The minimum degree matrix \(MD(G)\) of a graph \(G\) of order \(n\) is an \(n\times n\) symmetric matrix whose \((i,j)^{th}\) entry is \(min\{d_i,d_j\}\) whenever \(i\neq j,\) and zero otherwise, where \(d_i\) and \(d_j\) are the degrees of the \(i^{th}\) and \(j^{th}\) vertices of \(G\), respectively. In the present work, we obtain the minimum degree polynomial of the graphs obtained by some graph operators (generalized \(xyz\)-point-line transformation graphs).
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
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
Lemma 4.[16] If \(a, b, c\) and \(d\) are real numbers, then the determinant of the form
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].