Minimum degree polynomial of graphs obtained by some graph operators

1. Introduction

In the literature of graph theory, we can find several graph polynomials based on different matrices defined on the graph such as adjacency matrix [1], Laplacian matrix [1], signless Laplacian matrix [3, 4], distance matrix [5], degree sum matrix [6, 7], seidel matrix [8] etc. The purpose of this paper is to obtain the characteristic polynomial of the minimum degree matrix of a graph obtained by some graph operators (generalized \(xyz\)-point-line transformation graphs). For undefined graph theoretic terminologies and notions refer [1, 9, 10].

Let \(G=(n,m)\) be a simple, undirected graph. Let \(V(G)\) and \(E(G)\) be the vertex set and edge set of \(G\) respectively. The degree \(deg_G(v)\) (or \(d_G(v))\) of a vertex \(v\in V(G)\) is the number of edges incident to it in \(G\). The graph \(G\) is r-regular if the degree of each vertex in \(G\) is r. Let \(\{v_1,v_2,...,v_n\}\) be the vertices of \(G\) and let \(d_i=deg_G(v_i)\). The minimum degree matrix[11] of a graph \(G\) is an \(n\times n\) matrix \(MD(G)=[(md)_{ij}]\), whose elements are defined as $$(md)_{ij}=\left\{ \begin{array}{ll} min\{d_i,d_j\} & if\;\; i\neq j,\\ 0 & otherwise. \end{array} \right.$$

Let \(I\) be the identity matrix and \(J\) be the matrix whose all entries are equal to \(1\). The minimum degree polynomial of a graph \(G\) is defined as $$P_{MD(G)}(\xi)=det(\xi I-MD(G)).$$

The eigenvalues of the matrix \(MD(G)\), denoted by \(\xi_1,\xi_2,...,\xi_n\) are called the minimum degree eigenvalues of \(G\) and their collection is called the minimum degree spectra of \(G\). It is easy to see that if \(G\) is an r-regular graph, then \(MD(G)=r(J-I)\). Therefore, for an r-regular graph \(G\) of order \(n\),

The subdivision graph [9] \(S(G)\) of a graph \(G\) is a graph with the vertex set \(V(S(G))=V(G)\cup E(G)\) and two vertices of \(S(G)\) are adjacent whenever they are incident in \(G\). The partial complement of subdivision graph [12] \(\overline{S}(G)\) of a graph \(G\) is a graph with the vertex set \(V(\overline{S}(G))=V(G)\cup E(G)\) and two vertices of \(\overline{S}(G)\) are adjacent whenever they are nonincident in \(G\).

In [13], Wu Bayoindureng et al. introduced the total transformation graphs and obtained the basic properties of total transformation graphs. For a graph \(G=(V,E)\), let \(G^0\) be the graph with \(V(G^0)=V(G)\) and with no edges, \(G^1\) the complete graph with \(V(G^1)=V(G)\), \(G^+=G\), and \(G^-=\overline{G}\). Let \(\mathcal{G}\) denotes the set of simple graphs. The following graph operators depending on \(x, y, z \in \{0, 1, +, -\}\) induce functions \(T^{xyz}:\mathcal{G} \to \mathcal{G}.\) These operators are introduced by Deng et al. in [14]. They referred these resulting graphs as \(xyz\)-transformations of \(G\), denoted by \(T^{xyz}(G)=G^{xyz}\) and obtained the Laplacian characteristic polynomials and some other Laplacian parameters of \(xyz\)-transformations of an \(r\)-regular graph \(G\). Further, Basavanagoud [15]established the basic properties of these \(xyz\)-transformation graphs by calling them \(xyz\)-point-line transformation graphs.

Definition 1. [14] Given a graph \(G\) with vertex set \(V(G)\) and edge set \(E(G)\) and three variables \(x,y,z \in \{0,1,+,-\},\) the \(xyz\)-point-line transformation graph \(T^{xyz} (G)\) of \(G\) is the graph with vertex set \(V(T^{xyz}(G) )=V(G)\cup E(G)\) and the edge set \(E(T^{xyz}(G) )=E((G)^x)\cup E((L(G))^y)\cup E(W)\) where \(W=S(G)\) if \(z=+\), \(W=\overline{S}(G)\) if \(z=-\), \(W\) is the graph with \(V(W)=V(G)\cup E(G)\) and with no edges if \(z=0\) and \(W\) is the complete bipartite graph with parts \(V(G)\) and \(E(G)\) if \(z=1\).

Since there are 64 distinct \(3\)-permutations of \(\{0,1,+,-\}\). Thus obtained 64 kinds of generalized \(xyz\)-point-line transformation graphs. There are \(16\) different graphs for each case when \(z=0\), \(z=1\), \(z=+\), \(z=-.\)

For instance, the total graph \(T(G)\) is a graph with vertex set \(V(G)\cup E(G)\) and two vertices of \(T(G)\) are adjacent whenever they are adjacent or incident in \(G\). The \(xyz\)-point-line transformation graph \(T^{--+}(G)\) is a graph with vertex set \(V(G)\cup E(G)\) and two vertices of \(T^{--+}(G)\) are adjacent whenever they are nonadjacent or incident in \(G\).

The degree of vertices in the graphs \(T^{xyz}(G)\) are given in the following Theorems 2 and 3, which are helpful in proving our results.

Theorem 2. [15] Let \(G\) be a graph of order \(n\), size \(m\) and let \(v\) be the point-vertex of \(T^{xyz}\) corresponding to a vertex \(v\) of \(G\). Then

1. \(d_{T^{xy0}}(v)\) =\(\left\{ \begin{array}{ll} 0 & if \; x=0, y\in \{0,1, +, -\},\\ n-1 & if\; x=1, y\in \{0,1, +, -\}, \\ d_{G}(v) & if\; x=+, y\in \{0,1, +, -\}, \\ n-1-d_{G}(v) & if\; x=-, y\in \{0,1, +, -\}. \end{array} \right.\)
2. \(d_{T^{xy1}}(v)\) =\(\left\{ \begin{array}{ll} m & if \; x=0, y\in \{0,1, +, -\},\\ n+m-1 & if\; x=1, y\in \{0,1, +, -\}, \\ m+d_{G}(v) & if\; x=+, y\in \{0,1, +, -\},\\ n+m-1-d_{G}(v) & if\; x=-, y\in \{0,1, +, -\}. \end{array} \right.\)
3. \(d_{T^{xy+}}(v)\) =\(\left\{ \begin{array}{ll} d_{G}(v) & if \; x=0 , y\in \{0,1, +, -\},\\ n-1+d_{G}(v) & if\; x=1, y\in \{0,1, +, -\}, \\ 2d_{G}(v) & if\; x=+, y\in \{0,1, +, -\},\\ n-1 & if\; x=- , y\in \{0,1, +, -\}. \end{array} \right.\)
4. \(d_{T^{xy-}}(v)\) =\(\left\{ \begin{array}{ll} m- d_{G}(v) & if \; x=0, y\in \{0,1, +, -\},\\ n+m-1- d_{G}(v) & if\; x=1, y\in \{0,1, +, -\}, \\ m & if\; x=+ , y\in \{0,1, +, -\},\\ n+m-1-2d_{G}(v) & if\; x=-, y\in \{0,1, +, -\}. \end{array} \right.\)

Theorem 3.[15] Let \(G\) be a graph of order \(n\), size \(m\) and let \(e\) be the line-vertex of \(T^{xyz}\) corresponding to an edge \(e\) of \(G\). Then

1. \(d_{T^{xy0}}(e)\) =\(\left\{ \begin{array}{ll} 0 & if \; y=0, x\in \{0,1, +, -\},\\ m-1 & if\; y=1, x\in \{0,1, +, -\},\\ d_{G}(e)& if\; y=+, x\in \{0,1, +, -\},\\ m-1-d_{G}(e) & if\; y=-, x\in \{0,1, +, -\}. \end{array} \right.\)
2. \(d_{T^{xy1}}(e)\) =\(\left\{ \begin{array}{ll} n & if \; y=0, x\in \{0,1, +, -\},\\ n+m-1 & if\; y=1, x\in \{0,1, +, -\}, \\ n+d_{G}(e) & if\; y=+, x\in \{0,1, +, -\},\\ n+m-1-d_{G}(e) & if\; y=-, x\in \{0,1, +, -\}. \end{array} \right.\)
3. \(d_{T^{xy+}}(e)\) =\(\left\{ \begin{array}{ll} 2 & if \; y=0 , x\in \{0,1, +, -\},\\ m+1 & if\; y=1 , x\in \{0,1, +, -\}, \\ 2+d_{G}(e)& if\; y=+, x\in \{0,1, +, -\},\\ m+1-d_{G}(e) & if\; y=-, x\in \{0,1, +, -\}. \end{array} \right.\)
4. \(d_{T^{xy-}}(e)\) =\(\left\{ \begin{array}{ll} n-2 & if \; y=0, x\in \{0,1, +, -\},\\ n+m-3 & if\; y=1, x\in \{0,1, +, -\}, \\ n-2+d_{G}(e)& if\; y=+, x\in \{0,1, +, -\},\\ n+m-3-d_{G}(e) & if\; y=-, x\in \{0,1, +, -\}. \end{array} \right.\)

2. Degree exponent polynomial of graphs obtained by graph operations

In this section we obtain the minimum degree polynomial of graphs obtained by some graph operators. We use the following lemma in order to prove the following theorems.

Lemma 4.[16] If \(a, b, c\) and \(d\) are real numbers, then the determinant of the form

of order \(n_1+n_2\) can be expressed in the simplified form as $$(\xi+a)^{n_1-1}(\xi+b)^{n_2-1}\{[\xi-(n_1-1)a][\xi-(n_2-1)b]-n_1n_2cd\}.$$

Theorem 5. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{01+}(G))}(\xi)&=&(\xi+m+1)^{m-1}(\xi+r)^{n-1}\{\xi^{2}-[(n-1)r+(m-1)(m+1)]\xi\\ &&+(n-1)r(m-1)(m+1)-min\{m+1,r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{01+}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(r\) and the remaining \(m\) vertices are with degree \(m+1\). Hence $$MD(T^{01+}(G))=\left[ \begin{array}{cc} r(J_n-I_n) & \;\;min\{r,m+1\}J_{n\times m}\\ min\{r,m+1\}J_{m\times n} &\;\; (m+1)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{01+}(G))}(\xi)&=&|\xi I-MD(T^{01+}(G))|\\ &=&\left| \begin{array}{cc} (\xi+r)I_n-rJ_n & \;\;-min\{r,m+1\}J_{n\times m}\\ -min\{r,m+1\}J_{m\times n} &\;\; (\xi+m+1)I_m-(m+1)J_m \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 6. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{0-+}(G))}(\xi)&=&(\xi+r)^{n-1}(\xi+ {m+3-2r})^{m-1}\{\xi^2-[(n-1)r+(m-1)(m+3-2r)]\xi\\ &&+(n-1)(m-1)r(m+3-2r)-min\{r,m+3-2r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{0-+} (G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(r\) and the remaining \(m\) vertices are with degree \(m+3-2r\). Hence $$MD(T^{0-+}(G))=\left[ \begin{array}{cc} r(J_n-I_n) & \;\;min\{r,m+3-2r\}J_{n\times m}\\ min\{r,m+3-2r\}J_{m\times n} &\;\; (m+3-2r)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{0-+}(G))}(\xi)&=&|\xi I-MD(T^{0-+}(G))|\\ &=&\left| \begin{array}{cc} (\xi+r)I_n-rJ_n & \;-min\{r,m+3-2r\}J_{n\times m}\\ -min\{r,m+3-2r\}J_{m\times n} &\; (\xi+m+3-2r)I_m-(m+3-2r)J_m \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 7. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{10+}(G))}(\xi)&=&(\xi+n-1+r)^{n-1}(\xi+2)^{m-1}\{\xi^2-[(n-1)(n-1+r)+2(m-1)]\xi\\ &&+2(n-1)(m-1)(n-1+r)-min\{2,n-1+r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{10+} (G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(n-1+r\) and the remaining \(m\) vertices are with degree \(2\). Hence $$MD(T^{10+}(G))=\left[ \begin{array}{cc} (n-1+r)(J_n-I_n) & \;\;min\{2,n-1+r\}J_{n\times m}\\ min\{2,n-1+r\}J_{m\times n} &\;\; 2(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{10+}(G))}(\xi)&=&|\xi I-MD(T^{10+}(G))|\\ &=&\left| \begin{array}{cc} (\xi+n-1+r)I_n-(n-1+r)J_n & \;\;-min\{2,n-1+r\}J_{n\times m}\\ -min\{2,n-1+r\}J_{m\times n} &\;\; (\xi+2)I_m-2J_m \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 8. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{11+}(G))}(\xi)&=&(\xi+n-1+r)^{n-1}(\xi+{m+1})^{m-1}\{\xi^2-[(n-1)(n-1+r)+(m-1)(m+1)]\xi\\ &&+(n-1)(m-1)(n-1+r)(m+1)-min\{n-1+r,m+1\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{11+}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(n-1+r\) and the remaining \(m\) vertices are with degree \(m+1\). Hence $$MD(T^{11+}(G))=\left[ \begin{array}{cc} (n-1+r)(J_n-I_n) & \;\;min\{n-1+r,m+1\}J_{n\times m}\\ min\{n-1+r,m+1\}J_{m\times n} &\;\; (m+1)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{11+}(G))}(\xi)&=&|\xi I-MD(T^{11+}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+n-1+r)I_n-(n-1+r)J_n} & -min\{n-1+r,m+1\}J_{n\times m}\\ -min\{n-1+r,m+1\}J_{m\times n} & \small{(\xi+m+1)I_m-(m+1)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 9. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{1++}(G))}(\xi)&=&(\xi+n-1+r)^{n-1}(\xi+2r)^{m-1}\{\xi^2-[(n-1)(n-1+r)+(m-1)2r]\xi\\ &&+(n-1)(m-1)(n-1+r)2r-min\{2r,n-1+r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{1++}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(n-1+r\) and the remaining \(m\) vertices are with degree \(2r\). Hence $$MD(T^{1++}(G))=\left[ \begin{array}{cc} (n-1+r)(J_n-I_n) & \;\;min\{2r,n-1+r\}J_{n\times m}\\ min\{2r,n-1+r\}J_{m\times n} &\;\; 2r(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{1++}(G))}(\xi)&=&|\xi I-MD(T^{1++}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+n-1+r)I_n-(n-1+r)J_n} & -min\{2r,n-1+r\}J_{n\times m}\\ -min\{2r,n-1+r\}J_{m\times n} & \small{(\xi+2r)I_m-2rJ_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 10. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{1-+}(G))}(\xi)&=&(\xi+n-1+r)^{n-1}(\xi+m+3-2r)^{m-1}\{\xi^2-[(n-1)(n-1+r)\\ &&+(m-1)(m+3-2r)]\xi+(n-1)(m-1)(n-1+r)(m+3-2r)\\ &&-min\{m+3-2r,n-1+r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{1-+}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R_1=n-1+r\) and the remaining \(m\) vertices are with degree \(R_2=m+3-2r\). Hence $$MD(T^{1-+}(G))=\left[ \begin{array}{cc} R_1(J_n-I_n) & \;\;min\{R_1,R_2\}J_{n\times m}\\ min\{R_1,R_2\}J_{m\times n} &\;\; R_2(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{1-+}(G))}(\xi)&=&|\xi I-MD(T^{1-+}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R_1)I_n-R_1J_n} & -min\{R_1,R_2\}J_{n\times m}\\ -min\{R_1,R_2\}J_{m\times n} & \small{(\xi+R_2)I_m-R_2J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 11. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{+1+}(G))}(\xi)&=&(\xi+2r)^{n-1}(\xi+m+1)^{m-1}\{\xi^2-[(n-1)2r+(m-1)(m+1)]\xi\\ &&+(n-1)(m-1)2r(m+1)-min\{2r,m+1\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{+1+}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(2r\) and the remaining \(m\) vertices are with degree \(m+1\). Hence $$MD(T^{+1+}(G))=\left[ \begin{array}{cc} 2r(J_n-I_n) & \;\;min\{2r,m+1\}J_{n\times m}\\ min\{2r,m+1\}J_{m\times n} &\;\; (m+1)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{+1+}(G))}(\xi)&=&|\xi I-MD(T^{+1+}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+2r)I_n-2rJ_n} & -min\{2r,m+1\}J_{n\times m}\\ -min\{2r,m+1\}J_{m\times n} & \small{(\xi+m+1)I_m-(m+1)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 12. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{+-+}(G))}(\xi)&=&(\xi+2r)^{n-1}(\xi+m+3-2r)^{m-1}\{\xi^2-[(n-1)2r+(m-1)(m+3-2r)]\xi\\ &&+(n-1)(m-1)2r(m+3-2r)-min\{2r,m+3-2r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{+-+}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(2r\) and the remaining \(m\) vertices are with degree \(m+3-2r\). Hence $$MD(T^{+-+}(G))=\left[ \begin{array}{cc} 2r(J_n-I_n) & \;\;min\{2r,m+3-2r\}J_{n\times m}\\ min\{2r,m+3-2r\}J_{m\times n} &\;\; (m+3-2r)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{+-+}(G))}(\xi)&=&|\xi I-MD(T^{+-+}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+2r)I_n-2rJ_n} & -min\{2r,m+3-2r\}J_{n\times m}\\ -min\{2r,m+3-2r\}J_{m\times n} & \small{(\xi+m+3-2r)I_m-(m+3-2r)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 13. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{-0+}(G))}(\xi)&=&(\xi+n-1)^{n-1}(\xi+2)^{m-1}\{\xi^2-[(n-1)(n-1)+2(m-1)]\xi\\ &&+2(n-1)(m-1)(n-1)-min\{n-1,2\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{-0+}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(n-1\) and the remaining \(m\) vertices are with degree \(2\). Hence $$MD(T^{-0+}(G))=\left[ \begin{array}{cc} (n-1)(J_n-I_n) & \;\;min\{n-1,2\}J_{n\times m}\\ min\{n-1,2\}J_{m\times n} &\;\; 2(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{-0+}(G))}(\xi)&=&|\xi I-MD(T^{-0+}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+n-1)I_n-(n-1)J_n} &\;\; -min\{n-1,2\}J_{n\times m}\\ -min\{n-1,2\}J_{m\times n} &\;\; \small{(\xi+2)I_m-2J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 14. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{-1+}(G))}(\xi)&=&(\xi+n-1)^{n-1}(\xi+m+1)^{m-1}\{\xi^2-[(n-1)(n-1)+(m-1)(m+1)]\xi\\ &&+(n-1)(m-1)(n-1)(m+1)-min\{n-1,m+1\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{-1+}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(n-1\) and the remaining \(m\) vertices are with degree \(m+1\). Hence $$MD(T^{-1+}(G))=\left[ \begin{array}{cc} (n-1)(J_n-I_n) & \;\;min\{n-1,m+1\}J_{n\times m}\\ min\{n-1,m+1\}J_{m\times n} &\;\; (m+1)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{-1+}(G))}(\xi)&=&|\xi I-MD(T^{-1+}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+n-1)I_n-(n-1)J_n} &\;\; -min\{n-1,m+1\}J_{n\times m}\\ -min\{n-1,m+1\}J_{m\times n} &\;\; \small{(\xi+m+1)I_m-(m+1)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 15. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{-++}(G))}(\xi)&=&(\xi+n-1)^{n-1}(\xi+2r)^{m-1}\{\xi^2-[(n-1)(n-1)+(m-1)2r]\xi\\ &&+(n-1)(m-1)(n-1)2r-min\{n-1,2r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{-++}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(n-1\) and the remaining \(m\) vertices are with degree \(2r\). Hence $$MD(T^{-++}(G))=\left[ \begin{array}{cc} (n-1)(J_n-I_n) & \;\;min\{n-1,2r\}J_{n\times m}\\ min\{n-1,2r\}J_{m\times n} &\;\; 2r(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{-++}(G))}(\xi)&=&|\xi I-MD(T^{-++}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+n-1)I_n-(n-1)J_n} & -min\{n-1,2r\}J_{n\times m}\\ -min\{n-1,2r\}J_{m\times n} & \small{(\xi+2r)I_m-2rJ_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 16. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{--+}(G))}(\xi)&=&(\xi+n-1)^{n-1}(\xi+m+3-2r)^{m-1}\{\xi^2-[(n-1)(n-1)\\ &&+(m-1)(m+3-2r)]\xi+(n-1)(m-1)(n-1)(m+3-2r)\\ &&-min\{n-1,m+3-2r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{--+}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(n-1\) and the remaining \(m\) vertices are with degree \(R=m+3-2r\). Hence $$MD(T^{--+}(G))=\left[ \begin{array}{cc} (n-1)(J_n-I_n) & \;\;min\{n-1,R\}J_{n\times m}\\ min\{n-1,R\}J_{m\times n} &\;\; R(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{--+}(G))}(\xi)&=&|\xi I-MD(T^{--+}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+n-1)I_n-(n-1)J_n} & -min\{n-1,R\}J_{n\times m}\\ -min\{n-1,R\}J_{m\times n} & \small{(\xi+R)I_m-RJ_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 17. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{00-}(G))}(\xi)&=&(\xi+m-r)^{n-1}(\xi+n-2)^{m-1}\{\xi^2-[(n-1)(m-r)+(m-1)(n-2)]\xi\\ &&+(n-1)(m-1)(m-r)(n-2)-min\{m-r,n-2\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{00-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m-r\) and the remaining \(m\) vertices are with degree \(n-2\). Hence $$MD(T^{00-}(G))=\left[ \begin{array}{cc} (m-r)(J_n-I_n) & \;\;min\{m-r,n-2\}J_{n\times m}\\ min\{m-r,n-2\}J_{m\times n} &\;\; (n-2)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{00-}(G))}(\xi)&=&|\xi I-MD(T^{00-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m-r)I_n-(m-r)J_n} & -min\{m-r,n-2\}J_{n\times m}\\ -min\{m-r,n-2\}J_{m\times n} & \small{(\xi+n-2)I_m-(n-2)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 18. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{01-}(G))}(\xi)&=&(\xi+m-r)^{n-1}(\xi+n+m-3)^{m-1}\{\xi^2-[(n-1)(m-r)+(m-1)(n+m-3)]\xi\\ &&+(n-1)(m-1)(m-r)(n+m-3)-min\{m-r,n+m-3\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{01-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m-r\) and the remaining \(m\) vertices are with degree \(R=n+m-3\). Hence $$MD(T^{01-}(G))=\left[ \begin{array}{cc} (m-r)(J_n-I_n) & \;\;min\{m-r,R\}J_{n\times m}\\ min\{m-r,R\}J_{m\times n} &\;\; R(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{01-}(G))}(\xi)&=&|\xi I-MD(T^{01-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m-r)I_n-(m-r)J_n} & -min\{m-r,R\}J_{n\times m}\\ -min\{m-r,R\}J_{m\times n} & \small{(\xi+R)I_m-RJ_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 19. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{0+-}(G))}(\xi)&=&(\xi+m-r)^{n-1}(\xi+n-4+2r)^{m-1}\{\xi^2-[(n-1)(m-r)+(m-1)(n-4+2r)]\xi\\ &&+(n-1)(m-1)(m-r)(n-4+2r)-min\{m-r,n-4+2r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{0+-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m-r\) and the remaining \(m\) vertices are with degree \(R=n-4+2r\). Hence $$MD(T^{0+-}(G))=\left[ \begin{array}{cc} (m-r)(J_n-I_n) & \;\;min\{m-r,R\}J_{n\times m}\\ min\{m-r,R\}J_{m\times n} &\;\; R(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{0+-}(G))}(\xi)&=&|\xi I-MD(T^{0+-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m-r)I_n-(m-r)J_n} & -min\{m-r,R\}J_{n\times m}\\ -min\{m-r,R\}J_{m\times n} & \small{(\xi+R)I_m-RJ_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 20. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{0--}(G))}(\xi)&=&(\xi+m-r)^{n-1}(\xi+n+m-1-2r)^{m-1}\{\xi^2-[(n-1)(m-r)\\ &&+(m-1)(n+m-1-2r)]\xi+(n-1)(m-1)(m-r)(n+m-1-2r)\\ &&-min\{m-r,n+m-1-2r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{0--}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m-r\) and the remaining \(m\) vertices are with degree \(R=n+m-1-2r\). Hence $$MD(T^{0--}(G))=\left[ \begin{array}{cc} (m-r)(J_n-I_n) & \;\;min\{m-r,R\}J_{n\times m}\\ min\{m-r,R\}J_{m\times n} &\;\; R(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{0--}(G))}(\xi)&=&|\xi I-MD(T^{0--}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m-r)I_n-(m-r)J_n} & -min\{m-r,R\}J_{n\times m}\\ -min\{m-r,R\}J_{m\times n} & \small{(\xi+R)I_m-RJ_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 21. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{10-}(G))}(\xi)&=&(\xi+n+m-r-1)^{n-1}(\xi+n-2)^{m-1}\{\xi^2-[(n-1)(n+m-r-1)+(m-1)(n-2)]\xi\\ &&+(n-1)(m-1)(n+m-r-1)(n-2)-min\{n+m-r-1,n-2\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\(-point-line transformation graph \(T^{10-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R_1=n+m-r-1\) and the remaining \(m\) vertices are with degree \(R_2=n+m-1-2r\). Hence $$MD(T^{10-}(G))=\left[ \begin{array}{cc} R_1(J_n-I_n) & \;\;min\{R_1,R_2\}J_{n\times m}\\ min\{R_1,R_2\}J_{m\times n} &\;\; R_2(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{10-}(G))}(\xi)&=&|\xi I-MD(T^{10-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R_1)I_n-R_1J_n} & -min\{R_1,R_2\}J_{n\times m}\\ -min\{R_1,R_2\}J_{m\times n} & \small{(\xi+R_2)I_m-R_2J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 22. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{11-}(G))}(\xi)&=&(\xi+n+m-r-1)^{n-1}(\xi+n+m-3)^{m-1}\{\xi^2-[(n-1)(n+m-r-1)\\ &&+(m-1)(n+m-3)]\xi+(n-1)(m-1)(n+m-r-1)(n+m-3)\\ &&-min\{n+m-r-1,n+m-3\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{11-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R_1=n+m-r-1\) and the remaining \(m\) vertices are with degree \(R_2=n+m-3\). Hence $$MD(T^{11-}(G))=\left[ \begin{array}{cc} R_1(J_n-I_n) & \;\;min\{R_1,R_2\}J_{n\times m}\\ min\{R_1,R_2\}J_{m\times n} &\;\; R_2(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{11-}(G)}(\xi)&=&|\xi I-MD(T^{11-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R_1)I_n-R_1J_n} & -min\{R_1,R_2\}J_{n\times m}\\ -min\{R_1,R_2\}J_{m\times n} & \small{(\xi+R_2)I_m-R_2J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 23. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{1+-}(G))}(\xi)&=&(\xi+n+m-r-1)^{n-1}(\xi+n+2r-4)^{m-1}\{\xi^2-[(n-1)(n+m-r-1)\\ &&+(m-1)(n+2r-4)]\xi+(n-1)(m-1)(n+m-r-1)(n+2r-4)\\ &&-min\{n+m-r-1,n+2r-4\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{1+-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R_1=n+m-r-1\) and the remaining \(m\) vertices are with degree \(R_2=n+2r-4\). Hence $$MD(T^{1+-}(G))=\left[ \begin{array}{cc} R_1(J_n-I_n) & \;\;min\{R_1,R_2\}J_{n\times m}\\ min\{R_1,R_2\}J_{m\times n} &\;\; R_2(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{1+-}(G))}(\xi)&=&|\xi I-MD(T^{1+-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R_1)I_n-R_1J_n} & -min\{R_1,R_2\}J_{n\times m}\\ -min\{R_1,R_2\}J_{m\times n} & \small{(\xi+R_2)I_m-R_2J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 24. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{1--}(G))}(\xi)&=&(\xi+n+m-r-1)^{n-1}(\xi+n+m-2r-1)^{m-1}\{\xi^2-[(n-1)(n+m-r-1)\\ &&+(m-1)(n+m-2r-1)]\xi+(n-1)(m-1)(n+m-r-1)(n+m-2r-1)\\ &&-min\{n+m-r-1,n+m-2r-1\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{1--}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R_1=n+m-r-1\) and the remaining \(m\) vertices are with degree \(R_2=n+m-2r-1\). Hence $$MD(T^{1--}(G))=\left[ \begin{array}{cc} R_1(J_n-I_n) & \;\;min\{R_1,R_2\}J_{n\times m}\\ min\{R_1,R_2\}J_{m\times n} &\;\; R_2(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{1--}(G))}(\xi)&=&|\xi I-MD(T^{1--}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R_1)I_n-R_1J_n} & -min\{R_1,R_2\}J_{n\times m}\\ -min\{R_1,R_2\}J_{m\times n} & \small{(\xi+R_2)I_m-R_2J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 25. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{+0-}(G))}(\xi)&=&(\xi+m)^{n-1}(\xi+n-2)^{m-1}\{\xi^2-[(n-1)m+(m-1)(n-2)]\xi\\ &&+(n-1)(m-1)m(n-2)-min\{m,n-2\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{+0-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m\) and the remaining \(m\) vertices are with degree \(n-2\). Hence $$MD(T^{+0-}(G))=\left[ \begin{array}{cc} m(J_n-I_n) & \;\;min\{m,n-2\}J_{n\times m}\\ min\{m,n-2\}J_{m\times n} &\;\; (n-2)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{+0-}(G))}(\xi)&=&|\xi I-MD(T^{+0-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m)I_n-mJ_n} & -min\{m,n-2\}J_{n\times m}\\ -min\{m,n-2\}J_{m\times n} & \small{(\xi+n-2)I_m-(n-2)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 26. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{+1-}(G))}(\xi)&=&(\xi+m)^{n-1}(\xi+n+m-3)^{m-1}\{\xi^2-[(n-1)m+(m-1)(n+m-3)]\xi\\ &&+(n-1)(m-1)m(n+m-3)-min\{m,n+m-3\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{+1-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m\) and the remaining \(m\) vertices are with degree \(n+m-3\). Hence $$MD(T^{+1-}(G))=\left[ \begin{array}{cc} m(J_n-I_n) & \;\;min\{m,n+m-3\}J_{n\times m}\\ min\{m,n+m-3\}J_{m\times n} &\;\; (n+m-3)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{+1-}(G))}(\xi)&=&|\xi I-MD(T^{+1-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m)I_n-mJ_n} & -min\{m,n+m-3\}J_{n\times m}\\ -min\{m,n+m-3\}J_{m\times n} &\;\;\; \small{(\xi+n+m-3)I_m-(n+m-3)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 27. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{++-}(G))}(\xi)&=&(\xi+m)^{n-1}(\xi+n+2r-4)^{m-1}\{\xi^2-[(n-1)m+(m-1)(n+2r-4)]\xi\\ &&+(n-1)(m-1)m(n+2r-4)-min\{m,n+2r-4\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{++-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m\) and the remaining \(m\) vertices are with degree \(n+2r-4\). Hence $$MD(T^{++-}(G))=\left[ \begin{array}{cc} m(J_n-I_n) & \;\;min\{m,n+2r-4\}J_{n\times m}\\ min\{m,n+2r-4\}J_{m\times n} &\;\; (n+2r-4)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{++-}(G))}(\xi)&=&|\xi I-MD(T^{++-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m)I_n-mJ_n} & -min\{m,n+2r-4\}J_{n\times m}\\ -min\{m,n+2r-4\}J_{m\times n} &\;\;\; \small{(\xi+n+2r-4)I_m-(n+2r-4)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 28. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{+--}(G))}(\xi)&=&(\xi+m)^{n-1}(\xi+n+m-1-2r)^{m-1}\{\xi^2-[(n-1)m+(m-1)(n+m-1-2r)]\xi\\ &&+(n-1)(m-1)m(n+m-1-2r)-min\{n+m-1-2r,n+m-1-2r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{+--}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m\) and the remaining \(m\) vertices are with degree \(R=n+m-1-2r\). Hence $$MD(T^{+--}(G))=\left[ \begin{array}{cc} m(J_n-I_n) & \;\;min\{m,R\}J_{n\times m}\\ min\{m,R\}J_{m\times n} &\;\; R(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{+--}(G))}(\xi)&=&|\xi I-MD(T^{+--}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m)I_n-mJ_n} & -min\{m,R\}J_{n\times m}\\ -min\{m,R\}J_{m\times n} &\;\;\; \small{(\xi+R)I_m-RJ_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 29. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{-0-}(G))}(\xi)&=&(\xi+n+m+3-4r)^{n-1}(\xi+n-2)^{m-1}\{\xi^2-[(n-1)(n+m+3-4r)\\ &&+(m-1)(n-2)]\xi+(n-1)(m-1)(n+m+3-4r)(n-2)\\ &&-min\{n+m+3-4r,n-2\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{-0-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R=n+m+3-4r\) and the remaining \(m\) vertices are with degree \(n-2\). Hence $$MD(T^{-0-}(G))=\left[ \begin{array}{cc} R(J_n-I_n) & \;\;min\{n-2,R\}J_{n\times m}\\ min\{n-2,R\}J_{m\times n} &\;\; (n-2)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{-0-}(G))}(\xi)&=&|\xi I-MD(T^{-0-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R)I_n-RJ_n} & -min\{n-2,R\}J_{n\times m}\\ -min\{n-2,R\}J_{m\times n} &\;\; \small{(\xi+n-2)I_m-(n-2)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 30. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{-1-}(G))}(\xi)&=&(\xi+n+m+3-4r)^{n-1}(\xi+n+m-3)^{m-1}\{\xi^2-[(n-1)(n+m+3-4r)\\ &&+(m-1)(n+m-3)]\xi+(n-1)(m-1)(n+m+3-4r)(n+m-3)\\ &&-min\{n+m+3-4r,n+m-3\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{-1-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R=n+m+3-4r\) and the remaining \(m\) vertices are with degree \(n+m-3\). Hence $$MD(T^{-1-}(G))=\left[ \begin{array}{cc} R(J_n-I_n) & \;\;min\{n+m-3,R\}J_{n\times m}\\ min\{n+m-3,R\}J_{m\times n} &\;\; (n+m-3)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{-1-}(G))}(\xi)&=&|\xi I-MD(T^{-1-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R)I_n-RJ_n} & -min\{n+m-3,R\}J_{n\times m}\\ -min\{n+m-3,R\}J_{m\times n} & \small{(\xi+n+m-3)I_m-(n+m-3)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 31. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{-+-}(G))}(\xi)&=&(\xi+n+m+3-4r)^{n-1}(\xi+n+2r-4)^{m-1}\{\xi^2-[(n-1)(n+m+3-4r)\\ &&+(m-1)(n+2r-4)]\xi+(n-1)(m-1)(n+m+3-4r)(n+2r-4)\\ &&-min\{n+m+3-4r,n+2r-4\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{-+-}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R=n+m+3-4r\) and the remaining \(m\) vertices are with degree \(n+2r-4\). Hence $$MD(T^{-+-}(G))=\left[ \begin{array}{cc} R(J_n-I_n) & \;\;min\{n+2r-4,R\}J_{n\times m}\\ min\{n+2r-4,R\}J_{m\times n} &\;\; (n+2r-4)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{-+-}(G))}(\xi)&=&|\xi I-MD(T^{-+-}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R)I_n-RJ_n} & -min\{n+2r-4,R\}J_{n\times m}\\ -min\{n+2r-4,R\}J_{m\times n} & \small{(\xi+n+2r-4)I_m-(n+2r-4)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 32. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{---}(G))}(\xi)&=&(\xi+n+m+3-4r)^{n-1}(\xi+n+m-2r-1)^{m-1}\{\xi^2-[(n-1)(n+m+3-4r)\\ &&+(m-1)(n+m-2r-1)]\xi+(n-1)(m-1)(n+m+3-4r)(n+m-2r-1)\\ &&-min\{n+m+3-4r,n+m-2r-1\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{---}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R_1=n+m+3-4r\) and the remaining \(m\) vertices are with degree \(R_2=n+m-2r-1\). Hence $$MD(T^{---}(G))=\left[ \begin{array}{cc} R_1(J_n-I_n) & \;\;min\{R_1,R_2\}J_{n\times m}\\ min\{R_1,R_2\}J_{m\times n} &\;\; R_2(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{---}(G))}(\xi)&=&|\xi I-MD(T^{---}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R_1)I_n-R_1J_n} & -min\{R_1,R_2\}J_{n\times m}\\ -min\{R_1,R_2\}J_{m\times n} & \small{(\xi+R_2)I_m-R_2J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 33. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{001}(G))}(\xi)&=&(\xi+m)^{n-1}(\xi+n)^{m-1}\{\xi^2-[(n-1)m+(m-1)n]\xi\\ &&+(n-1)(m-1)mn-min\{m,n\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{001}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m\) and the remaining \(m\) vertices are with degree \(n\). Hence $$MD(T^{001}(G))=\left[ \begin{array}{cc} m(J_n-I_n) & \;\;min\{m,n\}J_{n\times m}\\ min\{m,n\}J_{m\times n} &\;\; n(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{001}(G))}(\xi)&=&|\xi I-MD(T^{001}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m)I_n-mJ_n} & -min\{m,n\}J_{n\times m}\\ -min\{m,n\}J_{m\times n} &\;\;\; \small{(\xi+n)I_m-nJ_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 34. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{011}(G))}(\xi)&=&(\xi+m)^{n-1}(\xi+n+m-3)^{m-1}\{\xi^2-[(n-1)m+(m-1)(n+m-3)]\xi\\ &&+(n-1)(m-1)m(n+m-3)-min\{m,n+m-3\}^{2}mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{011}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m\) and the remaining \(m\) vertices are with degree \(n+m-3\). Hence $$MD(T^{011}(G))=\left[ \begin{array}{cc} m(J_n-I_n) & \;\;min\{m,n+m-3\}J_{n\times m}\\ min\{m,n+m-3\}J_{m\times n} &\;\; (n+m-3)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{011}(G))}(\xi)&=&|\xi I-MD(T^{011}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m)I_n-mJ_n} & -min\{m,n+m-3\}J_{n\times m}\\ -min\{m,n+m-3\}J_{m\times n} &\;\;\; \small{(\xi+n+m-3)I_m-(n+m-3)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 35. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{0+1}(G))}(\xi)&=&(\xi+m)^{n-1}(\xi+n+2r-2)^{m-1}\{\xi^2-[(n-1)m+(m-1)(n+2r-2)]\xi\\ &&+(n-1)(m-1)m(n+2r-2)-min\{m,n+2r-2\}^{2}mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{0+1}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m\) and the remaining \(m\) vertices are with degree \(n+2r-2\). Hence $$MD(T^{0+1}(G))=\left[ \begin{array}{cc} m(J_n-I_n) & \;\;min\{m,n+2r-2\}J_{n\times m}\\ min\{m,n+2r-2\}J_{m\times n} &\;\; (n+2r-2)(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{0+1}(G))}(\xi)&=&|\xi I-MD(T^{0+1}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m)I_n-mJ_n} & -min\{m,n+2r-2\}J_{n\times m}\\ -min\{m,n+2r-2\}J_{m\times n} &\;\;\; \small{(\xi+n+2r-2)I_m-(n+2r-2)J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 36. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{0-1}(G))}(\xi)&=&(\xi+m)^{n-1}(\xi+n+m+1-2r)^{m-1}\{\xi^2-[(n-1)m+(m-1)(n+m+1-2r)]\xi\\ &&+(n-1)(m-1)m(n+m+1-2r)-min\{m,n+m+1-2r\}^{2}mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{0-1}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(m\) and the remaining \(m\) vertices are with degree \(R=n+m+1-2r\). Hence $$MD(T^{0-1}(G))=\left[ \begin{array}{cc} m(J_n-I_n) & \;\;min\{m,R\}J_{n\times m}\\ min\{m,R\}J_{m\times n} &\;\; R(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{0-1}(G))}(\xi)&=&|\xi I-MD(T^{0-1}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+m)I_n-mJ_n} & -min\{m,R\}J_{n\times m}\\ -min\{m,R\}J_{m\times n} &\;\;\; \small{(\xi+R)I_m-RJ_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 37. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{101}(G))}(\xi)&=&(\xi+n+m-1)^{n-1}(\xi+n)^{m-1}\{\xi^2-[(n-1)(n+m-1)\\ &&+(m-1)n]\xi+(n-1)(m-1)(n+m-1)n-min\{n+m-1,n\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{101}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(n+m-1\) and the remaining \(m\) vertices are with degree \(n\). Hence $$MD(T^{101}(G))=\left[ \begin{array}{cc} (n+m-1)(J_n-I_n) & \;\;min\{n+m-1,n\}J_{n\times m}\\ min\{n+m-1,n\}J_{m\times n} &\;\; n(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{101}(G))}(\xi)&=&|\xi I-MD(T^{101}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+n+m-1)I_n-(n+m-1)J_n} & -min\{n+m-1,n\}J_{n\times m}\\ -min\{n+m-1,n\}J_{m\times n} & \small{(\xi+n)I_m-nJ_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 38. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then the degree exponent polynomial of \(T^{111}(G)\) is $$P_{MD(T^{111}(G))}(\xi)=[\xi-(n+m-1)^{2}][\xi+(n+m-1)]^{n+m-1}.$$

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{111}(G)\) of a regular graph \(G\) of degree \(r\) is a regular graph of degree \(n+m-1\). Hence the result follows from (\ref{eq1.1}).

Theorem 39. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{1+1}(G))}(\xi)&=&(\xi+n+m-1)^{n-1}(\xi+n+2r-2)^{m-1}\{\xi^2-[(n-1)(n+m-1)+(m-1)(n+2r-2)]\xi\\ &&+(n-1)(m-1)(n+m-1)(n+2r-2)-min\{n+m-1,n+2r-2\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{1+1}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R_1=n+m-1\) and the remaining \(m\) vertices are with degree \(R_2=n+2r-2\). Hence $$MD(T^{1+1}(G))=\left[ \begin{array}{cc} R_1(J_n-I_n) & \;\;min\{R_1,R_2\}J_{n\times m}\\ min\{R_1,R_2\}J_{m\times n} &\;\; R_2(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{1+1}(G))}(\xi)&=&|\xi I-MD(T^{1+1}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R_1)I_n-R_1J_n} & -min\{R_1,R_2\}J_{n\times m}\\ -min\{R_1,R_2\}J_{m\times n} & \small{(\xi+R_2)I_m-R_2J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

Theorem 40. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{1-1}(G))}(\xi)&=&(\xi+n+m-1)^{n-1}(\xi+n+m+1-2r)^{m-1}\{\xi^2-[(n-1)(n+m-1)\\ &&+(m-1)(n+m+1-2r)]\xi+(n-1)(m-1)(n+m-1)(n+m+1-2r)\\ &&-min\{n+m-1,n+m+1-2r\}^2mn\}. \end{eqnarray*}

Proof. The generalized \(xyz\)-point-line transformation graph \(T^{1-1}(G)\) of a regular graph \(G\) of degree \(r\) has two types of vertices. The \(n\) vertices with degree \(R_1=n+m-1\) and the remaining \(m\) vertices are with degree \(R_2=n+m+1-2r\). Hence $$MD(T^{1-1}(G))=\left[ \begin{array}{cc} R_1(J_n-I_n) & \;\;min\{R_1,R_2\}J_{n\times m}\\ min\{R_1,R_2\}J_{m\times n} &\;\; R_2(J_m-I_m) \end{array}\right] .$$ Therefore, \begin{eqnarray*} P_{MD(T^{1-1}(G))}(\xi)&=&|\xi I-MD(T^{1-1}(G))|\\ &=&\left| \begin{array}{cc} \small{(\xi+R_1)I_n-R_1J_n} & -min\{R_1,R_2\}J_{n\times m}\\ -min\{R_1,R_2\}J_{m\times n} & \small{(\xi+R_2)I_m-R_2J_m} \end{array}\right|. \end{eqnarray*} Using Lemma 4, we get the required result.

The proof of the following theorems are analogous to that of the above.

Theorem 41. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{+01}(G))}(\xi)&=&(\xi+m+r)^{n-1}(\xi+n)^{m-1}\{\xi^2-[(n-1)(m+r)+(m-1)n]\xi\\ &&+(n-1)(m-1)(m+r)n-min\{n,m+r\}^2mn\}. \end{eqnarray*}

Theorem 42. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{+11}(G))}(\xi)&=&(\xi+m+r)^{n-1}(\xi+m+n-1))^{m-1}\{\xi^2-[(n-1)(m+r)+(m-1)(m+n-1)]\xi\\ &&+(n-1)(m-1)(m+r)(m+n-1)-min\{m+r,m+n-1\}^2mn\}. \end{eqnarray*}

Theorem 43. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{++1}(G))}(\xi)&=&(\xi+m+r)^{n-1}(\xi+n+2r-2)^{m-1}\{\xi^2-[(n-1)(m+r)+(m-1)(n+2r-2)]\xi\\ &&+(n-1)(m-1)(m+r)(n+2r-2)-min\{m+r,n+2r-2\}^2mn\}. \end{eqnarray*}

Theorem 44. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{+-1}(G))}(\xi)&=&(\xi+m+r)^{n-1}(\xi+n+m+1-2r)^{m-1}\{\xi^2-[(n-1)(m+r)+(m-1)(n+m+1-2r)]\xi\\ &&+(n-1)(m-1)(m+r)(n+m+1-2r)-min\{m+r,n+m+1-2r\}^2mn\}. \end{eqnarray*}

Theorem 45. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{-01}(G))}(\xi)&=&(\xi+n+m-1-r)^{n-1}(\xi+n)^{m-1}\{\xi^2-[(n-1)(n+m-1-r)+(m-1)n]\xi\\ &&+n(n-1)(m-1)(n+m-1-r)-min\{n,n+m-1-r\}^2mn\}. \end{eqnarray*}

Theorem 46. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{-11}(G))}(\xi)&=&(\xi+n+m-1-r)^{n-1}(\xi+n+m-1)^{m-1}\{\xi^2-[(n-1)(n+m-1-r)\\ &&+(m-1)(n+m-1)]\xi+(n-1)(m-1)(n+m-1-r)(n+m-1)-(n+m-1)^2mn\}. \end{eqnarray*}

Theorem 47. Let \(G\) be an r-regular graph of order \(n\) and size \(m\). Then \begin{eqnarray*} P_{MD(T^{-+1}(G))}(\xi)&=&(\xi+n+m-1-r)^{n-1}(\xi+n+2r-2)^{m-1}\{\xi^2-[(n-1)(n+m-1-r)\\ &&+(m-1)(n+2r-2)]\xi+(n-1)(m-1)(n+m-1-r)(n+2r-2)\\ &&-min\{n+m-1-r,n+2r-2\}^2mn\}. \end{eqnarray*}

The minimum degree polynomial of subdivision graph \((T^{00+})\) [9], total graph \(T^{+++}\) [9], semitotal point graph \((T^{+0+})\) [17], semitotal line graph \((T^{0++})\) [11] can be found in [18].

Acknowledgments

The authors are thankful to the referee for useful suggestions. The first author is thankful to University Grants Commission (UGC), Government of India, New Delhi, for the financial support through UGC-SAP DRS-III for 2016-2021: F.510/3/DRS-III/2016(SAP-I) dated: \(29^{th}\) Feb. 2016. The second author is thankful to Directorate of Minorities, Government of Karnataka, Bangalore, for the financial support through M. Phil/Ph. D. Fellowship 2017-18: No.DOM/FELLOWSHIP/CR-29/2017-18 dated: \(09^{th}\) Aug. 2017.

Competing Interests

The authors declare that they have no competing interests.