i \leq \dfrac{n+1}{3}, \\[2mm] f\left(v_{3{i-1}}\right)=3(2 i-1)-2, & 1 \leq i \leq \dfrac{n+1}{3}, \\[2mm] f\left(v_{3{i}}\right)=3(2 i-1)+2, & 1 \leq i \leq \dfrac{n-2}{3}. \end{array} \] For \(1\leq i \leq \left\lfloor {\dfrac{n}{3}} \right\rfloor\), we have \begin{equation*} \begin{aligned} \left(f\left(v_{3i-2}\right), f\left(v_{3i-1}\right)\right)&=(3(2 i-1), 3(2 i-1)-2)=1, \\ \left(f\left(v_{3i-1}\right), f\left(v_{3 i}\right)\right)&=(3(2 i-1)-2,3(2 i-1)+2)=1,\\ \left(f\left(v_{3 i-2}\right), f\left(v_{3 i}\right)\right)&=(3(2 i-1), 3(2 i-1)+2)=1. \end{aligned} \end{equation*} Checking the relativelilty prime of the other vertices, showing that the graph \(G\) is odd prime with. For \(n \equiv 2(\bmod 3),\) the set \(\left\{v_{1}, v_{4}, v_{7}, \ldots, v_{n-1}\right\}\) is the maximal independent set in the graph \(G .\) For the other cases we have \(\beta(G)=\beta\left(C_{n}^{2}\right) .\) That is in all cases we have \( \beta(G)=\left\lfloor {\dfrac{{n + 1}}{3}} \right\rfloor . \)

Now, as \(G\) is odd prime, then

For the other side of the inequality, since the set \(\left\{v_{i} \in V\left(M_{n}\right): \quad 1 \leq i \leq n, 3 \mid i\right\}\) is an independent set of vertices in \(M_{n},\) then From (1) and (2), the result follows.

The proof of the following theorem is similar to a proof given by Youssef [15], which concerns the prime labeling.

Theorem 4. For \(n\leq 2,\)

• (i) \(\omega\left(M_{n}\right)=\pi(2 n-1),\) where \(\omega(G)\) is the vertex clique number of a graph \(G\)
• (ii) \(\chi\left(M_{n}\right)=\pi(2 n-1)\), where \(\chi(G)\) is the chromatic number of a graph \(G\).

Proof.

• (i) Let \(p_{1}< p_{2}< \cdots< p_{\pi(2 n-1)}\) be the list of all odd prime numbers \(\leq 2 n-1\), then the induced subgraph of \(M_{n}\) generated by \(U=\left\{v_{i} \in V\left(M_{n}\right): i=1\right.\) or \(i=p_{j}\) for some \(\left.1 \leq j \leq \pi(2 n-1)\right\}\) is the graph \(K_{\pi(2 n-1)},\) hence \(\omega\left(M_{n}\right) \geq \pi(2 n-1) .\) Conversely, for \(1 \leq i \leq \pi(2 n-1)-1\) define, \(\displaystyle V_{i}=\left\{v_{j} \in V\left(M_{n}\right): p_{i} \mid j\right\},\) then \(V(G)=\displaystyle\left\{v_{1}\right\} \cup \bigcup_{i=1}^{\pi(2 n-1)-1} V_{i}=\left\{v_{1}\right\} \cup \bigcup_{i=1}^{\pi(2 n-1)-1}\left(V \backslash \bigcup_{ j< i} V_{j}\right)\) is a partition of \(V\left(M_{n}\right)\) into independent sets and let \(K_{\omega\left(M_{n}\right)}\) be a maximal complete subgraph of \(M_{n},\) then for \(\displaystyle 1 \leq i \leq \pi(2 n-1)-1,\left|V\left(K_{\omega\left(M_{n}\right)}\right) \cap\left(V \backslash \bigcup_{ j< i} V_{j}\right)\right| \leq 1,\) hence \(\left|V\left(K_{\omega\left(M_{n}\right)}\right)\right| \leq \pi(2 n-1),\) that is \(\omega\left(M_n\right)\leq \pi (2n-1).\)
• (ii) An argument similar to that in part \((i)\) shows that \(\chi(M_n)=\pi(2n-1)\).

The following corollary due to Theorem 3 and Theorem 4 gives some necessary conditions for a graph to be an odd prime graph.

Corollary 1. If \(G\) is an odd prime graph of order \(n\), then

• (i) \(\left| {E\left( G \right)} \right| \le \gamma \left( n \right),\)
• (ii) \(\beta \left( G \right) \ge \left\lfloor {\dfrac{{n + 1}}{3}} \right\rfloor ,\)
• (iii) \(\omega(G)\leq \pi (2n-1),\)
• (iv) \(\chi (G)\leq \pi (2n-1)\).

Proof. If \(G\) is an odd prime graph of order \(n\), then \(G\) is a spanning subgraph of \(M_{n}\). That is \(G \subseteq M_{n}\) and the results follow.

3. Odd prime labeling of some special graphs

In this section we deal with the odd prime labeling of some special graphs. The cycle \(C_{n}\) is odd prime for every \(n \geq 3,\) while the following result concerns \(K_{n}\).

Theorem 5. For \(n \geq 2, K_{n}\) is odd prime for every \(n \leq 4\).

Proof. If \(2 \leq n \leq n,\) we label the vertices of \(K_{n}\) by the numbers \(1,3, \ldots, 2 n-1 .\) Conversely, if \(n \geq 5,\) we have \(\beta \left( {{K_n}} \right) = 1 < \left\lfloor {\dfrac{{n + 1}}{3}} \right\rfloor \) and the graph is not odd prime by Corollary 1(ii).

Although the wheel \(W_{n}\) is prime if and only if \(n\) is even, we prove that all wheels are odd prime.

Theorem 6. \(\mathrm{W}_{n}\) is odd prime for every \(n \geq 3\).

Proof. Let \(V\left(W_{n}\right)=\left\{u_{0}, u_{1}, u_{2}, \ldots, u_{n}\right\},\) where \(u_{0}\) is the center vertex and let \(E\left(W_{n}\right)=\left\{u_{0} u_{i}: 1 \leq i \leq n\right\} \cup\left\{u_{i} u_{j}: j-i \equiv \pm 1(\bmod n)\right\}\).

Let \(f: V\left(W_{n}\right) \to \{1,3,5, \ldots, 2 n+1\}\). We have two cases to consider:

Case 1: \(n \not\equiv 1(\bmod 3)\)

Define \(f\) as follows: \(f\left(u_{i}\right)=2 i+1, \quad 0 \leq i \leq n\). As any integer is relatively prime to \(1\) and any two consecutive odd integers are relatively prime, we have to check only the relativity prime of \(f\left(u_{1}\right)\) and \(f\left(u_{n}\right) .\) We have \(\left(f\left(u_{1}\right), f\left(u_{n}\right)\right)=(3,2 n+1)=1,\) since \(2 n+1 \not\equiv 0(\bmod 3)\). Hence \(f\) is an odd prime labeling of \(W_{n}\) in this case.

Case 2: \(n \equiv 1(\bmod 3)\).

Define \(f\) as \(f\left(u_{i}\right)=2 i+1, \quad 0 \leq i \leq n-2, \quad f\left(u_{n-1}\right)=2 n+1, \quad\) and \(\quad f\left(u_{n}\right)=2 n-1\). We can easily check again that \(f\) is an odd prime labeling of \(W_{n}\) in that case.

Theorem 7. \(H_{n}\) is odd prime for every \(n \geq 3\).

Proof. Let \(V\left(H_{n}\right)=\left\{u_{o}, u_{1}, u_{2}, \ldots, u_{n}\right\} \cup\left\{v_{1}, v_{2}, \ldots, v_{n}\right\},\) where \(u_{o}\) is the center vertex and let \[ E\left(H_{n}\right)=\left\{u_{o} u_{i}: 1 \leq i \leq n\right\} \cup\left\{u_{i} u_{j}: i-j \equiv \pm 1(\bmod n)\right\} \cup\left\{u_{i} v_{i}: 1 \leq i \leq n\right\}\,. \] Let \(f: V\left(H_{n}\right) \to\{1,3,5, \ldots, 4 n+1\} .\) We have two cases to consider:

Case 1: \(n \not\equiv 1(\bmod 3)\). Define \(f\) as follows

\[ \begin{aligned} f\left(u_{o}\right)&=1, \\ f\left(u_{i}\right)&=4 i-1, \quad 1 \leq i \leq n, \\ f\left(v_{i}\right)&=4 j+1, \quad 1 \leq j \leq n. \end{aligned} \] As any integer is relatively prime to 1 and any two consecutive odd integers are relatively prime. Also \(\left(f\left(u_{i}\right), f\left(u_{i+1}\right)\right)=1, \quad 1 \leq i \leq n-1\) and \(\left(f\left(u_{1}\right), f\left(u_{n}\right)\right)=(3,4 n-1)=1,\) since \(4 n-1 \not\equiv 0(\bmod 3) .\) Hence \(f\) is an odd prime labeling of \(H_{n}\) in this case.

Case 2: \(n \equiv 1(\bmod 3)\). Define \(f\) as follows:

\[ \begin{aligned} f\left(u_{1}\right)&=1, & & \\ f\left(u_{i}\right)&=4 i-1, & 1 \leq i \leq n-1,& & f\left(u_{n}\right)=4 n+1,& \\ f\left(v_{j}\right)&=4 j+1, & 1 \leq j \leq n-1,& & f\left(v_{n}\right)=4 n-1.& \end{aligned} \] As in Case 1, it remains to check \(\left(f\left(u_{1}\right), f\left(u_{n}\right)\right)\) and \(\left(f\left(u_{n-1}\right), f\left(u_{n}\right)\right).\) Now \[\left(f\left(u_{1}\right), f\left(u_{n}\right)\right)=(3,4 n+1)=1, \;\text{since}\; 4 n+1 \equiv 2(\bmod 3), \] and \[\left(f\left(u_{n-1}\right), f\left(u_{n}\right)\right)=(4 n-5,4 n+1)=(4 n-5,6)=1,\; \text{since}\;4 n-5 \equiv 2(\bmod 3) . \] Thus \(f\) is an odd prime labeling of \(H_{n}\) in that case also.

It is clear that \(K_{1, n}, n \geq 1\) is odd prime for every \(n \geq 1\) and Bertrand's Postulate [3] guarantees the odd prime labeling of \(K_{2, n}\), \(n \geq 2\). Although the smallest values of \(m\) and \(n\) for which \(K_{m, n}\) is not prime is \((m, n)=(3,3)\), however, we show in the following results that the smallest values of \(m\) and \(n\) for which \(K_{m, n}\) is not odd prime is \((m, n)=(7,7)\). The proof of the result is similar to the one given by Fu and Huang [5] and therefore we omit it.

Theorem 8. \(K_{m, n}\), \(3 \leq m \leq n\) is odd prime if and only if \(m \leq \pi(2 m+2 n-1)-\pi\left(\dfrac{2 m+2 n-1}{3}\right)+1.\)

According to the above theorem and the odd prime labeling of \(K_{1, n}\) and \(K_{2, n}\), we can find that for \(2\le m+n\le 20\), \(K_{m,n}\) is odd prime if and only if \((m,n)\neq(7,7)\), \((8,9)\), \((8,10)\), \((9,9)\), \((9,10)\), \((8,12)\), \((9,11)\), and \((10,10)\). Figure 2 shows the odd prime labeling of the complete bipartite graph \(K_{8,8}\)

In 2002, Vilfred et al., [13] conjectured that all ladders \(L_{n}\) are prime. This conjectured was proved in \(2015\) by Dean [8], however in the next theorem we prove that all ladders have odd prime labeling.

Theorem 9. \(L_{n}\) is odd prime for every \(n \geq 1\).

Proof. Let \(V\left(L_{n}\right)=\left\{u_{1}, u_{2}, \ldots, u_{n}\right\} \cup\left\{v_{1}, v_{2}, \ldots, v_{n}\right\},\) and \(E\left(L_{n}\right)=\left\{u_{i} u_{i+1}: 1 \leq i \leq n-1\right\} \cup\left\{v_{i} v_{i+1}: 1 \leq i \leq n-1\right\} \cup\left\{u_{i} v_{i}: 1 \leq i \leq n\right\}.\) Let \(f: V\left(L_{n}\right) \to \{1,3,5, \ldots, 4 n-1\}\) be defined as \[f\left(u_{i}\right)=4 i-3, \quad 1 \leq i \leq n, \quad f\left(v_{j}\right)=4 j-1, \quad 1 \leq j \leq n. \] We have \[ \begin{aligned} \left(f\left(u_{i}\right), f\left(u_{i+1}\right)\right)&=(4 i-3,4 i+1)=(4 i-3,4)=1,& &1 \leq i \leq n-1,&\\ \left(f\left(v_{i}\right), f\left(v_{i+1}\right)\right)&=(4 i-1,4 i+3)=(4 i-1,4)=1,& &1 \leq i \leq n-1,&\\ \left(f\left(u_{i}\right), f\left(v_{i}\right)\right)&=(4 i-3,4 i-1)=(4 i-3,2)=1,& &1 \leq i \leq n.& \end{aligned} \] Hence \(f\) is an odd prime labeling.

Theorem 10. \(T_2 (n)\) is odd prime for every \(n\ge 2\).

Proof. Let the vertices of the ith level of \(T_{2}(n)\) be \(v_{i, 1}, v_{i, 2}, v_{i, 3} \cdots, v_{i, 2^{i}}\) where \(1 \leq i \leq n .\) Define a labeling function \(f\left(V\left(T_{2}(n)\right)\right) \to\left\{1,3, \ldots, 2^{n+1}-3\right\}\) as follows: \[ f\left(v_{p, q}\right)=\left\{\begin{aligned} &1,& & \text { if } p=n, q=2^{n-1},& \\ &(2 q-1)^{n+1-p}+1,& &\text { otherwise }.& \end{aligned}\right. \] To prove that this function is an odd labeling, we note that each vertex in a level \(p,\) say the vertex \(v_{p, q},\) is adjacent to the two vertices \(v_{p+1,2 q-1}\) and \(v_{p+1,2 q}\) where \(1 \leq p< n\). So we have to show that \(\left(f\left(v_{p, q}\right), f\left(v_{p+1,2 q-1}\right)\right)=1\) and \(\left(f\left(v_{p, q}\right), f\left(v_{p+1,2 q}\right)\right)=1\). For the first, we have \[ \begin{aligned} \left(\left(f\left(v_{p, q}\right), f\left(v_{p+1,2 q-1}\right)\right)\right.&=\left((2 q-1) 2^{n+1-p}+1,(4 q-3) 2^{n-p}+1\right) \\ &=\left((2 q-1) 2^{n+1-p}-(4 q-3) 2^{n-p},(4 q-3) 2^{n-p}+1\right) \\ &=\left(2^{n-p}((2 q-1) 2-(4 q-3)),(4 q-3) 2^{n-p}+1\right) \\ &=\left(2^{n-p},(4 q-3) 2^{n-p}+1\right)=1. \end{aligned} \] The second one can be treated similarly.

4. Odd prime labeling of disjoint union of graphs

The graph \(C_{m} \cup C_{n}\) is prime if and only if \(m\) or \(n\) is odd. However, we prove in the following theorem that the disjoint union of any two cycles is odd prime.

Theorem 11. \(\mathrm{C}_{m} \cup \mathrm{C}_{n}\) is odd prime for all \(m, n \geq 3\).

Proof. If \(m \not\equiv 1(\bmod 3)\), we label the vertices of \(C_{m}\) and \(C_{n}\) successively by the labels \(3,5,7, \ldots, 2 m+1\) and \(1,2 m+3,2 m+5, \ldots, 2 m+2 n-1 \) respectively. Otherwise, if \( m \equiv 1(\bmod 3)\), we label the vertices the vertices of \(C_{m}\) and \(C_{n}\) successively by the labels \(3,5,7, \ldots, 2 m-1,2 m+3\) and \( 1,2 m+1,2 m+5,2 m+7, \ldots, 2 m+2 n-1\) respectively. The reader can check easily that the graph is odd prime in the two cases.

Theorem 12. \(K_{m} \cup K_{m}\), \(m \leq n\) is odd prime if and only if \(m+n \leq 7\).

Proof. Necessity, let \(m+n>7\). Since \(\left\lfloor {\dfrac{{m + n + 1}}{3}} \right\rfloor \ge 3 > \beta \left( {{K_m} \cup {K_m}} \right) = 2\), then \(K_{m} \cup K_{m}\) is not odd prime. Sufficiency, let \(m+n \leq 7\), \(m \leq n\) and \(V\left(K_{m} \cup K_{m}\right)=\left\{x_{1}, x_{2}, \ldots, x_{m}\right\} \cup\left\{y_{1}, y_{2}, \ldots, y_{n}\right\}\). Define \(f: V\left(K_{m} \cup K_{m}\right) \to\{1,3, \ldots, 2 m+2 n-1\}\) as \[f\left(x_{i}\right)=2 i+1, \quad 1 \leq i \leq m, \quad f\left(y_{j}\right)=2 m+2 j+1, \quad 1 \leq i \leq m-1, \text{ and } f\left(y_{n}\right)=1. \] Since all positive odd integers not exceeding \(13\) except the integer \(3\) or \(9\) are pairwise relatively prime and since we put the label \(3\) in only one component as \(m\le 3\), this guarantees that \(f\) is an odd prime labeling of \(K_{m} \cup K_{m}\). This completes the proof.

Theorem 13. If \(G\) is odd prime graph of order \(n\), then \(G \cup P_{m}\) is odd prime.

Proof. Let \(V\left(P_{m}\right)=\left\{u_{1}, u_{2}, \ldots, u_{m}\right\}\) and \(f\) be an odd prime labeling for the graph \(G .\) Define a labeling \(g: V\left(G \cup P_{m}\right) \to\{1,3,5, \ldots, 2 n+2 m-1\}\) as follows: \(g(v)=f(v)\) for every vertex \(v \in V(G)\) and \(g\left(u_{i}\right)=2 n+2 i-1\), \(1 \leq i \leq m\). As every two consecutive odd integers are relatively prime, the result follows.

The following result shows that the disjoint union of any number of paths is odd prime.

Corollary 2. \(\displaystyle\bigcup_{i=1}^{n} P_{m_{i}}\) is odd prime.

Theorem 14. If \(G\) is odd prime graph of order \(n\), then \(G \cup K_{3}\) is odd prime.

Proof. Let \(V\left(K_{3}\right)=\left\{u_{1}, u_{2}, u_{3}\right\}\) and \(f\) be an odd prime labeling for the graph \(G .\) Define a labeling \(g: V\left(G \cup P_{m}\right) \to\{1,3,5, \ldots, 2 n+2 m-1\}\) as follows: \(g(v)=f(v)\) for every vertex \(v \in V(G)\) and \(g\left(u_{i}\right)=2 n+2 i-1\), \(1 \leq i \leq 3 .\) As every three consecutive odd integers are pairwise relatively prime, the result follows.

Corollary 3. \(mK_{3}\) is odd prime for every \(m \geq 1\).

The above theorem can be generalized by substitute \(K_3\) by the graph \(C_{2^m+1}\) for every positive integer \(m\).

Theorem 15. For \(m\ge 1\) and \(n\ge 2\), \(mK_n\) is odd prime if and only if (\(m \ge 1\) and \(n= 2\) or \(3\)) or (\(m= 1\) and \(n= 4\)).

Proof. Necessity, if \((m \geq 1\) and \(n \geq 5)\) or \((m \geq 2\) and \(n=4),\) we have \(\left\lfloor {\dfrac{{mn + 1}}{3}} \right\rfloor > m = \beta \left( {m{K_n}} \right).\) Then the graph is not odd prime by Corollary 1(ii). Sufficiency comes by Corollary 2 and Corollary 3.

Comment

In this paper, we showed that some families of graphs like \(K_{4}, W_{2 n+1},\) and \(C_{2 m+1} \cup C_{2 n+1}\) are odd prime, although they are not prime. Thus there are odd prime graphs that are not prime. You might have already asked whether the converse is true. That is there are prime graphs that are not prime? We expect that the answer is no and formalize this in the following conjecture.

Conjecture 1. Every prime graph is odd prime graph.

Authorcontributions

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

Conflictofinterests

The authors declare no conflict of interest.

1. Introduction

Fractional differential equations are more suitable for modeling many real-life phenomena than ordinary differential equations. Fractional calculus has seen significant development over the past few decades. Chaos is a very interesting nonlinear phenomenon which has been intensively studied due to its useful applications in science and technology [1,2,3]. Studying chaos in fractional-order dynamical systems is an interesting topic as well. The discrete fractional equations recently attracted researchers in the modeling of the biological populations and interaction of the species [4,5,6,7]. Harvesting has a strong impact on the dynamic evolution of a population subjected to it. The effects of predator harvesting were reported in [8,9], harvesting among two fish species and its effects were studied in [10].

In this paper, the predator-prey model is considered with harvesting effort affecting both species directly. We studied the following fractional-order predator-prey model.

By using discretization of fractional order using piecewise constant arguments methods [11,12] to (1) fractional model we have the following discrete model where \(0< \alpha \leq 1\) is the fractional-order and \(k\) is the step size of discretization, \(x(t)\) is the population of prey at time \(t\), \(y(t)\) is the population of predator at time \(t\), \(a\) is the rate of growth of prey, \(b\) is the interaction rate, \(c\) is the constant harvesting effort of prey, \(d\) is the mortality rate of predator, \(e\) is the conversion rate of prey, \(f\) is the constant harvesting effort of predator. All parameters are assumed to be positive.

The paper is organized as follows. In Section 2, we investigate the topological classification of the unique positive equilibrium point of the system (2). In Section 3, we discuss period-doubling bifurcation and in Section 4, we discuss Neimark-Sacker bifurcation at positive steady-state. In Section 5, some numerical examples are presented to verify our theoretical results. In Section 6, some concluding remarks are presented.

2. Local stability of positive equilibrium point

The system (2) has three equilibrium points. \[ E_0\left(0,0\right), E_1\left(\frac{a-c}{a},0\right), E_2\left(\frac{d+f}{e},\frac{ae-ad-af-ce}{be}\right). \] Note that if \(ae-ad-af-ce>0\), then \(E_2\) is the unique positive equilibrium point of (2). \(E_0\) is the mutual extinction, \(E_1\) is the predator extinction and \(E_3\) is coexistence equilibrium point of system (2). For bilogically meaningful we concentrated on coexistence equilibrium point \(E_2\). The jacobian matrix of (2) evaluated at \(E_2\) is given by \[ J(E_2)= \begin{bmatrix} 1-\frac{aM(d+f)}{e} & \frac{-bM(d+f)}{e} \\ -\frac{(ce+a(d-e+f))M}{b} & 1 \end{bmatrix}, \] where \(M=\frac{k^{\alpha}}{\Gamma({\alpha+1})}\). The corresponding characteristic polynomial is By simple computations we have \begin{eqnarray*} F(0)&=&1+M^2(d+f)(a-c)-\frac{aM(d+f)(1+M(d+f))}{e}, \\ F(1)&=&\frac{M^2(d+f)(ae-ad-af-ce)}{e}, \\ F(-1)&=&4+M^2(d+f)(a-c)-\frac{aM(d+f)(2+M(d+f))}{e}. \end{eqnarray*} Since \(ae-ad-af-ce>0\), therefore \(F(1)>0\). For stability analysis we use the following result.

Lemma 1. [13]: Let \(F(\lambda)=\lambda^2-A \lambda+B\), where \(A, B\) are constants. Suppose \(F(1)>0\) and \(\lambda_1, \lambda_2\) are roots of \(F(\lambda)=0\), then

• (i) \(|\lambda_1|< 1\) and \(|\lambda_2|< 1\) iff \(F(-1)>0\) and \(F(0)< 1\).
• (ii) \(|\lambda_1|< 1\) and \(|\lambda_2|>1\) iff \(F(-1)< 0\).
• (iii) \(|\lambda_1|>1\) and \(|\lambda_2|>1\) iff \(F(-1)>0\) and \(F(0)>1\).
• (iv) \(\lambda_1=-1\) and \(|\lambda_2|\neq 1\) iff \(F(-1)=0\) and \(F(0)\neq \pm 1\).
• (v) \(\lambda_1\) and \(\lambda_2\) are complex and \(|\lambda_1|=|\lambda_2|=1\) iff \(A^2-4B< 0\) and \(F(0)=1\).

Theorem 1. If \((a,b,c,d,e,f,M)\in \mathbb{R}_+^7, \ ae-ad-af-ce>0\) and \(\lambda_1, \lambda_2\) are the roots of \(F(\lambda)=0\) given by (3), then

• (i) \(|\lambda_1|< 1\) and \(|\lambda_2|< 1\) iff \[ a-\frac{a(1+dM+fM)}{eM}< c< a+\frac{4}{(d+f)M^2}-\frac{a(2+dM+fM)}{eM}, \]
• (ii) \(|\lambda_1|< 1\) and \(|\lambda_2|>1\) iff \[ c>a+\frac{4}{(d+f)M^2}-\frac{a(2+dM+fM)}{eM}, \]
• (iii) \(|\lambda_1|>1\) and \(|\lambda_2|>1\) iff \[ c< min \lbrace a-\frac{a(1+dM+fM)}{eM}, a+\frac{4}{(d+f)M^2}-\frac{a(2+dM+fM)}{eM} \rbrace, \]
• (iv) \(\lambda_1=-1\) and \(|\lambda_2|\neq 1\) iff \[ c=a+\frac{4}{(d+f)M^2}-\frac{a(2+dM+fM)}{eM}, \ c \neq a-\frac{a(1+dM+fM)}{eM}, \ c \neq a+\frac{2}{(d+f)M^2}-\frac{a(1+dM+fM)}{eM}, \]
• (v) \(\lambda_1\) and \(\lambda_2\) are complex and \(|\lambda_1|=|\lambda_2|=1\) iff \[ c=a-\frac{a(1+dM+fM)}{eM}, \ \frac{a(d+f)M}{e}< 4. \]
The following corollary is an immediate consequence of Theorem 1.

Corollary 1. If \((a,b,c,d,e,f,M)\in \mathbb{R}_+^7, \ ae-ad-af-ce>0\), then the unique positive equilibrium point \(E_2\) of (2) is:

• (i) a sink and therefore locally asymptotically stable if \[ a-\frac{a(1+dM+fM)}{eM}< c< a+\frac{4}{(d+f)M^2}-\frac{a(2+dM+fM)}{eM}, \]
• (ii) saddle point and therefore unstable if \[ c>a+\frac{4}{(d+f)M^2}-\frac{a(2+dM+fM)}{eM}, \]
• (iii) source or repeller and therefore unstable if \[ c< min \left\lbrace a-\frac{a(1+dM+fM)}{eM}, a+\frac{4}{(d+f)M^2}-\frac{a(2+dM+fM)}{eM} \right\rbrace, \] and
• (iv) non-hyperbolic if \[ c=a+\frac{4}{(d+f)M^2}-\frac{a(2+dM+fM)}{eM}, \ c \neq a-\frac{a(1+dM+fM)}{eM}, \ c \neq a+\frac{2}{(d+f)M^2}-\frac{a(1+dM+fM)}{eM}, \] or \[ c=a-\frac{a(1+dM+fM)}{eM}, \ \frac{a(d+f)M}{e}< 4. \]

3. Period doubling bifurcation

In this section, we investigate that system (2) undergoes period doubling bifurcation [14,15] at the positive equilibrium point \(E_2\). Consider \begin{eqnarray*} \Lambda&=&\left\lbrace (a,b,c,d,e,f,M) \in \mathbb{R}^7_+\right. \Big| ae-ad-af-ce>0,\;\; c=a+\frac{4}{(d+f)M^2}-\frac{a(2+dM+fM)}{eM}, \\&& c \neq a-\frac{a(1+dM+fM)}{eM},\;\; c \neq a+\frac{2}{(d+f)M^2}\left.-\frac{a(1+dM+fM)}{eM} \right\rbrace. \end{eqnarray*} We discuss the period doubling bifurcation of the system (2) at \(E_2\) when parameters vary in a small neighborhood of \(\Lambda\). Taking \(c\) as bifurcation parameter, we consider a perturbation of the system (2) as follows:
\begin{equation} \label{PD} \begin{cases} x(t+1)=x(t)+M\left[ax(t)(1-x(t))-bx(t)y(t)-(c+\delta)x(t)\right], \\ y(t+1)=y(t)+M\left[-dy(t)+ex(t)y(t)-fy(t)\right], \end{cases} \end{equation}
(4)
where \(M=\frac{k^{\alpha}}{\Gamma({\alpha+1})}\) and \(|\delta| \ll 1\) is small perturbation parameter.

We define a transformation by \(\xi(t)=x(t)-\frac{d+f}{e}\) and \(\eta(t)=y(t)-\frac{a(e-d-f)-(c+\delta)e}{be}\) to transfer the equilibrium point \(E_2\) to \((0,0)\). Under this transformation system (4) will become

\begin{equation} \label{PD1} \begin{bmatrix} \xi(t+1) \\ \eta(t+1) \end{bmatrix}= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} &1 \end{bmatrix} \begin{bmatrix} \xi(t) \\ \eta(t) \end{bmatrix}+ \begin{bmatrix} a_1 \xi^2(t)+a_2 \xi(t) \eta(t) \\ b_1 \xi(t) \delta+b_2 \xi(t) \eta(t) \end{bmatrix}, \end{equation}
(5)
where \( a_{11}=1-\frac{a(d+f)M}{e} , \ a_{12}=-\frac{bM(d+f)}{e}, \ a_{21}=\frac{2(aM(d+f)-2e)}{bM(d+f)}, a_1=-a M, \ a_2= -bM, \ b_1=-\frac{eM}{b}, \ b_2=eM. \) Eigenvalues of linearized part of (5) are \(-1\) and \(\lambda=3-\frac{aM(d+f)}{e}\). We construct an invertible matrix \(T\) as \( T=\begin{bmatrix} \frac{bM(d+f)}{2e-aM(d+f)} & -\frac{bM(d+f)}{2e} \\ 1 & 1 \end{bmatrix}. \) Now we apply the transformation \( \begin{bmatrix} \xi(t) \\ \eta(t) \end{bmatrix}= T \begin{bmatrix} u(t) \\ v(t) \end{bmatrix}, \) to the system (5) and we get
\begin{equation} \label{PD2} \begin{bmatrix} u(t+1) \\ v(t+1) \end{bmatrix}= \begin{bmatrix} -1 & 0 \\ 0 & \lambda \end{bmatrix} \begin{bmatrix} u(t) \\ v(t) \end{bmatrix}+ \begin{bmatrix} F(u(t),v(t),\delta) \\ G(u(t),v(t),\delta) \end{bmatrix}, \end{equation}
(6)
where \begin{eqnarray*} F(u(t),v(t),\delta)=a_1 u^2(t)+a_2 v^2(t)+a_3 u(t)v(t)+a_4 u(t) \delta+a_5 v(t) \delta, \\ G(u(t),v(t),\delta)=b_1 u^2(t)+b_2 v^2(t)+b_3 u(t) v(t)+b_4 u(t) \delta+b_5 v(t) \delta, \end{eqnarray*} with \begin{align*} a_1&=\frac{beM(-aM^2(d+f)^2+2e(-2+dM+fM))}{(2e-a(d+f)M)(4e-a(d+f)M)}, \\ a_2&=\frac{bM(-2e+a(d+f)M)(a(d+f)M+e(-2+dM+fM))}{2e(4e-a(d+f)M)}, \\ a_3&=\frac{ab(d+f)M^2(2+dM+fM)}{8e-2a(d+f)M} ,\\ a_4&=\frac{e(d+f)M^2}{aM(d+f)-4e}, \\ a_5&=-\frac{(d+f)M^2(-2e+a(d+f)M)}{8e-2a(d+f)M},\\ b_1&=\frac{2be^2M(2+dM+fM)}{(2e-a(d+f)M)(4e-a(d+f)M)}, \\ b_2&=-\frac{bM(-4ae(d+f)M+a^2(d+f)^2M^2+2e^2(2+dM+fM))}{2e(4e-a(d+f)M)}, \\ b_3&=\frac{ab(d+f)M^2(a(d+f)M+e(-2+dM+fM))}{(2e-a(d+f)M)(4e-a(d+f)M)},\\ b_4&=-\frac{2e^2M^2(d+f)}{(2e-a(d+f)M)(4e-a(d+f)M)}, \\ b_5&=\frac{e(d+f)M^2}{4e-a(d+f)M}. \end{align*} Next we determine the center manifold \(W^c(0,0)\) of the system (6) at the equilibrium point \((0,0)\) in a small neighbourhood of \(\delta=0\). By using center manifold theory [16,17,18], we know that there exist a center manifold \(W^c(0,0)\), which can be approximately represented as follows: \[ W^c(0,0)=\left\lbrace (u(t),v(t)) \in \mathbb{R}^2 \vert v(t)=W(u(t),\delta)=m_1 u^2(t)+m_2 \delta u(t) +m_3 \delta^2+ O ((|u(t)|+|\delta|)^3) \right\rbrace. \] By using first equation of system (6), we have
\begin{align} v(t+1)=W(u(t+1),\delta)=-m_2 \delta u(t)+m_1 u^2(t)+m_3 \delta^2+O((|u(t)|+|\delta|)^3). \label{PD3} \end{align}
(7)
By using second equation of system (6), we have
\begin{align} v(t+1)=\lambda v(t)+G(u(t),v(t),\delta)=(b_4+\lambda m_2) \delta u(t)+(b_1+\lambda m_1)u^2(t)+\lambda m_3 \delta^2+O((|u(t)|+|\delta|)^3). \label{PD4} \end{align}
(8)
Comparing the coefficients for (7) and (8), we have \[ m_1=\frac{b_1}{1-\lambda}, \ m_2=-\frac{b_4}{1+\lambda}, \ m_3=0.\] Then the center manifold can be approximated as follows
\begin{equation} \label{PD5} v(t)=\frac{b_1}{1-\lambda}u^2(t)-\frac{b_4}{1+\lambda}\delta u(t)+O((|u(t)|+|\delta|)^3). \end{equation}
(9)
Substituting (9) in the first equation of the system (6), we have \begin{align} u(t+1)&=-u(t)+a_1 u^2(t)+a_4 u(t) \delta+\frac{a_3 b_1}{1-\lambda} u^3(t)+ (\frac{a_5 b_1}{1-\lambda}-\frac{a_3 b_4}{1+\lambda}) u^2(t) \delta \nonumber \\ & \;\;\;-\frac{a_5 b_4}{1+\lambda} u(t) \delta^2+O((|u(t)|+|\delta|)^3)=\tilde{F}(u(t),\delta). \nonumber \end{align} For period doubling bifurcation we need the following two quantities \[ L=\left( \frac{\partial^2 \tilde{F}}{\partial u(t) \partial \delta} +\frac{1}{2}\frac{\partial \tilde{F}}{\partial \delta} \frac{\partial^2 \tilde{F}}{\partial u^2(t)}\right)_{u(t)=0,\delta=0}, \] and \[ M=\left(\frac{1}{6}\frac{\partial^3 \tilde{F}}{\partial u^3(t)}+\left(\frac{1}{2}\frac{\partial^2 \tilde{F}}{\partial u^2(t)}\right)^2 \right)_{u(t)=0,\delta=0}. \] By simple computations, we have \[ L=a_4 , \ M=a_1^2+\frac{a_3b_1}{1-\lambda}. \] By using above calculations we have the following result for period doubling bifurcation of the system (2).

Theorem 2. If \((a,b,c,d,e,f,M)\in \Lambda\) and \(L \neq 0, M \neq 0\) then the system (2) goes under period doubling bifurcation at the equilibrium point \(E_2\). Moreover, if \(M>0\) (or \(M< 0\)) then period-2 orbits that bifurcate from the equilibrium point \(E_2\) are stable (or unstable).

4. Neimark-Sacker bifurcation

In this section, we investigate that system (2) undergoes Neimark-Sacker bifurcation [19,20] at the positive equilibrium point \(E_2\). Consider \[ \Omega=\left\lbrace (a,b,c,d,e,f,M) \in \mathbb{R}^7_+ \mid ae-ad-af-ce>0, \ c=a-\frac{a(1+dM+fM)}{eM}, \ \frac{a(d+f)M}{e}< 4 \right\rbrace, \] where \(M=\frac{k^{\alpha}}{\Gamma(\alpha+1)}\).

We discuss the Neimark-Sacker bifurcation of the system (2) at \(E_2\) when parameters vary in a small neighbourhood of \(\Omega\). Taking \(c\) as bifurcation parameter, we consider a perturbation of the system (2) as follows:

\begin{equation} \label{NS1} \begin{cases} x(t+1)=x(t)+M\left[ax(t)(1-x(t))-bx(t)y(t)-(c+\delta)x(t)\right], \\ y(t+1)=y(t)+M\left[-dy(t)+ex(t)y(t)-fy(t)\right], \end{cases} \end{equation}
(10)
where \(M=\frac{k^{\alpha}}{\Gamma({\alpha+1})}\) and \(|\delta| \ll 1\) is small perturbation parameter.

We define a transformation by \(\xi(t)=x(t)-\frac{d+f}{e}\) and \(\eta(t)=y(t)-\frac{a(e-d-f)-(c+\delta)e}{be}\) to transfer the equilibrium point \(E_2\) to \((0,0)\). Under this transformation system (10) will become

\begin{equation} \label{NS2} \begin{bmatrix} \xi(t+1) \\ \eta(t+1) \end{bmatrix}= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & 1 \end{bmatrix} \begin{bmatrix} \xi(t) \\ \eta(t) \end{bmatrix}+ \begin{bmatrix} -bM \xi(t) \eta(t)-aM \xi^2(t) \\ eM \xi(t) \eta(t) \end{bmatrix}, \end{equation}
(11)
where \( a_{11}=1-\frac{aM(d+f)}{e}, \ a_{12}=-\frac{bM(d+f)}{e}, \ a_{21}=\frac{a-eM \delta}{b}. \)

The characteristic equation of the linearized part of the system (11) at equilibrium point \((0,0)\) is

\begin{equation} \label{NS3} \lambda^2-p(\delta) \lambda+q(\delta)=0, \end{equation}
(12)
where \(p(\delta)=2-\frac{aM(d+f)}{e}, \) and \( q(\delta)=1-dM^2 \delta-fM^2 \delta. \)

The roots of (12) are complex with \(|\lambda_{1,2}|=1\) which are given by \( \lambda_{1,2}=\frac{p(\delta)\pm i \sqrt{4q(\delta)-p^2(\delta)}}{2}. \) By computations, we obtain \( |\lambda_1|=|\lambda_2|=\sqrt{q(\delta)}, \) and \( \left(\frac{d|\lambda_1|}{d \delta}\right)_{\delta=0}=\left(\frac{d|\lambda_2|}{d \delta}\right)_{\delta=0}=-\frac{(d+f)M^2}{2}< 0. \)

By simple computations, we can check that \(p(0)=2-\frac{a(d+f)M}{e}\). Since \(\frac{a(d+f)M}{e}< 4\), therefore we have \(-2< p(0)< 2\). If we set \(p(0)=0\), then we obtain \(\frac{a(d+f)M}{e}=2\) and if we set \(p(0)=1\), then we obtain \(\frac{a(d+f)M}{e}=1\). Since \(\lambda_1^m,\lambda_2^m \neq1\) for \(m=1,2,3,4\) at \(\delta=0\) is equivalent to \(p(0) \neq \pm 2, 0,1\), therefore \(\lambda_1^m,\lambda_2^m \neq1\) for \(m=1,2,3,4\) at \(\delta=0\) if \((a,b,c,d,e,f,M)\in \Omega\) and following conditions are satisfied:

\begin{equation} \label{NS3A} \frac{a(d+f)M}{e} \neq 1, 2. \end{equation}
(13)
Now we use the following transformation to convert the linear part of (11) into its canonical form at \(\delta=0\).
\begin{equation} \label{NS4} \begin{bmatrix} \xi(t) \\ \eta(t) \end{bmatrix}= \begin{bmatrix} -\frac{b(d+f)M}{e} & 0 \\ \frac{aM(d+f)}{2e} & -\frac{1}{2} \sqrt{4-(-2+\frac{aM(d+f)}{e})^2} \end{bmatrix} \begin{bmatrix} u(t) \\ v(t) \end{bmatrix}. \end{equation}
(14)
Under the transformation (14), the system (11) will become
\begin{equation} \label{NS5} \begin{bmatrix} u(t+1) \\ v(t+1) \end{bmatrix}= \begin{bmatrix} \alpha & -\beta \\ \beta & \alpha \end{bmatrix} \begin{bmatrix} u(t) \\ v(t) \end{bmatrix}+ \begin{bmatrix} f(u(t),v(t)) \\ g(u(t),v(t)) \end{bmatrix}, \end{equation}
(15)
where \begin{align*} \alpha&=1-\frac{aM(d+f)}{2e}, \\ \beta&=\frac{1}{2}\sqrt{4-(-2+\frac{aM(d+f)}{e})^2}, \\ f(u,v)&=\frac{abM^2(d+f)}{2e} u^2+\frac{bM}{2}\sqrt{4-(-2+\frac{aM(d+f)}{e})^2} u v, \\ g(u,v)&=\frac{abM^3(a+2e)(d+f)^2}{2e\sqrt{aM(d+f)(4e-aM(d+f))}} u^2+\frac{b(a-2e)(d+f)M^2}{2e} u v. \end{align*} We define the following real number \(L\), which analyses the direction in which the closed invariant curve occurs in a system undergoing Neimark-Sacker bifurcation method: \[L=\left( \left[-Re \left( \frac{(1-2 \lambda_1)\lambda_2^2}{1-\lambda_1}\eta_{20}\eta_{11} \right) - \frac{1}{2} \vert \eta_{11} \vert^2 - \vert \eta_{02} \vert^2 + Re(\lambda_2 \eta_{21}) \right] \right)_{\delta=0},\] where \begin{align*} \eta_{20}&=\frac{1}{8}\left[ f_{uu} - f_{vv} + 2g_{uv} + i (g_{uu} - g_{vv} -2f_{uv})\right], \\ \eta_{11}&=\frac{1}{4}\left[ f_{uu} + f_{vv} + i (g_{uu} + g_{vv})\right], \\ \eta_{02}&=\frac{1}{8}\left[ f_{uu} - f_{vv} - 2g_{uv} + i (g_{uu} - g_{vv} + 2f_{uv})\right], \\ \eta_{21}&=\frac{1}{16}\left[ f_{uuu} + f_{uvv} + f_{uuv} + g_{vvv} + i (g_{uuu} + g_{uvv} - f_{uuv} - f_{vvv})\right]. \end{align*} From calculations the computed partial derivatives are \begin{align*} f_{uu}&=\frac{abM^2(d+f)}{e}, \\ f_{uv}&=\frac{bM}{2}\sqrt{4-(-2+\frac{aM(d+f)}{e})^2}, \\ f_{vv}&=f_{uuu}=f_{vvv}=f_{uvv}=f_{uuv}=0, \\ g_{vv}&=g_{uuu}=g_{vvv}=g_{uvv}=g_{uuv}=0, \\ g_{uu}&=\frac{abM^3(a+2e)(d+f)^2}{e\sqrt{aM(d+f)(4e-aM(d+f))}}, \\ g_{uv}&=\frac{bM^2(d+f)(a-2e)}{2e}. \end{align*} Due to above calculations, we have the following result for existence and direction of Neimark-Sacker bifurcation.

Theorem 3. Assume that \((a,b,c,d,e,f,M)\in \Omega\) and conditions (13) are satisfied. If \(L \neq 0\), then the system (2) goes under Neimark-Sacker bifurcation at the unique positive equilibrium point \(E_2\) when the parameter \(c\) varies in a small neighbourhood of \(c=a-\frac{a(1+dM+fM)}{eM}\). Furthermore, if \(L< 0\), then an attracting invariant closed curve bifurcates from the equilibrium point for \(c>a-\frac{a(1+dM+fM)}{eM}\), and if \(L>0\), then a repelling invariant closed curve bifurcates from the equilibrium point for \(c< a-\frac{a(1+dM+fM)}{eM}\).

5. Numerical examples

In this section some interesting numerical examples are provided to validate our theoretical discussions on various qualitative aspects of the model.

Example 1. We select the parameters values as \(a=200, b=0.8, d=2.3, e=40,f=0.5, k=0.15,\alpha=0.95\) and initial conditions \(x(0)=0.1, y(0)=8\). For these values the unique positive equilibrium point of (2) is \((0.07,11.2316)\). The eigenvalues of \(J(E_2)\) for these values are \(\lambda_1=-1, \lambda_2=0.643641\), which confirms that the system (2) undergoes period doubling bifurcation at \((0.07,11.2316)\) as bifurcation parameter passes through \(c=177.015\). We plot bifurcation diagrams for both prey and predator populations for \(c \in [176,180]\). (see Figure 1)

Example 2. We select the parameters values as \(a=200, b=0.8, d=2.3, e=40,f=0.5, k=0.15,\alpha=0.95\) and initial conditions \(x(0)=0.1, y(0)=35\). For these values the unique positive equilibrium point of (2) is \((0.07,37.1336)\). The eigenvalues of \(J(E_2)\) for these values are \(\lambda_1=-0.17818+0.983998i, \lambda_2=-0.17818-0.983998i\) with \(|\lambda_{1,2}|=1\), which confirms that the system (2) undergoes Neimark-Sacker bifurcation at \((0.07,37.1336)\) as bifurcation parameter passes through \(c=156.293\). We plot bifurcation diagrams for both prey and predator populations for \(c \in [156,156.7]\). (see Figure 2 )

Example 3. We select the parameters values as \(a=20, b=10, d=2, e=40, f=1, k=0.11, \alpha=0.9\) and the initial conditions \(x(0)=0.073, y(0)=0.35\). The positive equilibrium point for these values is \((0.075,0.35058)\). For these values the equilibrium point is locally asymptotically stable iff \(14.9942< c< 77.0381\). We plot phase portraits for \(c=14, 14.9942, 15, 15.2 \). Clearly one can see that for \(c=14\), the equilibrium point \((0.35058,0.142621)\) is unstable, whereas for \(c=15\) and \(c=15.2\) it is stable. For \(c=14.9942\), the system undergoes bifurcation. (See figure 3 )

6. Conclusion

We studied a fractional-order predator-prey model with harvesting in both species. We study the existence and local stability of coexistence equilibrium point \(E_2\) of system (2) and their dependence in the form of constant harvesting effort of prey. Moreover, the system (2) goes under period-doubling bifurcation and Neimark-Sacker bifurcation under certain conditions on constant harvesting effort of prey. Numerical examples are presented to support our theoretical results.

conflictofinterests

The author declares no conflict of interest.

1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode's analysis of feedback amplifiers, signal and image processing, capacitor theory, electrical circuits, electron-analytical chemistry, biology, ow in porous media, aerodynamics, viscoelasticity, control theory, fitting of experimental data, and so forth, and involves derivatives of fractional order. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes (for details, see [1,2,3,4,5,6,7,8,9]). The fractional differential equations under various conditions have been studied by ([10,11,12,13]), etc. The three point boundary value problem given by a coupled system of FDE on the interval \((0,1)\) was studied by Bashir [10]

where \(1< \alpha\), \(\beta< 2\), \(p, q, a>0\), \(0< \xi< 1\), \(\alpha-q\geq1\), \(\beta-p\geq1\), \(a\xi^{\alpha-1}< 1\) and \(a\xi^{\beta-1}< 1\). \(D\) is the standard Riemann-Liouville fractional derivative operator and \(f: [0,1]\times \mathbf{R}^{2}\longrightarrow \mathbf{R}^{2}\).

Infinite systems of ODE's was first studied by Persidskii [14] with the aid of classical tools such as successive approximation and the classical Banach fixed point principle. The infinite systems of differential equations emerge in study of various topics of nonlinear analysis. For example semidiscretization of certain parabolic partial differential equation leads to an infinite system of ODE [15], while modeling certain physical phenomenon in theory of neural sets, branching process and mechanics ([16,17]), where the infinite system can be represented as an ordinary differential equation. Consider the following infinite system of fractional differential equations [18]

where each \(u_{i}(t)\) is a differentiable function of class \(C^{[\alpha]+1}\). We will denote the sequence \(\{u_{i}(t)\}^{\infty}_{i=1}=u(t)\), \(\{u_{i}(0)\}^{\infty}_{i=1}=u_{0}\), \(\{u_{i}(\xi)\}^{\infty}_{i=1}=u(\xi)\) and \(\{f_{i}(t,u(t))\}^{\infty}_{i=1}=f(t,u(t))\) which is an element of some Banach sequence space \((E,\|.\|)\).

Motivated by the above works, the aim of this paper is to establish some sufficient conditions for the existence of nontrivial solution for the fractional differential equations (FDE) as follows

where \(1< \alpha< 2\); \(\nu, a>0\), \(\xi\in (0,T)\); \(\alpha-\mu\geq1\) and \(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\). \(D\) is the standard Riemann-Liouville fractional derivative operator and \(f\in C([0,1]\times\mathbf{R}^{2},\mathbf{R})\).

This paper is organized as follows. In Section 2, we present some definitions and lemmas that will be used to prove the results. Then, in Section 3, we present and prove our main results which consists of existence theorems and corollary for nontrivial solution of the FDE 3, and we establish some existence criteria of at least one solution by using the Leray-Schauder nonlinear alternative. Finally, in Section 4, as an application, we give some examples to illustrate the results we obtained.

2. Preliminaries

In this section, we introduce some necessary definitions and lemmas of fractional calculus to facilitate the analysis of the Problem (3). These details can be found in the recent literature, see ([3,7,19,20,21,22,23]) and the references therein.

Definition 1. Let \(\alpha> 0\), \(n-1< \alpha< n\), \(n=[\alpha]+1\) and \(u\in C([0,1), \mathbf{R})\). The Caputo derivative of fractional order \(\alpha\) for the function \(u\) is defined by \[^{c}D^{\alpha}u(t)=\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}(t-s)^{n-\alpha-1}u^{(n)}(s)ds,\] where \(\Gamma(\cdot)\) is the Gamma function.

Definition 2. The Riemann-Liouville fractional integral of order \(\alpha>0\) of a function \(u: (0, \infty)\longrightarrow \mathbf{R}\) is given by \[I^{\alpha}u(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}u(s)ds,~~t>0,\] where \(\Gamma(\cdot)\) is the Gamma function, provided that the right side is pointwise defined on \((0, \infty).\)

Lemma 1.([23]) Let \(\alpha, \beta>0\) and \(u\in L^{p}(0,1)\), \(1\leq p\leq +\infty\). Then the next formulas hold;

• (i) \((I^{\beta}I^{\alpha}u)(t)=I^{\alpha+\beta}u(t),\)
• (ii) \((D^{\beta}I^{\alpha}u)(t)=I^{\alpha-\beta}u(t),\)
• (iii) \((D^{\alpha}I^{\alpha}u)(t)=u(t).\)

Lemma 2. Let \(\alpha>0\), \(n-1< \alpha< n\) and the function \(g: [0; T]\longrightarrow \mathbf{R}\) be continuous for each \(T>0\). Then, the general solution of the fractional differential equation \(^{c}D^{\alpha}g(t)=0\) is given by \[g(t)=c_{0}+c_{1}t+...+c_{n-1}t^{n-1},\] where \(c_{0}, c_{1},..., c_{n-1}\) are real constants and \(n=[\alpha]+1.\)

Lemma 3. Assume that \(u\in C[0,1]\cap L^{1}(0,1)\) with a Caputo fractional derivative of order \(\alpha>0\) that belongs to \(u\in C^{n}[0, 1]\), then \[I^{\alpha}~^{c}D^{\alpha}u(t)=u(t)+c_{0}+c_{1}t+...+c_{n-1}t^{n-1},\] where \(c_{0}, c_{1},..., c_{n-1}\) are real constants and \(n=[\alpha]+1.\)

Lemma 4. For \(\alpha>0\), the general solution of the fractional differential equation \(D^{\alpha}u(t)=0\) with \(u\in C[0,1]\cap L^{1}(0,1)\) is given by \[u(t)=c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2}+...+c_{n}t^{\alpha-n},\] where \(c_{i}\in\mathbf{R},~~i=1,2,...,n\). Hence for \(u\in C[0,1]\cap L^{1}(0,1)\), we have \[I^{\alpha}D^{\alpha}u(t)=u(t)+c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2}+...+c_{n}t^{\alpha-n}.\]

Lemma 5. Let \(y\in C([0,T])\), \(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\). Then FDE \[ \begin{cases} D^{\alpha}u(t)=y(t),& t\in(0,T)\\ u(0)=0,\; u(T)=au(\xi), & \end{cases} \] has a unique solution \begin{eqnarray*}u(t)&=&\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\left[(t-s)^{\alpha-1}-\frac{(t(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right]y(s)ds-\frac{1}{(T^{\alpha-1} -a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t}^{T}(t(T-s))^{\alpha-1}y(s)ds \\&&+\frac{a}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(t(\xi-s))^{\alpha-1}y(s)ds.\end{eqnarray*}

Proof. ([10])The general solution of FDE is \[u(t)=I^{\alpha}y(t)+c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2},~where~~c_{1}, c_{2}\in\mathbf{R}.\] Using the boundary conditions, we find that \(c_{2}=0\) and \[c_{1}=-\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})}\left[\int_{0}^{T}\frac{y(s)ds}{(T-s)^{\alpha-1}\Gamma(\alpha)}-a\int_{0}^{\xi}\frac{y(s)ds}{(\xi-s)^{\alpha-1}\Gamma(\alpha)}\right].\] Substituting \(c_{1}\) and \(c_{2}\) by their values in \(u(t)\), we obtain the solution in the statement of the lemma. This completes the proof.

Define the integral operator \(F: E\rightarrow E\), by

\begin{eqnarray*}Fu(t)&=&\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\left[(t-s)^{\alpha-1}-\frac{(t(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right]f(s,v(s),D^{\nu}v(s))ds\\ &&-\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t}^{T}(t(T-s))^{\alpha-1}f(s,v(s),D^{\nu}v(s))ds\\ &&+\frac{a}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(t(\xi-s))^{\alpha-1}f(s,v(s),D^{\nu}v(s))ds.\end{eqnarray*} By Lemma 5, the FDE (3) has a solution if and only if the operator \(F\) has a fixed point in \(E\). So we only need to seek a fixed point of \(F\) in \(E\). By Ascoli-Arzela theorem, we can prove that \(F\) is a completely continuous operator. Now we cite the Leray-Schauder nonlinear alternative.

Lemma 6. Let \(E\) be a Banach space and \(\Omega\) be a bounded open subset of \(E\), \(0\in\Omega\). \(F:\overline{\Omega}\rightarrow E\) be a completely continuous operator. Then, either

• (i) there exists \(u\in \partial \Omega\) and \(\lambda>1\) such that \(F(u)=\lambda u\), or
• (ii) there exists a fixed point \(u^{\ast}\in \overline {\Omega}\) of \(F\).

3. Main results

In this section, we prove the existence of a nontrivial solution for the FDE (3). Let \(E=C([0,T])\) with the norm \(\|v\|=\max_{t\in[0,T]}\{|v(t)|,|D^{\nu}v(t)|\}\) for any \(v\in E\), \(f\in C([0,T]\times\mathbf{R}^{2},\mathbf{R}).\)

Theorem 1. Suppose that \(f(t,0,0)\neq 0\), \(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\), and there exist nonnegative functions \(k,h,l \in L^{1}[0,T]\) such that \[|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x,y)\in[0,T]\times \mathbf{R}^{2},\] and \[\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))ds< 1.\] Then the FDE (3) has at least one nontrivial solution \(u^{\ast}\in C([0,T]).\)

Proof. Let \[A=\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))ds,\] and \[B=\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}l(s)ds+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}l(s)ds,\] then \(A< 1\). Since \(f(t,0,0)\neq 0\), there exists an interval \([a,b]\subset [0,1]\) such that \(\min_{a\leq t\leq b}|f(t,0,0)|>0\), and, as \(l(t)\geq |f(t,0,0)|\), a.e., and \(t\in [0,T]\), so \(B>0\).

Let \(C=B(1-A)^{-1}\) and \(\Omega=\{(u, v)\in E^{2}: \|(u, v)\|_{E^{2}}< C\}\). Assume that \(u\in \partial \Omega\) and \(\lambda>1\) such that \(Fu=\lambda u\), then

\begin{eqnarray*}\lambda C&=&\lambda \|u\|=\|Fu\|=\max_{0\leq t\leq T}|(Fu)(t)|\\ &\leq&\frac{1}{\Gamma(\alpha)}\max_{t\in[0,T]}\int_{0}^{t}\left|(t-s)^{\alpha-1}-\frac{(t(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right||f(s,v(s),D^{\nu}v(s))|ds\\ &&+\max_{t\in[0,T]}\frac{1}{|T^{\alpha-1}-a\xi^{\alpha-1}|\Gamma(\alpha)}\int_{t}^{T}(t(T-s))^{\alpha-1}|f(s,v(s),D^{\nu}v(s))|ds\\ &&+\max_{t\in[0,T]}\frac{a}{|T^{\alpha-1}-a\xi^{\alpha-1}|\Gamma(\alpha)}\int_{0}^{\xi}(t(\xi-s))^{\alpha-1}|f(s,v(s),D^{\nu}v(s))|ds\\ &\leq&\frac{1}{\Gamma(\alpha)}\int_{0}^{T}\left[(T-s)^{\alpha-1}+\frac{(T(T-s))^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})}\right]|f(s,v(s),D^{\nu}v(s))|ds\\ &&+\frac{a}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(T(\xi-s))^{\alpha-1}|f(s,v(s),D^{\nu}v(s))|ds\\ &\leq&\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)|v(s)|+h(s)|D^{\nu}v(s)|+l(s))ds\\ &&+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)|v(s)|+h(s)|D^{\nu}v(s)|+l(s))ds\\ &\leq&\left[\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))\|v\|ds\right.\\ &&\left.+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))\|v\|ds\right]\\ &&+\left[\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}l(s)ds\right.\left.+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}l(s)ds\right]\\ &=&A\|v\|+B.\end{eqnarray*} Therefore, \(\lambda \leq A+\frac{B}{C}=A+\frac{B}{B(1-A)^{-1}}=A+(1-A)=1.\) This contradicts \(\lambda>1\). By Lemma 6, \(F\) has a fixed point \(u^{\ast}\in\overline{\Omega}\). In view of \(f(t,0,0)\neq0\), the FDE (3) has a nontrivial solution \(u^{\ast}\in E\).

Now, we prove that the operator \(F\) is completely continuous, we have \(B_{C}=\{v \in E: ||v||\leq C\}\) is a bounded closed convex set of \(E\). We shall prove that \(F(B_{C})\) is relatively compact. The proof will be done is some steps.

• (i) Let \(v\in B_{C}\), we have \(|Fu(t)|\leq A\|v\|+B.\) Consequently \(F(B_{C})\) is uniformly bounded.
• (ii) Let us prove that \(F(B_{C})\) is equicontinuous. Let \(t_{1}, t_{2}\in[0, T], \;\text{with}\;t_{1}< t_{2},\;\text{and}\; v\in B_{C}\), we have

\begin{eqnarray*}\bigg|Fu(t_{1})-Fu(t_{2})\bigg|&=&\bigg|\frac{1}{\Gamma(\alpha)} \int_{0}^{t_{1}}\left[(t_{1}-s)^{\alpha-1}-\frac{(t_{1}(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right] f(s,v(s), D^{\nu}v(s))ds \\ &&-\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{1}}^{T} (t_{1}(T-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\\ &&+\frac{a}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi} (t_{1}(\xi-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\\ &&-\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{2}}\left[(t_{2}-s)^{\alpha-1}-\frac{(t_{2}(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right]f(s,v(s), D^{\nu}v(s))ds \\ &&+\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{2}}^{T} (t_{2}(T-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\\ &&-\frac{a}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi} (t_{2}(\xi-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\bigg|\\ &=&\bigg|\frac{1}{\Gamma(\alpha)} \int_{0}^{t_{1}}\left[(t_{1}-s)^{\alpha-1}-\frac{(t_{1}(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right] f(s,v(s), D^{\nu}v(s))ds \\ &&-\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{1}}^{T} (t_{1}(T-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\\ &&-\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{2}}\left[(t_{2}-s)^{\alpha-1}-\frac{(t_{2}(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right]f(s,v(s), D^{\nu}v(s))ds \\ &&+\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{2}}^{T} (t_{2}(T-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\\ &&+\frac{a(t_{1}^{\alpha-1}-t_{2}^{\alpha-1})}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\bigg|\\ &\leq&\frac{1}{\Gamma(\alpha)} \int_{0}^{t_{1}}\left[(t_{1}-s)^{\alpha-1}+\frac{(t_{1}(T-s))^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})}\right] |f(s,v(s), D^{\nu}v(s))|ds \\ &&+\frac{1}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{1}}^{T} (t_{1}(T-s))^{\alpha-1} |f(s,v(s), D^{\nu}v(s))|ds\\ &&+\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{2}}\left[(t_{2}-s)^{\alpha-1}+\frac{(t_{2}(T-s))^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})}\right]|f(s,v(s), D^{\nu}v(s))|ds\\ &&+\frac{1}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{2}}^{T} (t_{2}(T-s))^{\alpha-1} |f(s,v(s), D^{\nu}v(s))|ds\\ &&+\frac{a|t_{1}^{\alpha-1}-t_{2}^{\alpha-1}|}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1} |f(s,v(s), D^{\nu}v(s))|ds.\end{eqnarray*} Therefore, \begin{eqnarray*}&&\bigg|Fu(t_{1})-Fu(t_{2})\bigg|\leq\frac{1}{\Gamma(\alpha)} \int_{t_{2}}^{t_{1}}\left[((t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha-1})\right.\left.+\frac{[(t_{1}(T-s))^{\alpha-1}-(t_{2}(T-s))^{\alpha-1}]} {(T^{\alpha-1}+a\xi^{\alpha-1})}\right]\\&&\times|f(s,v(s), D^{\nu}v(s))|ds +\frac{1}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{1}}^{t_{2}} [(t_{1}(T-s))^{\alpha-1}-(t_{2}(T-s))^{\alpha-1}]\\&&\times |f(s,v(s), D^{\nu}v(s))|ds+\frac{a|t_{1}^{\alpha-1}-t_{2}^{\alpha-1}|}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1} |f(s,v(s), D^{\nu}v(s))|ds.\end{eqnarray*} Letting \(t_{1}\rightarrow t_{2}\), then \(|Fu(t_{1})-Fu(t_{2})|\) tends to \(0\). Consequently \(F( B_{C})\) is equicontinuous. From Ascoli-Arzela theorem, we deduce that \(F\) is a completely continuous. This completes the proof.

Theorem 2. Suppose that \(f(t,0,0)\neq0\), \(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\), and there exist nonnegative functions \(k,h,l\in L^{1}[0,T]\) such that \(|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x,y)\in [0,T]\times\mathbf{R}^{2}.\) If one of the following conditions is fulfilled;

1. There exists a constant \(p>1\) such that \[\int_{0}^{1}(k(s)+h(s))^{p}ds< \left[\frac{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}} {(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}+aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}\right]^{p},~\frac{1}{p}+\frac{1}{q}=1,\]
2. \(k(s)+h(s)\) satisfies \[k(s)+h(s)\leq\frac{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}},\quad a.e.~~~s\in [0,T],\] \[meas\left\{s\in[0,T] : k(s)+h(s)< \frac{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}}\right\}>0.\] Then the FDE (3) has at least one nontrivial solution \(u^{\ast}\in E.\)

Proof. Let \(A\) be defined as in the proof of Theorem 1. To prove Theorem 2, we only need to prove that \(A< 1\). Since \(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\), we have \[A=\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds +\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))ds.\]

1. Using the Hölder inequality, we have \begin{align*}A&\leq\left[\int_{0}^{1}(k(s)+h(s))^{p}ds\right]^{1/p}\left\{\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\left[\int_{0}^{T}\left((T-s)^{\alpha-1}\right)^{q}ds\right]^{1/q}\right.\\ &\;\;\;\left.+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\left[\int_{0}^{\xi}\left((\xi-s)^{\alpha-1}\right)^{q}ds\right]^{1/q}\right\}\\ &\leq\left[\int_{0}^{1}(k(s)+h(s))^{p}ds\right]^{1/p}\left\{\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\left[\frac{T^{1+q(\alpha-1)}}{(1+q(\alpha-1))}\right]^{1/q}\right.\\ &\;\;\;\left.+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\left[\frac{\xi^{1+q(\alpha-1)}}{1+q(\alpha-1)}\right]^{1/q}\right\}\\ &\leq\left[\int_{0}^{1}(k(s)+h(s))^{p}ds\right]^{1/p}\left[\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}\right. \left.+\frac{aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}\right]\\ &\leq\frac{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}+aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}\times \frac{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}+aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}\\ &=1. \end{align*}
2. In this case, we have \begin{align*}A&\leq\frac{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}}\left[\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}ds\right.\\ &\;\;\;\left.+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}ds\right] \end{align*} \begin{align*}&\leq\frac{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}}\left[\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}}{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})} +\frac{aT^{\alpha-1}\xi^{\alpha}}{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}\right]\\ &\leq\frac{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}} .\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}}{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}=1.\end{align*} This completes the proof.

Corollary 1. Suppose \(f(t,0,0)\neq0\), \((1+a)T^{\alpha-1}\neq0\), and there exist nonnegative functions \(k, h, l\in L^{1}[0,T]\) such that \(|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x,y)\in[0,T]\times \mathbf{R}^{2}.\) If one of following conditions is fulfilled;

1. There exists a constant \(p>1\) such that \[\int_{0}^{1}(k(s)+h(s))^{p}ds< \left[\frac{(1+a)T^{\alpha-1}\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}{2(1+a)T^{\alpha-1}T^{(1+q(\alpha-1))/q}}\right]^{p},~\frac{1}{p}+\frac{1}{q}=1.\]
2. \(k(s)+h(s)\) satisfies \[k(s)+h(s)\leq\frac{\alpha\Gamma(\alpha)}{2T^{\alpha}},\quad a.e.~~~s\in [0,T],\] \[meas\left\{s\in[0,T] : k(s)+h(s)< \frac{\alpha\Gamma(\alpha)}{2T^{\alpha}}\right\}>0.\] Then, the FDE (3) has at least one nontrivial solution \(u^{\ast}\in E.\)

Proof. In this case, we have \begin{align*}A&=\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds +\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))ds\\ &\leq\frac{2T^{\alpha-1}+aT^{\alpha-1}}{(T^{\alpha-1}+aT^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds +\frac{aT^{\alpha-1}}{(T^{\alpha-1}+aT^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds\\ &=\frac{2(1+a)T^{\alpha-1}}{(1+a)T^{\alpha-1}\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds.\end{align*} Proof of this Corollary 1 is similar to the proof Theorem 2. This completes the proof.

4. Applications

In order to illustrate the above results, we consider some examples.

Example 1. Consider the following system of FDE

Set \(\alpha=3/2\), \(a=2\), \(\xi=T/2\), and \[f(t,x,y)=\frac{t}{207}x(t)+\frac{t+2}{100}y(t)+t^{2}-1,\] \[k(t)=\frac{t}{100},\quad h(t)=\frac{t+2}{100},\quad l(t)=t^{2}.\] It is easy to prove that \(k, h, l\in L^{1}[0,T]\) are nonnegative functions, and \[|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x,y)\in [0,T]\times\mathbf{R}^{2},\] and \[T^{\alpha-1}+a\xi^{\alpha-1}=(1+\frac{2}{2^{1/2}})T^{1/2}\neq0.\] Moreover, we have \[A=\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))ds,\] \[A\approx 13.10^{-3}.T^{3/2}+4.10^{-3}.T^{5/2}< 1.\] Hence, by Theorem 1, the FDE (4) has at least one nontrivial solution \(u^{\ast}\) in \(E.\)

Example 2. Consider the following system of FDE

Set \(\alpha=1/2\), \(a=4\), \(\xi=T/3\), and \[f(t,x,y)=\frac{\sqrt[3]{1+t^{5}}}{20}x(t)\sin x(t)+\frac{\sqrt[3]{1+t^{5}}}{5}y(t)+\cos t-e^{t},\] \[k(t)=\sqrt[3]{1+t^{5}}/10,\quad h(t)=\sqrt[3]{1+t^{5}}/4,\quad l(t)=\cos t+e^{t}.\] It is easy to prove that \(k, h, l\in L^{1}[0,T]\) are nonnegative functions, and \[|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x,y)\in [0,T]\times\mathbf{R}^{2},\] and \[T^{\alpha-1}+a\xi^{\alpha-1}=(1+\frac{4}{3^{-1/2}})T^{-1/2}\neq0.\] Let \(p=3,~q=3/2\), such that \(\frac{1}{p}+\frac{1}{q}=1,\) then \[\int_{0}^{1}(k(s)+h(s))^{p}ds=\frac{2401}{48000}\approx 0.05.\] Moreover, we have \[\left[\frac{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}} {(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}+aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}\right]^{p}\approx0.51.T^{-1/2}.\] Therefore, \[\int_{0}^{1}(k(s)+h(s))^{p}ds< \left[\frac{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}} {(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}+aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}\right]^{p}.\] Hence, by Theorem 2(1), the FDE (5) has at least one nontrivial solution \(u^{\ast}\) in \(E.\)

Example 3. Consider the following system of FDE

1. Introduction

Fibonacci polynomials [1] are special cases of Chebyshev polynomials and are defined recursively by

for all \(n\geq2\) with \(F_0(x)=0\) and \(F_1(x)=1.\)

Lucas polynomials [1] are defined by

for all \(n\geq2\) with \(L_0(x)=2\) and \( L_1(x)=x.\)

Pell polynomials [2] are defined by

for all \(n\geq 2\) with \(P_0(x)=0\) and \(P_1(x)=1.\)

Generating function of Fibonacci polynomials is given by

Explicit sum formula for Fibonacci polynomials is given by Horadam polynomials sequence [3] \(h_n(x),\) for \(n\geq3\) is defined by recurrence relations \(h_n(x)=pxh_{n-1}(x)+qh_{n-2}(x),\) with initial conditions \(h_1(x)=a\) and \(h_2(x)=bx,\) where \(p,\) \(q,\) \(a\) and \(b\) are integers.

Generalized Fibonacci polynomials and some identities [4] are defined by

\[w_n(x)=xw_{n-1}(x)+w_{n-2}(x),\] for \(n\geq2\) with \(w_0(x)=2b\) and \(w_1(x)=a+b,\) where \(a\) and \(b\) are integers.

Generalized Fibonacci-Like polynomials [5] are defined by the recurrence relation:

\[m_n(x)=xm_{n-1}(x)+m_{n-2}(x),\] with \(m_0(x)=2s\) and \(m_1(x)=1+s,\) where \(s\) is an integer.

Fibonacci-like polynomials [6] are defined by the recurrence relation

\[S_n(x)=xS_{n-1}(x)+S_{n-2}(x),\] \(n\geq2\) with \(S_0(x)=2\) and \(S_1(x)=2x. \)

Generalized Fibonacci polynomials [7] are defined by the recurrence relation

\[b_n(x)=xb_{n-1}(x)+b_{n-2}(x),\] \(n\geq2\) with \(b_0(x)=2b\) and \(b_1(x)=s,\) where \(b\) and \(s\) are integer.

In this paper, generalized Fibonacci polynomials is studied by varying both the recurrence relation and initial conditions. The properties of these polynomials are derived by means of Binet's formula and generating function. Few terms of generalized Fibonacci polynomials and characteristic equation of the recurrence relation are presented in Section 2. In Section 3 Binet's formula is obtained and generating function is also obtained in Section 4. Further some properties of these polynomials are presented in Section 5 and finally in Section 6 conclusion is given.

2. Generalized Fibonacci Polynomials

We define generalized Fibonacci polynomials by the recurrence relation with initial conditions \(R_0(x)=2p\) and \(R_1(x)=px+q\) where \(a\) and \(b\) are positive integers and \(p\) and \(q\) are non negative integers.

Few terms of generalized Fibonacci sequence are as follows:

\begin{align*} R_0(x)&=2p,\\ R_1(x)&=px+q,\\ R_2(x)&=apx^2+aqx+2bp,\\ R_3(x)&=a^2px^3+a^2qx^2+2abpx+bpx+bq,\\ R_4(x)&=a^3px^4+a^3qx^3+2a^2bpx^2+abpx^2+abqx+abpx^2+abqx+2b^2p \end{align*} and so on.

The characteristic equation for the recurrence relation (6) is

This equation has two real and distinct roots; \[\alpha(x)=\dfrac{ax+\sqrt{a^2x^2+4b}}{2}\] and \[\beta(x)=\dfrac{ax-\sqrt{a^2x^2+4b}}{2}.\] Also from these roots, we have \begin{align*} \alpha(x)\beta(x)&=-b,\\ \alpha(x)+\beta(x)&=ax,\\ \alpha(x)-\beta(x)&=\sqrt{a^2x^2+4b},\\ \alpha^2(x)+\beta^2(x)&=a^2x^2+2b,\\ \alpha^2(x)&=ax\alpha(x)+b,\\ \beta^2(x)&=ax\beta(x)+b. \end{align*} Generalized Fibonacci polynomials (6) generalizes, Fibonacci polynomials, Lucas polynomials and Pell polynomials at different values of \(a,\) \(b,\) \(p\) and \(q.\)

• For \(a=b=q=1\) and \(p=0,\) we obtain Fibonacci polynomials.
• For \(a=b=p=1\) and \(q=0,\) we obtain Lucas polynomials.
• For \(a=2,\) \(b=q=1\) and \(p=0,\) we obtain Pell polynomials.

Further for \(x=1\) we obtain the corresponding sequences of these polynomials.

3. Binet's formula for generalized Fibonacci polynomials.

Theorem [Binet's formula] The \(n^{th}\) term of generalized Fibonacci polynomials is given by

Proof. The characteristics Equation (7) has real and distinct roots. The solution of the recurrence relation (6) is therefore of the form

where \(A\) and \(B\) are constants and \(\alpha(x)=\dfrac{ax+\sqrt{a^2x^2+4b}}{2}\) and \(\beta(x)=\dfrac{ax-\sqrt{a^2x^2+4b}}{2}\).

Setting \(n=0\) and \(n=1\) in (9), we obtain

\[A+B=R_0(x)\] and \[A\alpha(x)+B\beta(x)=R_1(x)\] respectively.

Solving these equations simultaneously, we obtain

\[A=\dfrac{R_1(x)- \beta(x)R_0(x)}{\alpha(x)-\beta(x)}\] and \[B=\dfrac{\alpha(x)R_0(x)-R_1(x)}{\alpha(x)-\beta(x)}.\] Substituting for \(A\) and \(B\) in (9), we get \begin{align*} R_n(x)&={\left(\dfrac{R_1(x)-\beta(x)R_0(x)}{\alpha(x)-\beta(x)}\right)}\alpha^n(x)-{\left(\dfrac{R_1(x)-\alpha(x)R_0(x)}{\alpha(x)-\beta(x)}\right)}\beta^n(x)\\ &=\dfrac{1}{\alpha(x)-\beta(x)}\left[R_1(x)(\alpha^n(x)-\beta^n(x))+bR_0(x)(\alpha^{n-1}(x)-\beta^{n-1}(x))\right]. \end{align*} Hence the proof.

Remark 1. We have that

and

4. Generating function for generalized Fibonacci polynomials

Theorem 2. [Generating function] Generating function for generalized Fibonacci polynomials is given by

Proof. Applying power series to the generalized Fibonacci polynomial \(\displaystyle\sum_{n=0}^{\infty}R_n(x)t^n,\) we have \[2p+(px+q)t+(ax^2p+axq+2bp)t^2+\cdots=\displaystyle\sum_{n=0}^{\infty}R_n(x)t^n. \] Now, multiplying the generating series by \((1-axt-bt^2),\) where \((1-axt-bt^2) \neq 0,\) we get \begin{align*} &(1-axt-bt^2)\displaystyle\sum_{n=0}^{\infty}R_n(x)t^n=\displaystyle\sum_{n=0}^{\infty}R_n(x)t^n-ax\displaystyle\sum_{n=0}^{\infty}R_n(x)t^{n+1}-b\displaystyle\sum_{n=0}^{\infty}R_n(x)t^{n+2}\\ &={\left[R_0(x)+R_1(x)t +\displaystyle\sum_{n=2}^{\infty}R_n(x)t^n\right]}-ax{\left[R_0(x)t+\displaystyle\sum_{n=2}^{\infty}R_{n-1}(x)t^n\right]}-b\displaystyle\sum_{n=2}^{\infty}R_{n-2}(x)t^n\\ &=R_0(x)+{\left[R_1(x)-axR_0(x)\right]}t+\displaystyle\sum_{n=2}^{\infty}{\left[R_n(x)-axR_{n-1}(x)-bR_{n-2}(x)\right]}t^n\\ &=2p+{\left[(px+q)-2apx\right]}t+\displaystyle\sum_{n=2}^{\infty}{\left[axR_{n-1}(x)+bR_{n-2}(x)-axR_{n-1}(x)-bR_{n-2}(x)\right]}\\ &=2p+{\left[(px+q)-2apx\right]t}. \end{align*} Therefore \[{\left(1-axt-bt^2\right)}\displaystyle\sum_{n=0}^{\infty}R_n(x)t^n=2p+{\left[(px+q)-2apx\right]t}.\] Hence \[\displaystyle\sum_{n=0}^{\infty}R_n(x)t^n=\dfrac{2p+[(px+q)-2apx]t}{1-axt-bt^2}.\]

Remark 2. If \(a=b=q=1\) and \(p=0\) in (15), we obtain generating functions for Fibonacci polynomials (4).

5. Some properties of generalized Fibonacci polynomials

In this section, we obtain some properties of generalized polynomials by means of Binet's formula and generating function.

Proposition 1. [Explicit sum formula] Let \(R_n(x)\) be the \(n^{th}\) generalized Fibonacci polynomials, then

where \(\left\lfloor n \right\rfloor\) is the greatest integer less than or equal to \(n.\)

Proof. By generating function (15), we have \begin{align*} \displaystyle\sum_{n=0}^{\infty}R_n(x)t^n&=\dfrac{2p+(px+q-2apx)t}{1-axt-bt^2}\\ &={\left[2p+(px+q-2apx)t\right]}{\left[1-(ax+bt)t\right]^{-1}}\\ &={\left[2p+(px+q-2apx)t\right]}\displaystyle\sum_{n=0}^{\infty}(ax+bt)^nt^n\\ &={\left[2p+(px+q-2apx)t\right]}\displaystyle\sum_{n=0}^{\infty}t^n\displaystyle\sum_{k=0}^n{n\choose k}(ax)^{n-k}(bt)^k\\ &={\left[2p+(px+q-2apx)t\right]}\displaystyle\sum_{n=0}^{\infty}\displaystyle\sum_{k=0}^n \dfrac{n!}{k!(n-k)!}(ax)^{n-k}b^kt^{n+k}. \end{align*} Now replacing \(n\) with \(n+k, \) we get \begin{align*} \displaystyle\sum_{n=0}^{\infty}R_n(x)t^n&={\left[2p+(px+q-2apx)t\right]}\displaystyle\sum_{n=0}^{\infty}\displaystyle\sum_{k=0}^{\infty} \dfrac{(n+k)!}{k!n!}(ax)^n b^k t^{n+2k}\\ &={\left[2p+(px+q-2apx)t\right]}\displaystyle\sum_{n=0}^{\infty}\displaystyle\sum_{n=0}^{\left\lfloor\dfrac{n}{2}\right\rfloor} \dfrac{(n-k)!}{k!(n-2k)!} (ax)^{n-2k} b^k t^n. \end{align*} Thus the sum equals to \(\displaystyle\sum_{n=0}^{\infty}{\left[2p\displaystyle\sum_{k=0}^{\left\lfloor\dfrac{n}{2}\right\rfloor} \dfrac{(n-k)!}{k!(n-2k)!} (ax)^{n-2k} b^k\right]t^n}+\displaystyle\sum_{n=0}^{\infty}{\left[(px+q-2apx)\displaystyle\sum_{k=0}^{\left\lfloor\dfrac{n}{2}\right\rfloor}\dfrac{(n-k)!}{k!(n-2k)!}(ax)^{n-2k} b^k\right]t^{n+1}}.\) Equating the coefficient of \(t^n\) on both sides, we obtain \(R_n(x)=2p\displaystyle\sum_{k=0}^{\left\lfloor\dfrac{n}{2}\right\rfloor}{n-k\choose k}(ax)^{n-2k} b^k +{\left(px+q-2apx\right)}\displaystyle\sum_{k=0}^{\left\lfloor\dfrac{n-1}{2}\right\rfloor} {n-k-1 \choose k}(ax)^{n-2k-1} b^k.\)

Proposition 2. [Sum of first \(n\) terms] The sum of the first \(n\) terms of generalized Fibonacci polynomials is given by \[\displaystyle\sum_{k=0}^{n-1}R_k(x)=\dfrac{R_n(x)+bR_{n-1}(x)-(R_1(x)-axR_0(x))-R_0(x)}{ax+b-1}.\]

Proof. Using Binet's formula (8), we have \begin{align*} \displaystyle\sum_{k=0}^{n-1}R_k(x)&=\displaystyle\sum_{k=0}^{n-1}{\left(A\alpha^{k}(x)+B\beta^{k}(x)\right)}, \end{align*} where \(A=\dfrac{R_1(x)- \beta(x)R_0(x)}{\alpha(x)-\beta(x)}\) and \(B=\dfrac{\alpha(x)R_0(x)-R_1(x)}{\alpha(x)-\beta(x)}.\) It follows that \begin{align*} \displaystyle\sum_{k=0}^{n-1}R_k(x)&=A\displaystyle\sum_{k=0}^{n-1}\alpha^k(x)+B\displaystyle\sum_{k=0}^{n-1}\beta^k(x)\\ &=\dfrac{A\left(\alpha^n(x)-1\right)}{\alpha(x)-1}+\dfrac{B\left(\beta^n(x)-1\right)}{\beta(x)-1}\\ &={\dfrac{A+B-(A\beta(x)+B\alpha(x))-(A\alpha^{n}(x)+B\beta^{n}(x))}{\alpha(x)\beta(x)-\alpha(x)-\beta(x)+1}} +{\dfrac{\alpha(x)\beta(x)(A\alpha^{n-1}(x) +B\beta^{n-1}(x))}{\alpha(x)\beta(x)-\alpha(x)-\beta(x)+1}}. \end{align*} Since \(\alpha(x)+\beta(x)=ax\) and \(\alpha(x)\beta(x)=-b\) and using (8),(10) and (12), we obtain \[\displaystyle\sum_{k=0}^{n-1}R_k(x)=\dfrac{R_n(x)+bR_{n-1}(x)-(R_1(x)-axR_0(x))-R_0(x)}{ax+b-1}.\]

Proposition 3. [Sum of first \(n\) terms with odd indices] The sum of first \(n\) terms with odd indices of generalized Fibonacci polynomials is given by \[\displaystyle\sum_{k=0}^{n-1}R_{2k+1}(x)=\dfrac{R_{2n+1}(x)-b^2R_{2n-1}(x)+b{\left(R_1(x)-axR_0(x)\right)}-R_1(x)}{a^2x^2-b^2+2b-1}.\]

Proof. Using Binet's formula (8), we have \begin{align*} \displaystyle\sum_{k=0}^{n-1}R_{2k+1}(x)&=\displaystyle\sum_{k=0}^{n-1}{\left(A\alpha^{2k+1}(x)+B\beta^{2k+1}(x)\right)}\\ &=A\displaystyle\sum_{k=0}^{n-1}\alpha^{2k+1}(x)+B\displaystyle\sum_{k=0}^{n-1}\beta^{2k+1}(x)\\ &=\dfrac{A\left(\alpha^{2n+1}(x)-\alpha(x)\right)}{\alpha^{2}(x)-1}+\dfrac{B\left(\beta^{2n+1}(x)-\beta(x)\right)}{\beta^{2}(x)-1}. \end{align*} Thus \begin{align*} \displaystyle\sum_{k=0}^{n-1}R_{2k+1}(x)&={\frac{A\alpha(x)+B\beta(x)-\alpha(x)\beta(x)(A\beta(x)+B\alpha(x))}{(\alpha(x)\beta(x))^2-\alpha^2(x)-\beta^2(x)+1}}\\ &\;\;\;-{\frac{A\alpha^{2n+1}(x)+B\beta^{2n+1}(x)+(\alpha(x)\beta(x))^2(A\alpha^{2n-1}(x)+B\beta^{2n-1}(x))}{(\alpha(x)\beta(x))^2-\alpha^2(x)-\beta^2(x)+1}}. \end{align*} Since \(\alpha(x)\beta(x)=-b\) and \(\alpha^2(x)+\beta^2(x)=a^2x^2+2b\) then using (8), (11) and (12), we obtain \begin{align*} \displaystyle\sum_{k=0}^{n-1}R_{2k+1}(x)&=\dfrac{R_{2n+1}(x)-b^2R_{2n-1}(x)+b{\left(R_1(x)-axR_0(x)\right)}-R_1(x)}{a^2x^2-b^2+2b-1}. \end{align*}

Proposition 4. [Sum of first \(n\) terms with even indices] The sum of first {n} terms of generalized Fibonacci sequences with even indices is given by \[\displaystyle\sum_{k=0}^{n-1} R_{2k}(x)=\dfrac{R_{2n}(x)-b^2R_{2n-2}(x)+(a^2x^2R_0(x)-axR_1(x)+bR_0(x))-R_0(x)}{a^2x^2-b^2+2b-1}.\]

Proof. Using Binet's formula (8), we have \begin{align*} \displaystyle\sum_{k=0}^{n-1}R_{2k}(x)&=\displaystyle\sum_{k=0}^{n-1}{\left(A\alpha^{2k}(x)+B\beta^{2k}(x)\right)}=A\displaystyle\sum_{k=0}^{n-1}\alpha^{2k}(x)+B\displaystyle\sum_{k=0}^{n-1}\beta^{2k}(x)=\dfrac{A\left(\alpha^{2n}(x)-1\right)}{\alpha^2(x)-1}+{\dfrac{B\left(\beta^{2n}(x)-1\right)}{\beta^2(x)-1}}. \end{align*} Hence \begin{align*} \displaystyle\sum_{k=0}^{n-1}R_{2k}(x)&={\dfrac{A+B-(A\beta^2(x)+B\alpha^2(x))-(A\alpha^{2n}(x)+B\beta^{2n}(x))}{(\alpha(x)\beta(x))^2-\alpha^2(x)-\beta^2(x)+1}}+{\dfrac{(\alpha(x)\beta(x))^2(A\alpha^{2n-2}(x)+B\beta^{2n-2}(x))}{(\alpha(x)\beta(x))^2-\alpha^2(x)-\beta^2(x)+1}}. \end{align*} Since \(\alpha^2(x)+\beta^2(x)=a^2x^2+2b,\) and \(\alpha(x)\beta(x)=-b,\) then using (8), (10) and (13) , we obtain \[\displaystyle\sum_{k=0}^{n-1} R_{2k}(x)=\dfrac{R_{2n}(x)-b^2R_{2n-2}(x)+(a^2x^2R_0(x)-axR_1(x)+bR_0(x))-R_0(x)}{a^2x^2-b^2+2b-1}.\]

Proposition 5. For every positive integer \(n,\) we have \[\displaystyle\sum_{k=1}^{n}R_{3k}(x)=\dfrac{R_{3n+3}(x)+b^3R_{3n}(x)-R_{3}(x)-bR_0(x)}{a^3x^3+b^3+3abx-1}.\]

Proof. By Binet's formula (8), we have \begin{align*} \displaystyle\sum_{k=1}^{n}R_{3k}(x)&=\displaystyle\sum_{k=1}^{n}{\left(A\alpha^{3k}(x)+B\beta^{3k}(x)\right)} =\dfrac{A\alpha^3(x)\left(\alpha^{3n}(x)-1\right)}{\alpha^3(x)-1}+{\dfrac{B\beta^3(x)\left(\beta^{3n}(x)-1\right)}{\beta^3(x)-1}}. \end{align*} Thus \begin{align*} \displaystyle\sum_{k=0}^{n}R_{3k}(x)&={\frac{(A\alpha^3(x)+B\beta^3(x))-(A\alpha^3\beta^3+B\beta^3\alpha^3)}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}\\ &\;\;\;+{\frac{(A\alpha^{3n+3}(x)\beta^3(x)+B\beta^{3n+3}(x)\alpha^3(x))-(A\alpha^{3n+3}(x)+B\beta^{3n+3}(x))}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}.\\ &={\frac{(A\alpha^3(x)+B\beta^3(x))-\alpha^3(x)\beta^3(x)(A+B)}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}\\ &\;\;\;+{\frac{\alpha^3(x)\beta^3(x)(A\alpha^{3n}(x)+B\beta^{3n}(x))-(A\alpha^{3n+3}(x)+B\beta^{3n+3}(x))}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}. \end{align*} Since \(\alpha^3(x)+\beta^3(x)=a^3x^3+3abx,\) and \(\alpha(x)\beta(x)=-b,\) then by Equations (8) and (10), we get \[\displaystyle\sum_{k=1}^{n} R_{3k}(x)=\dfrac{R_{3n+3}(x)+b^3R_{3n}(x)-R_3(x)-bR_0(x)}{a^3x^3+b^3+3abx-1}.\] Hence the proof.

Proposition 6. For every positive integer \(n,\) we have \[\displaystyle\sum_{k=1}^{n} R_{3k-1}(x)=\dfrac{R_{3n+2}(x)+b^3R_{3n-1}(x)+b^2(axR_0(x)-R_1(x))-R_{2}(x)}{a^3x^3+b^3+3abx-1}.\]

Proof. By Binet's formula (8), we have \begin{align*} \displaystyle\sum_{k=1}^{n}R_{3k-1}(x)&=\displaystyle\sum_{k=1}^{n}{\left(A\alpha^{3k-1}(x)+B\beta^{3k-1}(x)\right)}=\dfrac{A\alpha^2(x)\left(\alpha^{3n}(x)-1\right)}{\alpha^3(x)-1}+{\dfrac{B\beta^2(x)\left(\beta^{3n}(x)-1\right)}{\beta^3(x)-1}}. \end{align*} This sum gives \begin{align*} \displaystyle\sum_{k=1}^{n}R_{3k-1}(x)&={\dfrac{A\alpha^2(x)+B\beta^2(x)-(A\alpha^2(x)\beta^3(x)+B\alpha^3(x)\beta^2(x))}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}\\ &\;\;\;+{\dfrac{A\alpha^{3n+2}(x)\beta^3(x)+B\beta^{3n+2}(x)\alpha^3(x)- A\alpha^{3n+2}(x)+B\beta^{3n+2}(x)}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}\\ &={\dfrac{A\alpha^2(x)+B\beta^2(x)-(\alpha(x)\beta(x))^2(A\beta(x)+B\alpha(x))}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}\\ &\;\;\;+{\dfrac{(\alpha(x)\beta(x))^3(A\alpha^{3n-1}(x)+B\beta^{3n-1}(x))- A\alpha^{3n+2}(x)+B\beta^{3n+2}(x)}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}. \end{align*} Since \(\alpha^3(x)+\beta^3(x)=a^3x^3+3abx,\) and \(\alpha(x)\beta(x)=-b,\) then making use of (8) and (12) , we obtain \[\displaystyle\sum_{k=1}^{n} R_{3k-1}(x)=\dfrac{R_{3n+2}(x)+b^3R_{3n-1}(x)+b^2(axR_0(x)-R_1(x))-R_{2}(x)}{a^3x^3+b^3+3abx-1}.\]

Proposition 7. For every positive integer \(n,\) we have \[\displaystyle\sum_{k=1}^{n} R_{3k-2}(x)=\dfrac{R_{3n+1}(x)+b^3R_{3n-2}(x)-b{\left(a^2x^2R_0(x)-axR_1(x)+bR_0(x)\right)-R_1(x)}}{a^3x^3+b^3+3abx-1}.\]

Proof. By Binet's formula (8), we get \begin{align*} \displaystyle\sum_{k=1}^{n}R_{3k-2}(x)&=\displaystyle\sum_{k=1}^{n}{\left(A\alpha^{3k-2}(x)+B\beta^{3k-2}(x)\right)}\\ &=A\displaystyle\sum_{k=1}^{n}\alpha^{3k-2}(x)+B\displaystyle\sum_{k=1}^{n}\beta^{3k-2}(x)\\ &=\dfrac{A\alpha(x)\left(\alpha^{3n}(x)-1\right)}{\alpha^3(x)-1}+{\dfrac{B\beta(x)\left(\beta^{3n}(x)-1\right)}{\beta^3(x)-1}}. \end{align*} This Sum gives \begin{align*} \displaystyle\sum_{k=1}^{n}R_{3k-2}(x)&={\dfrac{A\alpha(x)+B\beta(x)-A\alpha\beta^3(x)+B\beta\alpha^3(x)}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}\\ &\;\;\;+{\dfrac{A\alpha^{3n+1}(x)\beta^3(x)+B\beta^{3n+1}(x)\alpha^3(x)- A\alpha^{3n+1}(x)+B\beta^{3n+1}(x)}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}\\ &={\dfrac{A\alpha(x)+B\beta(x)-\alpha(x)\beta(x)(A\alpha\beta^2(x)+B\alpha^2(x))}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}\\ &\;\;\;+{\dfrac{(\alpha(x)\beta(x))^3(A\alpha^{3n-2}(x)+B\beta^{3n-2}(x))- A\alpha^{3n+1}(x)+B\beta^{3n+1}(x)}{(\alpha(x)\beta(x))^3-\alpha^3(x)-\beta^3(x)+1}}. \end{align*} Since \(\alpha^3(x)+\beta^3(x)=a^3x^3+3abx,\) and \(\alpha(x)\beta(x)=-b,\) then making use of (8) and (13), we obtain \[\displaystyle\sum_{k=1}^{n} R_{3k-2}(x)=\dfrac{R_{3n+1}(x)+b^3R_{3n-2}(x)-b{\left(a^2x^2R_0(x)-axR_1(x)+bR_0(x)\right)-R_1(x)}}{a^3x^3+b^3+3abx-1}.\]

Theorem 3. [Generalized identity]Let \(R_n(x)\) be the \(n^{th}\) generalized Fibonacci polynomials. Then

where \(n>m \geq k\geq 1.\)

Proof. Using Binet's formula (8) to the left hand side, we have \begin{align*} \text{LHS}&={\left(A\alpha^m(x)+B\beta^m(x)\right)}{\left(A\alpha^n(x)+B\beta^n(x)\right)}-{\left(A\alpha^{m-k}(x)+B\beta^{m-k}(x) \right)}{\left(A\alpha^{n+k}(x)+B\beta^{n+k}(x)\right)}\\ &=AB{\left(\alpha^k(x)-\beta^k(x)\right)} {\left[\dfrac{\alpha^m(x)\beta^n(x)}{\alpha^k(x)}-\dfrac{\alpha^n(x)\beta^m(x)}{\beta^k(x)}\right]}\\ &=AB\dfrac{(\alpha^k(x)-\beta^k(x))}{(\alpha(x)\beta(x))^k}{\left(\alpha^m(x)\beta^{n+k}(x)-\alpha^{n+k}(x)\beta^m(x)\right)}\\ &=-AB(-b)^{m-k}{\left(\alpha^k(x)-\beta^k(x)\right)}{\left(\alpha^{n-m+k}(x)-\beta^{n-m+k}(x)\right)}. \end{align*} Since \(-AB=\dfrac{R_1^2(x)-R_0(x)R_2(x)}{(\alpha(x)-\beta(x))^2}\) by (14), then \begin{align*} &R_m(x)R_n(x)-R_{m-k}(x)R_{n+k}(x)=\dfrac{R_1^2(x)-R_0(x)R_2(x)}{(\alpha(x)-\beta(x))^2}(-b)^{m-k}{\left[(\alpha^k(x)-\beta^k(x))(\alpha^{n-m+k}(x)-\beta^{n-m+k}(x))\right]}\\ &={\left(R_1^2(x)-R_0(x)R_2(x)\right)}(-b)^{m-k}{\left[{\left(\dfrac{\alpha^k(x)-\beta^k(x)}{\alpha(x)-\beta(x)}\right)}{\left(\dfrac{\alpha^{n-m+k}(x)-\beta^{n-m+k}(x)}{\alpha(x)-\beta(x)}\right)}\right]}. \end{align*} From \[\dfrac{\alpha^k(x)-\beta^k(x)}{\alpha(x)-\beta(x)}=\dfrac{R_1(x)R_k(x)-R_0(x)R_{k+1}(x)}{R_1^2(x)-R_0(x)R_2(x)}\] and \[\dfrac{\alpha^{n-m+k}(x)-\beta^{n-m+k}(x)}{\alpha(x)-\beta(x)}=\dfrac{R_1(x)R_{n-m+k}(x)-R_0(x)R_{n-m+k+1}(x)}{R_1^2(x)-R_0(x)R_2(x)},\] we obtain our desired result.

Corollary 1. [Catalan's identity] If \(m=n\) in the generalized identity (17), we obtain \[R_n^2(x)-R_{n-k}(x)R_{n+k}(x)=\dfrac{(-b)^{m-k}}{R_1^2(x)-R_0(x)R_2(x)}{\left[R_1(x)R_{k}(x)-R_0(x)R_{k+1}(x)\right]^2},\] where \(n>k\geq1.\)

Corollary 2. [Cassini's identity] If \(m=n\) and \(k=1\) in the generalized identity (17), we obtain \[R_1^2(x)-R_{n-1}(x)R_{n+1}(x)=(-b)^{n-1}{\left[R_1^2(x)-R_0(x)R_2(x)\right]},\] for \(n\geq1.\)

Corollary 3. [d'Ocagne's identity] If \(n=m,\) \(m=n+1\) and \(k=1\) in the generalized identity (17), we obtain \[R_m(x)R_{n+1}(x)-R_{m+1}(x)R_n(x)=(-b)^n{\left[R_1(x)R_{m-n}(x)-R_0(x)R_{m-n+1}(x)\right]},\] where \(m>n \geq1.\)

6. Conclusion

In this paper generalized Fibonacci polynomials is defined by recurrence relations (6). Binet's formula (8) and generating function of these polynomials (15) are derived. Further explicit sum formula, sum of first \(n\) terms, sum of first \(n\) terms with (odd or even )indices and generalized identity (17) from which we obtain Catalan's identity, Cassini's identity and d'Ocagne's identity are also derived.

Conflict of Interests

The author declares no conflict of interest.

1. Introduction

Since, the foundation paper titled, Degree Tolerant Coloring of Graphs is under review, sufficient results and concepts will be recalled to makes this paper digestible. It is assumed that the reader is familiar with the definition and properties of finite Albenian groups over the operation addition \(+\), (or, multiplication \(\times\)). For ease of reference we recall that, depending on the characteristics of the elements of a non-empty set \(X\), the operations \(+\) or \(\times\) can be abstract operations. Having said that, we recall that if \(X\) is a non-empty set and \(f\) is a binary operation on \(X\) i.e., \(f:X\times X\rightarrow X\) we can denote, \(f((a,b))=a\circ b\). The ordered pair \((X,f)\) is a group if:

• (i) \(f\) is associative i.e. \(a\circ(b\circ c)=(a\circ b)\circ c\), \(\forall a,b,c\in X\).
• (ii) A \(e\in X\) exists such that, \(a\circ e=a\), \(\forall a\in X\) (called a right identity of \((X,f)\)).
• (iii) A \(b \in X\) exists for each \(a\in X\) such that, \(a\circ b=e\) (called a right inverse of \(a\)).

If for all \(a,b \in X\) we have \(a\circ b=b\circ a\) then \((X,f)\) is an Albenian group (also called a commutative group). A good introduction is found amongst others, in [1]. In this paper we will consider only a specific finite Albenian group.

It is also assumed that the reader is familiar with most of the classical concepts in graph theory. Throughout only finite, connected simple graphs will be considered. For more on general notation and concepts in graphs see [2,3,4]. It is also assumed that the reader is familiar with the concept of graph coloring. In a proper coloring of \(G\) all edges are said to be good i.e. \(\forall~uv \in E(G)\), \(c(u)\neq c(v)\). The set of colors assigned in a graph coloring is denoted by \(\mathcal{C}\) and a subset of colors assigned to a subset of vertices \(X\subseteq V(G)\) is denoted by \(c(X)\). In an improper (or defect) coloring it is permitted that for some \(uv \in E(G)\), the coloring is \(c(u)=c(v)\).

Recall that for a degree tolerant coloring abbreviated as, \(DT\)-coloring of a graph \(G\) the following condition is set:

• (i) If \(uv \in E(G)\) and \(deg(u)=deg(v)\) then, \(c(u)=c(v)\) else, \(c(u)\neq c(v)\).

Alternative formulation for condition (i). If \(uv \in E(G)\) then, \(c(u)=c(v)\) if and only if \(deg(u)=deg(v)\). The minimum number of colors which yields a \(DT\)-coloring is called the degree tolerant chromatic number of \(G\) and is denoted by, \(\chi_{dt}(G)\). A salient condition which is implied is, if \(uv \notin E(G)\) then, either \(c(u)=c(v)\) or \(c(u)\neq c(v)\). For \(K_n\), \(n\geq 1\) it easily follows that, \(\chi_{dt}(K_n) =1 \leq \chi(K_n)\). In fact, for \(n\geq 2\) it follows that, \(\chi_{dt}(K_n) < \chi(K_n)\). Furthermore, for paths \(P_1\), \(P_2\) we find \(\chi_{dt}(P_1) = \chi_{dt}(P_2) =1\). On the other hand for \(n \geq 3\) we have, \(\chi_{dt}(P_n) = \chi(P_n) = 2\). In the aforesaid it is only the coloring assignment which differs.

Consider the set \(R_u= \{deg(v): v\in N[u], u\in V(G)\}\). The degree tolerant index of \(u \in V(G)\) is defined by \(\rho(u) = |R_u|\). Note that since repetition in a set is not permitted, \(\rho(u) \leq |N[u]|\). Let the degree tolerant index of a graph \(G\) be \(\rho(G) = \max\{\rho(u):\) over all \(u\in V(G)\}\). We recall a result of interest.

Theorem 1. For a graph \(G\), it follows that \(\chi_{dt}(G)\leq \rho(G)\).

2. Albenian group \(\mathbb{Z}_n\) under addition modulo \(n\): \(n\in \mathbb{N}\)

Denote the finite Albenian group \(\mathbb{Z}_n\) under addition modulo \(n\), \(n\in \mathbb{N}\) by, \((\mathbb{Z}_{n},+_{n})\). We have \(\mathbb{Z}_n=\{0,1,2,\dots,n-1\}\) and the generator set, \(\mathbb{Z}_{gen}=\{m:m\) relatively prime to \(n\}\). Let \(\mathbb{P} =\{\)prime numbers\(\}\).

Since it is known that that any non-prime positive integer \(n\geq 4\) can be written as a product of prime numbers, begin by considering a product of two distinct primes i.e. \(n=p_1p_2\), \(p_1,p_2 \in \mathbb{P}\). It is agreed that the terms prime and prime number may be used interchangeably.

Lemma 1. For distinct positive integers \(m_1,m_2\), there are exactly \(m_1-1\) multiples of \(m_2\) and exactly \(m_2-1\) multiples of \(m_1\) in the integer range \([0,m_1m_2-1]\).

Proof. The number of multiples of \(m_1\) and \(m_2\) in the integer range \([0,m_1m_2-1]\) is given by \(\lfloor \frac{m_1p_m-1}{p_1}\rfloor\) and \(\lfloor \frac{m_1m_2-1}{m_2}\rfloor\) respectively. Further through simplification $$\lfloor \frac{m_1m_2-1}{m_1}\rfloor =m_2-1$$ and $$\lfloor \frac{m_1m_2-1}{m_2}\rfloor = m_1-1$$.

The respective sets of multiples of the primes \(p_1\) and \(p_2\) in the integer range \([0,p_1p_2-1]\) are denoted by \(M_{p_1}\) and \(M_{p_2}\). The vertex set of the power graph denoted by \(\mathcal{P}((\mathbb{Z}_{n},+_{n}))\) is \(V(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = \{v_0,v_1,v_2,\dots,v_{p_1p_2-1}\}\). Alternatively, \(V_{\mathcal{P}}(\mathbb{Z}_n)\) (for brevity). The power graph has edge set \(E(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = \{v_iv_j: i\in \mathbb{Z}_{gen}\cup \{0\}, v_j \in V_{\mathcal{P}}(\mathbb{Z}_n), i\neq j\} \cup \{v_kv_\ell: k,\ell \in M_{p_1}, k \neq \ell\} \cup \{v_mv_\tau: m,\tau \in M_{p_2}, m \neq \tau\} = E_{\mathcal{P}}(\mathbb{Z}_n)\), (for brevity).

Remark 1. Conventionally the power graph is a directed graph. We only consider the undirected case, hence the underlying power graph.

Note that a result stemming from the Euler \(\varphi\)-function (see Theorem 2.13.5, [1]) yields \(|\mathbb{Z}_{gen}|=(p_1-1)(p_2-1)\). Since \(|V(\mathcal{P}((\mathbb{Z}_{n},+_{n})))| = p_1p_2\), the above implies that the vertex set of the power graph \(\mathcal{P}((\mathbb{Z}_{n},+_{n}))\) can be partitioned into:

\(|V_1|= (p_2-1)\)-set and \(deg (v_i)=p_1(p_2-1)\), \(v_i\in V_1\),

\(|V_2|= (p_1-1)\)-set and \(deg(v_j)= p_2(p_1-1)\), \(v_j\in V_2\),

\(|V_{gen}\cup \{v_0\}| = (p_1p_2 -p_1-p_2+2)\)-set and \(deg(v_k)= p_1p_2-1\), \(v_k \in V_{gen}\cup \{v_0\}\).

The Figure 1 depicts \(\mathcal{P}((\mathbb{Z}_{n},+_{n}))\), \(p_1=2\), \(p_2=3\).

2.1. Degree tolerant chromatic number of \((\mathbb{Z}_{n},+_{n})\)

Lemma 2. For a positive integer \(n=p_1p_2\), \(p_1,p_2 \in \mathbb{P}\), \(p_1\neq p_2\), we have that \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = 2\).

Proof. Since \(\rho(v_i)=2\), \(v_i\in V_1\), \(\rho(v_j)=2\), \(v_j \in V_2\) and \(\rho(v_k)=3\), \(v_k \in V_{gen}\cup \{v_0\}\), it follows that \(\rho(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = 3\). From Theorem 1, \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) \leq 3\). Since \(V_1\cap V_2=\emptyset\) the coloring \(c(V_1)=c(V_2)\) is permissible in a minimal \(DT\)-coloring. Hence \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) \leq 2\). However \(\mathcal{P}((\mathbb{Z}_{n},+_{n}))\) is not regular, so \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) \geq 2\). This settles the result.

Now consider \(n=p_1p_2p_3\), \(p_1,p_2,p_3 \in \mathbb{P}\) and \(p_i\neq p_j\) for all pairs.

Lemma 3. In the integer range \([0,p_1p_2p_3-1]\), let \(V_1 = \{\)multiples of \(p_1\}\), \(V_2 = \{\)multiples of \(p_2\}\), \(V_3= \{\)multiples of \(p_3\}\). Then \(|V_1 \cap V_2|=p_3-1\), \(|V_1\cap V_3|=p_2-1\) and \(|V_2\cap V_3|=p_1-1\).

Proof. Noting that without loss of generality \(V_i\cap V_j=\lfloor \frac{1}{p_i}\cdot\lfloor \frac{p_ip_jp_k -1}{p_j}\rfloor \rfloor = p_k-1\), the result follows.

Lemma 4. For a positive integer \(n=p_1p_2p_3\) and \(p_1,p_2,p_3 \in \mathbb{P}\), we have \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = 3\).

Proof. Similar to the vertex partition for the case \(n=p_1p_2\), the vertex set \(V(\mathcal{P}((\mathbb{Z}_{n},+_{n})))\) for the case \(n=p_1p_2p_3\) can be partitioned into \(V_1\), \(V_2\),\(V_3\) and \(V_{gen}\cup \{v_0\}\). Clearly, by similar reasoning found in the proof of Lemma 3, \(\chi_{dt}((\mathbb{Z}_{n},+_{n})) \leq 4\). Since no multiple of \(p_1p_2p_3\) exists, it follows that no induced \(K_3\) exists on a triple \(v_i,v_j,v_k\), \(v_i \in V_1\), \(v_j\in V_2\), \(v_k\in V_3\). Therefore \(\chi_{dt}((\mathbb{Z}_{n},+_{n})) \leq 3\).

Since \(p_i,p_j,p_k\geq 2\), it follows that \(V_i \cap V_j \neq \emptyset\). Therefore, an induced \(K_3\) exists in \(\mathcal{P}((\mathbb{Z}_{n},+_{n}))\) such that each vertex has degree distinct from the other. Note that one of the vertices will be in \(V_{gen}\cup \{v_0\}\). By condition (ii), \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n})))\geq 3\). Hence the result.

We are ready to present the main result.

Theorem 2. For a positive integer \(n= \prod\limits_{i=1}^{k}p_i\), \(p_i\in \mathbb{P}\) and \(p_i \neq p_j\) iff \(i\neq j\), we have \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) = k\).

Proof. The result follows by Lemmas 1,2,3,4 and the utilization of induction.

3. Conclusion

It is known that for some graphs \(\chi_{dt}(G)< \chi(G)\).

Problem 1. Show that for a positive integer \(n= \prod\limits_{i=1}^{k}p_i\), \(p_i\in \mathbb{P}\) and \(p_i \neq p_j\) iff \(i\neq j\) that \(\chi_{dt}(\mathcal{P}((\mathbb{Z}_{n},+_{n}))) < \chi(\mathcal{P}((\mathbb{Z}_{n},+_{n})))\).

The numerous finite Albenian groups offer a wide scope for further research.

Acknowledgments

The author would like to thank the anonymous referees for their constructive comments, which helped to improve on the elegance of this short paper.

Conflict of Interests

''The author declares no conflict of interest.''