1. Introduction

The concept of degree is analogous to the concept of valence in organic chemistry with a limitation that degree of any vertex in any chemical graph is at most 4. This gives graph theory a broad way to chemistry. It is always interesting to find some properties of graphs or molecular graphs which are invariant. Topological indices and polynomials are foremost among them. Over the last decade there are numerous research papers devoted to topological indices and polynomials. Several topological indices have been defined in the literature. For various topological indices one can refer to [1, 2, 3, 4, 5, 6, 7,8].

Let \(G=(V,E)\) be a simple, undirected graph. Let \(V(G)\) be the vertex set and \(E(G)\) be the edge set of the graph \(G\), respectively. The degree \(d_G(v)\) of a vertex \(v\in V(G)\) is the number of edges incident to it in \(G\). Let \(\{v_1,v_2,...,v_n\}\) be the vertices of \(G\) and let \(d_i=d_G(v_i)\). A graph \(G\) is said to be \(r-\) regular if degree of each vertex in \(G\) is \(r\). A graph is called cycle if it is \(2-\) regular. A cactus graph is a connected graph in which any two simple cycles have at most one vertex in common. Every cycle of cactus graph is chordless and every block of a cactus graph is either an edge or a cycle. If all blocks of a cactus graph are triangular then it is called triangular cacti. If all the triangles of a triangular cactus graph has at most two cut-vertices and each cut-vertex is shared by exactly two triangles then we say that triangular cactus graph is a chain triangular cactus. In chain triangular cactus if we replace triangles by cycles of length 4 then we obtain cacti whose every block is \(C_{4}\), such cacti are called square cacti. For ortho-chain square cactus the cut vertices are adjacent and a para-chain square cactus their cut vertices are not adjacent. Recent study on some cactus chain can be found in [9, 10, 11] and references cited therein. For undefined graph theoretic terminology used in this paper can be found in [12].

The general form of degree-based topological index of a graph is given by $$TI(G)=\sum_{e=uv\in G}f(d_G(u),d_G(v))$$ where \(f=f(x,y)\) is a function appropriately chosen for the computation. Table 1 gives the standard topological indices defined by \(f(x,y)\).

The \(M-polynomial\) [13] was introduced in 2015 by Deutch and Klav\(\check{\text{z}}\)ar and is found useful in determining many degree-based topological indices (listed in Table 1).

Definition 1.[13] Let \(G\) be a graph. Then \(M-polynomial\) of \(G\) is defined as $$M(G;x,y)=\sum _{i\le j}m_{ij}(G)x^iy^j,$$ where \(m_{ij}, i, j\ge1\), is the number of edges \(uv\) of \(G\) such that \(\{d_G(u), d_G(v)\}=\{i,j\}\) [14].

Recently, the study of \(M-polynomial\) are reported in [15, 16, 17, 18, 19]. The Table 1 was given by Deutch and Klav\(\check{\text{z}}\)ar to derive degree based topological indices from the \(M-\) polynomial.

Table 1. [13] Derivation of some degree-based topological indices from \(M-\)polynomial.

Notation	Topological Index	\(f(x,y)\)	Derivation from \(M(G;x,y)\)
\(M_1(G)\)	First Zagreb	\(x+y\)	\((D_x+D_y)(M(G;x,y)) |_{x=y=1}\)
\(M_2(G)\)	Second Zagreb	\(xy\)	\((D_xD_y)(M(G;x,y)) |_{x=y=1}\)
\(^{m}M_2(G)\)	Second modified Zagreb	\(\frac{1}{xy}\)	\((S_xS_y)(M(G;x,y)) |_{x=y=1}\)
\(S_{D}(G)\)	Symmetric division index	\(\frac{x^2+y^2}{xy}\)	\((D_xS_y+D_yS_x)(M(G;x,y)) |_{x=y=1}\)
\(H(G)\)	Harmonic	\(\frac{2}{x+y}\)	\(2S_xJ(M(G;x,y)) |_{x=1}\)
\(I_{n}(G)\)	Inverse sum index	\(\frac{xy}{x+y}\)	\(S_xJD_xD_y(M(G;x,y)) |_{x=1}\)
\(R_{\alpha}(G)\)	General Randi\'{c} index	\((xy)^\alpha\)	\(D_xD_y(M(G;x,y)) |_{x=y=1}\)

Where \(D_x=x\frac{\partial f(x,y)}{\partial x}\), \(D_y=y\frac{\partial f(x,y)}{\partial y}\), \(S_x=\int_{0}^{x}\frac{f(t,y)}{t}dt\), \(S_y=\int_{0}^{y}\frac{f(x,t)}{t}dt\) and \(J(f(x,y))=f(x,x)\) are the operators. The Table 2 is given by us in [20] which gives operators to derive general sum connectivity index and the first general Zagreb index from the \(M-\)polynomial.

Table 2.[20] New operator to derive degree-based topological indices from \(M-polynomial\).

Notation	Topological Index	\(f(x,y)\)	Derivation from \(M(G;x,y)\)
\(\chi_{\alpha}(G)\)	General sum connectivity [7]	\((x+y)^\alpha\)	\(D^{\alpha}_x(J(M(G;x,y))) |_{x=1}\)
\(M^{\alpha}_1(G)\)	First general Zagreb [21]	\(x^{\alpha-1}+y^{\alpha-1}\)	\((D^{\alpha-1}_x+D^{\alpha-1}_y)(M(G;x,y)) |_{x=y=1}\)

Note 1. Hyper Zagreb index is obtained by taking \(\alpha=2\) in general sum connectivity index.
Note 2. Taking \(\alpha=2,3\) in first general Zagreb index, first Zagreb and forgotten \((F-index)\) topological indices are obtained respectively. The general Zagreb index or \((a,b)-\) Zagreb index was introduced by Azari et al. [22], which is a generalized version of vertex-degree-based topological index and is defined as $$M_{(a,b)}(G)=\sum _{uv\in E(G)} (d_{G}(u)^{a} d_{G}(v)^{b}+d_{G}(u)^{b}d_{G}(v)^{a}).$$ The importance of general Zagreb index is that from this index, one can derive seven more topological indices as given in [9]. But in fact, general Zagreb index can also be obtained from the \(M-\)polynomial with suitable operator, which we present in the Table 3.

Table 3. Operator to derive general Zagreb index from \(M-polynomial\).

Notation	Topological Index	\(f(x,y)\)	Derivation from \(M(G;x,y)\)
\(M_{(a,b)}(G)\)	General Zagreb index [22]	\(x^{a}y^{b}+x^{b}y^{a}\)	\((D^{a}_xD^{b}_y+D^{b}_xD^{a}_y)(M(G;x,y)) |_{x=y=1}\)

Thus, general Zagreb index can be derived from the \(M-\) polynomial. To show this we derive the \(M-\) polynomial of cactus graphs in the next section and compare with the results of Nilanjan De obtained in [9].

2. \(M-\)Polynomials of cactus chains

In this section, we obtain \(M-\) polynomials of two general cactus chains namely para cacti chain and ortho-cacti chain of cycles. We first compute \(M-\) polynomial of para cacti chain of cycles denoted by \(C^{n}_{m}\) where \(m\) is the length of each cycle and \(n\) is the length of the chain. Every block in \(C^{n}_{m}\) is a cycle \(C_{m}\).

The following theorem gives the \(M-\) polynomial of \(C^{n}_{m}.\)

Theorem 2. Let \(C^{n}_{m}\) be para cacti chain of cycles for \(m\ge 3, n\ge 2.\) Then \begin{eqnarray*} M(C^{n}_{m};x,y)&=&(mn-4n+4)x^2y^2+4(n-1)x^2y^4. \end{eqnarray*}

Proof. The para cacti chain of cycles \(C^{n}_{m}\) has \(mn-n+1\) vertices and \(mn\) edges. The edge set of \(C^{n}_{m}\) can be partitioned as, \begin{eqnarray*} |E_{\{2,2\}}|&=&|\{uv\in E(C^{n}_{m}): d_{C^{n}_{m}}(u)=2 \;\; and \;\; d_{C^{n}_{m}}(v)=2\}|=(mn-4n+4).\\ |E_{\{2,4\}}|&=&|\{uv\in E(C^{n}_{m}): d_{C^{n}_{m}}(u)=2 \;\; and \;\; d_{C^{n}_{m}}(v)=4\}|=4(n-1). \end{eqnarray*} Thus, by using definition of \(M-polynomial\) we have, $$ M(C^{n}_{m};x,y)=(mn-4n+4)x^2y^2+4(n-1)x^2y^4. $$

Corollary 3. Let \(C^{n}_{m}\) be para cacti chain of cycles for \(m\ge 3, n\ge 2.\) Then $$M_{a,b}(C^{n}_{m})=2(mn-4n+4)2^{a+b}+4(n-1)2^{a+b}(2^{a}+2^{b}).$$

Proof. To derive general Zagreb index from \(M-\)polynomial we use the operator given in Table 3. Now, \begin{eqnarray*} M_{a,b}(C^{n}_{m})&=&(D^{a}_xD^{b}_y+D^{b}_xD^{a}_y)(M(C^{n}_{m};x,y)) |_{x=y=1}\\ &=& (D^{a}_xD^{b}_y+D^{b}_xD^{a}_y)((mn-4n+4)x^2y^2+4(n-1)x^2y^4)|_{x=y=1}\\ &=& (D^{a}_xD^{b}_y)((mn-4n+4)x^2y^2+4(n-1)x^2y^4)\\&&+(D^{b}_xD^{a}_y)((mn-4n+4)x^2y^2+4(n-1)x^2y^4)|_{x=y=1}\\ &=&2(mn-4n+4)2^{a+b}+4(n-1)2^{a+b}(2^{a}+2^{b}). \end{eqnarray*}

The expression obtained above, i.e., $$M_{a,b}(C^{n}_{m})=2(mn-4n+4)2^{a+b}+4(n-1)2^{a+b}(2^{a}+2^{b})$$ is same expression obtained in [9], Theorem 1. This shows that the \(M-\)polynomial has an extra advantage than the general Zagreb index as one can derive about 10 (listed in Tables 1, 2 and 3) degree-based topological indices from the \(M-\)polynomial including general Zagreb index so far. The following corollary gives the several topological indices of para cacti chain of cycles derived from \(M-\)polynomial.

Corollary 4. Let \(C^{n}_{m}\) be para cacti chain of cycles for \(m\ge 3, n\ge 2.\) Then

1. \(M_1(C^{n}_{m})=4mn+8n-8.\)
2. \(M_2(C^{n}_{m})=4mn+16n-16.\)
3. \(^{m}M_2(C^{n}_{m})= \frac{mn-2n+2}{4}.\)
4. \(S_{D}(C^{n}_{m})= 2mn+2n-2.\)
5. \(H(C^{n}_{m})=\frac{3mn-8n-8}{6}.\)
6. \(I_{n}(C^{n}_{m})= \frac{3mn+4n-4}{3}.\)
7. \(\chi_{\alpha}(C^{n}_{m})=(mn-4n+4)4^{\alpha}+4(n-1)6^{\alpha}.\)
8. \(M^{\alpha}_{1}(C^{n}_{m})=(mn-4n+4)2^{\alpha}+(n-1)2^{2\alpha+1}(2^{\alpha-1}+1).\)
9. \(R_{\alpha}(C^{n}_{m})=(mn-4n+4)2^{2\alpha}+(n-1)2^{3\alpha+2}.\)

Proof. Using the above Theorem 2 and column 4 of Table 1, we get the desired results.

Corollary 5. Let \(Q_{n}\) be para-chain square cactus graph for \( n\ge 2.\) Then \begin{eqnarray*} M(Q_{n};x,y)&=&4x^2y^2+4(n-1)x^2y^4. \end{eqnarray*}

Proof. Taking \(m=4\) in the Theorem 2, we get the desired result.

Corollary 6. Let \(Q_{n}\) be para-chain square cactus graph for \( n\ge 2.\) Then

1. \(M_1(Q_{n})=24n-8.\)
2. \(M_2(Q_{n})=32n-16.\)
3. \(^{m}M_2(Q_{n})= \frac{n+1}{2}.\)
4. \(S_{D}(Q_{n})= 10n-2.\)
5. \(H(Q_{n})=\frac{4n-8}{6}.\)
6. \(I_{n}(Q_{n})=\frac{16n-4}{3}.\)
7. \(\chi_{\alpha}(Q_{n})=4^{\alpha+1}+4(n-1)6^{\alpha}.\)
8. \(M^{\alpha}_{1}(Q_{n})=2^{\alpha+2}+(n-1)2^{2\alpha+1}(2^{\alpha-1}+1).\)
9. \(R_{\alpha}(Q_{n})=2^{2\alpha+2}+(n-1)2^{3\alpha+2}.\)

Corollary 7. Let \(Q_{n}\) be para-chain square cactus graph for \( n\ge 2.\) Then \begin{eqnarray*} M(Q_{n};x,y)&=&(2n+4)x^2y^2+4(n-1)x^2y^4. \end{eqnarray*}

Proof. Taking \(m=6\) in the Theorem 2, we get the desired result.

Corollary 8. Let \(L_{n}\) be para-chain hexagonal cactus graph for \( n\ge 3.\) Then

1. \(M_1(L_{n})=32n-8.\)
2. \(M_2(L_{n})=40n-16.\)
3. \(^{m}M_2(L_{n})= \frac{2n+1}{2}.\)
4. \(S_{D}(L_{n})= 14n-2.\)
5. \(H(L_{n})=\frac{5n-4}{3}.\)
6. \(I_{n}(L_{n})=\frac{22n-4}{3}.\)
7. \(\chi_{\alpha}(L_{n})=(2n+4)4^{\alpha}+4(n-1)6^{\alpha}.\)
8. \(M^{\alpha}_{1}(L_{n})=(2n+4)2^{\alpha}+(n-1)2^{2\alpha+1}(2^{\alpha-1}+1).\)
9. \(R_{\alpha}(L_{n})=(2n+4)2^{2\alpha}+(n-1)2^{3\alpha+2}.\)

We now consider the ortho-chain cycles with cut-vertices are adjacent. Let \(CO^{n}_{m}\) be ortho-chain cactus graph, where \(m\) is the length of each cycle and \(n\) is the length of the chain. It is easy to see that \(|V(CO^{n}_{m})|=mn-n+1\) and \(|E(CO^{n}_{m})|=mn.\) In the following theorem we obtain \(M-\)polynomial of \(CO^{n}_{m}.\)

Theorem 9. Let \(CO^{n}_{m}\) be ortho cacti chain of cycles for \(m\ge 3, n\ge 2.\) Then \begin{eqnarray*} M(CO^{n}_{m};x,y)&=&(mn-3n+2)x^2y^2+2nx^2y^4+(n-1)x^4y^4. \end{eqnarray*}

Proof. The \(CO^{n}_{m}\) be ortho-chain cacti of cycles has \(mn-n+1\) vertices and \(mn\) edges. The edge partition of \(CO^{n}_{m}\) is given by, \begin{eqnarray*} E_{\{2,2\}}&=&{\{uv\in E(CO^{n}_{m}): d_{CO^{n}_{m}}(u)=2 \;\; and \;\; d_{CO^{n}_{m}}(v)=2\}},\\ E_{\{2,4\}}&=&{\{uv\in E(CO^{n}_{m}): d_{CO^{n}_{m}}(u)=2 \;\; and \;\; d_{CO^{n}_{m}}(v)=4\}},\\ E_{\{4,4\}}&=&{\{uv\in E(CO^{n}_{m}): d_{CO^{n}_{m}}(u)=4 \;\; and \;\; d_{CO^{n}_{m}}(v)=4\}},\\ \text{Now,}\;\; |E_{\{2,2\}}|&=&mn-3m+2,\\ |E_{\{2,4\}}|&=&2n,\\ |E_{\{4,4\}}|&=&n-1. \end{eqnarray*} Thus, the \(M-polynomial\) of \(CO^{n}_{m}\) is $$ M(CO^{n}_{m};x,y)=(mn-3n+2)x^2y^2+2nx^2y^4+(n-1)x^4y^4.$$

Corollary 10. Let \(CO^{n}_{m}\) be ortho cacti chain of cycles for \(m\ge 3, n\ge 2.\) Then

1. \(M_1(CO^{n}_{m})=4mn+8n-8.\)
2. \(M_2(CO^{n}_{m})=4mn+20n-24.\)
3. \(^{m}M_2(CO^{n}_{m})= \frac{4mn-7n+7}{16}.\)
4. \(S_{D}(CO^{n}_{m})= 2mn+n.\)
5. \(H(CO^{n}_{m})=\frac{6mn-7n+9}{12}.\)
6. \(I_{n}(CO^{n}_{m})= \frac{3mn+5n}{3}.\)
7. \(\chi_{\alpha}(CO^{n}_{m})=(mn-3n+2)4^{\alpha}+2n6^{\alpha}+(n-1)8^\alpha.\)
8. \(M^{\alpha}_{1}(CO^{n}_{m})=(mn-3n+2)2^{\alpha}+(n-1)2^{2\alpha-1}+n2^{\alpha}(2^{\alpha-1}+1).\)
9. \(R_{\alpha}(CO^{n}_{m})=(mn-3n+2)2^{2\alpha}+n2^{3\alpha+1}+(n-1)4^{2\alpha}.\)

Proof. Using the Theorem 9 and column 4 of Table 1, we get the desired results.

Now, we consider chain triangular cactus as shown in Figure 3, denoted by \(T_{n}\), where \(n\) is the length of the \(T_{n}\). \(T_{n}\) is special case of \(CO^{n}_{m}\) for \(m=3.\)

Corollary 11. Let \(T_{n}\) be the chain triangular cactus for \(n\ge 2\). Then \begin{eqnarray*} M(T_{n};x,y)&=&2x^2y^2+2nx^2y^4+(n-1)x^4y^4. \end{eqnarray*}

Proof. The proof follows by substituting \(m=3\) in Theorem 9.

Corollary 12. Let \(T_{n}\) be the chain triangular cactus for \(n\ge 2\). Then

1. \(M_1(T_{n})=20n-8.\)
2. \(M_2(T_{n})=32n-24.\)
3. \(^{m}M_2(T_{n})= \frac{5n+7}{16}.\)
4. \(S_{D}(T_{n})= 7n.\)
5. \(H(T_{n})=\frac{11n+9}{12}.\)
6. \(I_{n}(T_{n})= \frac{14n}{3}.\)
7. \(\chi_{\alpha}(T_{n})=2^{2\alpha+1}+2n6^{\alpha}+(n-1)8^\alpha.\)
8. \(M^{\alpha}_{1}(T_{n})=2^{\alpha+1}+(n-1)2^{2\alpha-1}+n2^{\alpha}(2^{\alpha-1}+1).\)
9. \(R_{\alpha}(T_{n})=2^{2\alpha+1}+n2^{3\alpha+1}+(n-1)4^{2\alpha}.\)

Corollary 13. Let \(O_{n}\) be the ortho-chain square cactus for \(n\ge 2\). Then \begin{eqnarray*} M(O_{n};x,y)&=&(n+2)x^2y^2+2nx^2y^4+(n-1)x^4y^4. \end{eqnarray*}

Proof. The proof follows by substituting \(m=4\) in Theorem 9.

Corollary 14. Let \(O_{n}\) be the ortho-chain square cactus for \(n\ge 2\). Then

1. \(M_1(O_{n})=24n-8.\)
2. \(M_2(O_{n})=36n-24.\)
3. \(^{m}M_2(O_{n})= \frac{9n+7}{16}.\)
4. \(S_{D}(O_{n})= 9n.\)
5. \(H(O_{n})=\frac{17n+9}{12}.\)
6. \(I_{n}(O_{n})= \frac{17n}{3}.\)
7. \(\chi_{\alpha}(O_{n})=(n+2)4^{\alpha}+2n6^{\alpha}+(n-1)8^{\alpha}.\)
8. \(M^{\alpha}_{1}(O_{n})=(n+2)2^{\alpha}+(n-1)2^{2\alpha-1}+n2^{\alpha}(2^{\alpha-1}+1).\)
9. \(R_{\alpha}(O_{n})=(n+2)2^{2\alpha}+n2^{3\alpha+1}+(n-1)4^{2\alpha}.\)

The graph \(Q(m,n)\) is derived from \(K_{m}\) and \(m\) copies of \(K_{n}\) by identifying every vertex of \(K_{m}\) with a vertex of one \(K_{n}\) [11]. Here we compute the \(M-\) polynomial of the graph \(Q(m,n)\) and derive some other topological indices from it. The graph \(Q(m,n)\) is depicted in Figure 5.

Theorem 15. Let \(Q(m,n)\) be ortho-chain for \(m, n\ge 2.\) Then \begin{eqnarray*} M(Q(m,n);x,y)&=&\frac{m(n-1)(n-2)}{2}x^{n-1}y^{n-1}+m(n-1)x^{n-1}y^{m+n-2}+ \frac{m(n-1)}{2}x^{m+n-2}y^{m+n-2}. \end{eqnarray*}

Proof. The edge partition of \(Q(m,n)\) is partitioned into the following subsets, \begin{eqnarray*} E_{1}&=&{\{uv\in E(Q(m,n)): d_{Q(m,n)}(u)=d_{Q(m,n)}(v)=(n-1)\}},\\ E_{2}&=&{\{uv\in E(Q(m,n)): d_{Q(m,n)}(u)=(n-1) \;\; and \;\; d_{Q(m,n)}(v)=(m+n-2)\}},\\ E_{3}&=&{\{uv\in E(Q(m,n)): d_{Q(m,n)}(u)=d_{Q(m,n)}(v)=(m+n-2)\}},\\ \text{Now,}\;\; |E_{1}|&=&\frac{m(n-1)(n-2)}{2},\\ |E_{2}|&=&m(n-1),\\ |E_{3}|&=&\frac{m(n-1)}{2}. \end{eqnarray*} Thus, the \(M-polynomial\) of \(Q(m,n)\) is $$M(Q(m,n);x,y)=\frac{m(n-1)(n-2)}{2}x^{n-1}y^{n-1}+m(n-1)x^{n-1}y^{m+n-2}+ \frac{m(n-1)}{2}x^{m+n-2}y^{m+n-2}.$$

In the para-cacti chain \(C^{n}_{m}\), if we join a new vertex and each cycle of length \(m \ge 3\), \((C_{m}+K_{1})\) then we call it as wheel chain, denoted by \(W^{n}_{m}.\) The number of vertices and edges of \(W^{n}_{m}\) are \(mn+1\) and \(2mn\), respectively. In the following theorem we calculate the \(M-\)polynomial of \(W^{n}_{m}.\)

Theorem 16. Let \(W^{n}_{m}\) be wheel chain for \(m \ge 3, n\ge 2.\) Then \begin{eqnarray*} M(W^{n}_{m};x,y)&=&2(m(n-1)-3n+4)x^{3}y^{3}+4(n-1)x^{3}y^{6}+2(m-1)x^{3}y^{m}+2(n-1)x^{6}y^{m}. \end{eqnarray*}

Proof. The edge partition of \(W^{n}_{m}\) is partitioned into the following subsets, \begin{eqnarray*} E_{1}&=&{\{uv\in E(W^{n}_{m}): d_{W^{n}_{m}}(u)=d_{W^{n}_{m}}(v)=3\}},\\ E_{2}&=&{\{uv\in E(W^{n}_{m}): d_{W^{n}_{m}}(u)=3 \;\; and \;\; d_{W^{n}_{m}}(v)=6\}},\\ E_{3}&=&{\{uv\in E(W^{n}_{m}): d_{W^{n}_{m}}(u)=3 \;\; and \;\; d_{W^{n}_{m}}(v)=m\}},\\ E_{4}&=&{\{uv\in E(W^{n}_{m}): d_{W^{n}_{m}}(u)=6 \;\; and \;\; d_{W^{n}_{m}}(v)=m\}},\\ \text{Now,}\;\; |E_{1}|&=&mn-4n+4,\\ |E_{2}|&=&4(n-1),\\ |E_{3}|&=&mn-2n+2,\\ |E_{4}|&=&2(n-1). \end{eqnarray*} Thus, the \(M-polynomial\) of \(W^{n}_{m}\) is $$M(W^{n}_{m};x,y)=(mn-4n+4)x^{3}y^{3}+4(n-1)x^{3}y^{6}+(mn-2n+2)x^{3}y^{m}+2(n-1)x^{6}y^{m}.$$

Taking \(m=4\) in the Theorem 16, we get the following corollary for \(M-\) polynomial of \(W^{n}_{4}\). The graph \(W^{n}_{4}\) is shown in Figure 6.

Corollary 17. Let \(W^{n}_{4}\) be the wheel chain graph for \(n\ge 2\). Then \begin{eqnarray*} M(W^{n}_{4};x,y)&=&4x^{3}y^{3}+4(n-1)x^{3}y^{6}+2(n+1)x^{3}y^{4}+2(n-1)x^{6}y^{4}. \end{eqnarray*}

Using Theorem 16 and Tables 1, 2 and 3 one can easily obtain topological indices of wheel chain graph \(W^{n}_{4}\). Since it is a routine task, we omit the calculation here.

3. Conclusion

In this paper, we have obtained \(M-\)polynomials of some cactus graphs. We have shown that the general Zagreb index can also be derived from the \(M-\)polynomial of a graph. As an example, we have derived general Zagreb index of cactus graphs from the \(M-\) polynomial of cactus graphs. The Theorem 1 obtained in [9] is the same as the corollary 3 proved in this paper. Thus generalising the results of general Zagreb index of some cactus graphs.

Author Contributions

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

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

Let \(G=(V(G),E(G))\) (or \(G=(V,E)\) for short if not ambiguous) be a simple, finite and undirected \((p,q)\)-graph of order \(|V|=p\) and size \(|E| = q\). For integers \(a,b\) with \(a\le b\), let \([a,b]=\{n\in Z \,|\, a\le n\le b\}\). All notation not defined in this paper can be found in [1].

The concept of prime labeling was originated by Entringer and it was introduced in a paper by Tout et al. [2]. A graph \(G\) with \(p\) vertices and \(q\) edges is said to have a prime labeling if function \(f : V \to [1,p]\) is bijective and for every edge \(e = uv\) of \(G\), \(GCD(f(u), f(v)) = 1\). For simplicity, we will use \((a,b)\) to denote \(GCD(a,b)\). Currently, the two most prominent open conjectures involving prime labelings are the following:

1. All tree graphs have a prime labeling (Entringer-Tout Conjecture);
2. All unicyclic graphs have a prime labeling (Seoud and Youssef [3]).

In 2011, Haxell and Pikhurko [4] proved that all large trees are prime. In 1991, Deretsky et al. [5] introduced the notion of dual of prime labeling which is known as vertex prime labeling. A graph with \(q\) edges has vertex prime labeling if its edges can be labeled with distinct integers \([1,q]\) such that for each vertex of degree at least 2, the greatest common divisor of the labels on its incident edges is 1. A conjecture: ``Any 2-regular graph has a vertex prime labeling if and only if it does not have two odd cycles." was proposed.

An excellent survey on graph labeling is maintained by Gallian [6]. In [7], we introduce another prime labeling of graphs.

Definition 1. Let \(G= (V,E)\) be a \((p,q)\)-graph. A bijection \(f: E \to [1,q]\) is called an edge-prime labeling if for each edge \(uv\) in \(E\), we have \((f^+(u),f^+(v))=1\), where \(f^+(u) = \sum_{uw\in E} f(uw)\). A graph that admits an edge-prime labeling is called an edge-prime graph.

Among others, we proved that all 2-regular graphs, complete bipartite graphs \(K(2,n)\), the bipartite graph \(K(2,n)+ K(2,n)\) \((n\ge 2)\), disjoint union of paths with at most one \(P_2\) component, the generalized theta graph having \(n\ge 3\) internally disjoint paths of length 3 (respectively, 4), or having 3 internally disjoint paths of length \(n\ge 2\) and certain family of trees are edge-prime. It is a conjecture that all trees of diameter at least 3 are edge-prime.

Let \(n(G)\) (or \(nG\) if no ambiguity) denote the disjoint union of \(n\) copies of graph \(G\). In this paper we obtained some sufficient conditions for graphs with regular component(s) to admit or not admit an edge-prime labeling. Consequently, we proved that if \(G\) is a cubic graph with every component is of order \(4, 6\) or \(8\), then \(G\) is edge-prime if and only if \(G\not\cong K_4\) or \(nK(3,3)\), \(n\equiv2,3\pmod{4}\). In what follows, we only consider cubic graphs unless specified otherwise.

It is clear that an edge labeling \(f: E(G)\to [1,|E(G)|]\) such that for each edge \(uv\), \(f^+(u)\) and \(f^+(v)\) are not both even, and \(|f^+(u) - f^+(v)| = 2^m, m\ge 0\) is an edge-prime labeling.

Suppose \(f\) is an edge labeling of a graph \(H=(V,E)\). For each edge \(xy\) of \(H\), let \(d_{xy}=|f^+(x) - f^+(y)|\). We shall use this notation throughout this paper.

1. We say \(f\) has Property (A) if \(d_{xy}=2^m\) for some \(m\ge 0\), and each \(xy\in E(H)\).
2. Let \(G\) be an edge-prime graph of size \(q\). Suppose \(H\) is \(r\)-regular with an edge-prime labeling \(f\). We say \(f\) has Property (B) if \((d_{xy}, rq)=d_{xy}\), for each \(xy\in E(H)\). Moreover, \(f\) has Property (C) if for each \(xy\in E(H)\), \(d_{xy}=2^m\) for some \(m\ge 0\) or \((d_{xy}, rq)=d_{xy}\).

Theorem 2. Let \(G\) be an edge-prime graph. Suppose \(H\) is an \(r\)-regular graph that admits an edge-prime labeling having Property (A).

1. If \(r|E(G)|\) is even or \(|E(G+H)|\) is odd, then \(G+H\) is edge-prime.
2. If \(r|E(G)|\) is odd and \(|E(H)|\) is even, then \(G+H\) is edge-prime.

Proof. Let \(f_1\) and \(f_2\) be edge-prime labelings of \(G\) and \(H\) respectively with \(f_2\) satisfying the given condition.

(a). Suppose \(r|E(G)|\) is even. Define \(g\) such that \(g(e) = f_1(e)\) for \(e\in E(G)\), and \(g(e) = f_2(e) + |E(G)|\) for \(e\in E(H)\). Consider an edge \(uv\in E(G+H)\). It suffices to consider \(uv\in E(H)\). Without loss of generality, assume \(f^+_2(u)\) is odd. Now, \(g^+(u)=r|E(G)|+f_2^+(u)\) is odd. \[(g^+(u),g^+(v)) = (r|E(G)|+ f^+_1(u), r|E(G)|+ f^+_2(v)) = (g^+(u), f^+_2(u)- f^+_2(v)) = (g^+(u), 2^m) = 1.\] Suppose \(|E(G+H)|=q\) is odd. Define \(g\) such that \(g(e) = f_1(e)\) for \(e\in E(G)\), and \(g(e) = q+1- f_2(e)\) for \(e\in E(H)\). Again, we only need to consider \(uv\in E(H)\). Without loss of generality, assume \(f^+_2(u)\) is odd. Now, \(g^+(u)=r(q+1)-f_2^+(u)\) is odd. We have \[(g^+(u),g^+(v)) = (rq - f^+_2(u)+r, rq - f^+_2(v)+r) = (g^+(u), f^+_2(u)- f^+_2(v)) = (g^+(u), 2^m) = 1.\] \[(g^+(u),g^+(v)) = (rq - f^+_2(u)+r, rq - f^+_2(v)+r) = (g^+(u), f^+_2(u)- f^+_2(v)) = (g^+(u), 2^m) = 1.\] Hence, \(g\) is also an edge-prime labeling.

(b). Suppose \(r|E(G)|\) is odd and \(|E(H)|\) is even. Clearly, \(|E(G)|\) is odd, and thus \(|E(G+H)|\) is odd. From (a), we get that \(G+H\) is also edge-prime.

Theorem 3. Suppose \(G\) is a graph of even size \(q\) such that every component of \(G\) is regular. If \(G\) admits an edge-prime labeling \(f\) having Property (A), then \(nG\) is edge-prime for \(n\ge 2\).

Proof. For \(k\ge 1\), define a partial edge labeling \(g_k\) for the \(k\)-th copy of \(G\) such that \(g_k(e)=f(e) + (k-1)q\). Clearly, every induced vertex label in \(G\) and the corresponding vertex in the \(k\)-th copy of \(nG\) have the same parity. Moreover, every 2 induced adjacent vertex labels differ by \(2^m, m\ge 0\). Hence, \(nG\) is edge-prime.

Theorem 4. Suppose every component of \(G\) is an even regular graph. If \(G\) admits an edge-prime labeling \(f\) having Property (A), then \(nG\) is edge-prime for \(n\ge 2\).

Proof. Let \(H\) be a component of \(G\), which is \(r\)-regular. By Theorem 2 (a), since \(r\) is even, \(G+H\) is edge-prime. We may repeat this procedure for each component of \(G\) to show that \(2G\) is edge-prime and so is \(nG\).

Lemma 5. Suppose \(G\) is a graph of size \(q\) with an edge-prime labeling \(f\). If \(H\) is an \(r\)-regular graph with an edge-prime labeling \(g\) having Property (B), then \(G+H\) is edge-prime.

Proof. We define an edge labeling \(h\) for \(G+H\) by \[h(e)=\begin{cases}f(e),&\mbox{ if }e\in E(G);\\ g(e)+q,&\mbox{ if }e\in E(H).\end{cases}\] We only need to consider edge \(uv\in E(H)\). Now, \(h^+(u) = g^+(u) + rq\) and \(h^+(v) = g^+(v) + rq\). Since \((d_{uv},g^+(v))=(g^+(u), g^+(v))=1\) and \(rq\) is a multiple of \(d_{uv}\), we have \((h^+(u), h^+(v)) = (g^+(u) + rq, g^+(v) + rq) = (d_{uv},g^+(v) + rq) = 1\). Hence \(h\) is an edge-prime labeling and \(G+H\) is edge-prime.

When \(rq\) is even, or \(rq\) is odd and \(|E(H)|\) is even, we may relax the condition of Lemma 5 to have the following corollary.

Corollary 6. Suppose \(G\) is a graph of size \(q\) with an edge-prime labeling \(f\), and \(H\) is an \(r\)-regular graph with an edge-prime labeling \(g\).

1. Suppose \(rq\) is even. If \(g\) has Property (C), then \(G+H\) is edge-prime.
2. Suppose \(rq\) is odd and \(|E(H)|\) is even. If \(g\) has Property (C), then \(G+H\) is edge-prime.

Proof.

(a). We define an edge labeling \(h\) for \(G+H\) as in Lemma 5. By the proof of Lemma 5, we only need to consider the case when \(d_{uv}=2^m\) with \(m\ge 0\), where \(uv\in E(H)\). Without loss of generality, we may assume \(g^+(v)\) is odd. This implies that \(h^+(v)\) is odd. By the same computation in the proof of Lemma 5, we have \((h^+(u), h^+(v)) =(d_{uv}, h^+(v))=1\). Hence \(G+H\) is edge-prime.

(b). Suppose \(rq\) is odd and \(|E(H)|\) is even. Clearly, \(q\) is odd and \(|E(G+H)|\) is odd. Similar to (a), we only need to consider the case when \(d_{uv} = 2^m\) with \(m\ge 0\), where \(uv\in E(H)\). By Theorem 2 (b), we have \(G+H\) is edge-prime.

Theorem 7. Suppose \(G\) is an \(r\)-regular graph of size \(q\) with an edge-prime labeling \(f\). If \(f\) has Property (B), then \(nG\) is edge-prime.

Proof. The theorem follows by applying Lemma 5 repeatedly.

Corollary 8. Suppose \(G\) is an \(r\)-regular graph of size \(q\) with an edge-prime labeling \(f\), where \(rq\) is even. If \(f\) has Property (C), then \(nG\) is edge-prime.

Proof. This follows by applying Corollary 6 repeatedly.

Remark 1. Suppose \(G\) is an edge-prime graph of size \(q\), where \(q\equiv 0\pmod{6}\). Suppose \(H\) is a cubic graph with an edge-prime labeling \(g\). If \(d_{xy}\in\{1,2,3,4,6,8\}\) for all \(xy\in E(H)\), then \(g\) has Property (C).

Remark 2. Note that the induced vertex labeling of each new edge labeling defined at each theorem or corollary above is difference preserved, i.e., all \(d_{xy}\) remain unchanged.

We next show the possible existence of non-edge-prime regular graphs.

Theorem 9.

Let \(G\) be a \((p,q)\)-graph containing \(t\) component(s) such that every component of \(G\) is of size \(e_j\) with \(e_j \equiv 1 \pmod{4}\), \(1\le j \le t\). Let \(f: E(G)\to [1,q]\) be any bijection. Suppose no \(2\) adjacent vertex labels of \(G\) under \(f^+\) are even implies that every component of \(G\) receives odd number of odd edge labels under \(f\). If \(t \equiv 2\) or \(3 \pmod{4}\), then \(G\) is not edge-prime.

Proof. Let \(f\) be any bijective edge labeling of \(G\). Suppose \(t\equiv 2\pmod{4}\). Now \(q \equiv 2 \pmod{4}\). Hence, there are odd number of odd edge labels. There is a component receives even number of odd edge labels under \(f\). By the hypothesis, there is a component containing two adjacent vertices whose labels are even. Hence \(f\) is not an edge prime labeling. That is, \(G\) does not admit an edge-prime labeling.

Similarly, if \(t\equiv 3\pmod{4}\), then \(q \equiv 3 \pmod{4}\). Hence, there are even number of odd edge labels. There is a component receives even number of odd edge labels under \(f\). Hence, \(G\) does not admit an edge prime labeling.

2. Cubic graphs with same order components

Lemma 10. If \(h\) is an edge labeling of \(G=(V,E)\), then there are even number of odd vertex labels.

Proof. The lemma follows from \begin{equation}\label{eq-sum} \sum\limits_{u\in V} h^+(v)=2\sum\limits_{e\in E} h(e). \end{equation}

Corollary 11. Suppose \(f\) is an edge labeling of a graph \(G\) such that no adjacent vertex labels are even.

1. Suppose \(G\) contains a component \(H =K_n\) for \(n\ge 3\). If \(n\) is even, then all vertex labels in \(H\) are odd. If \(n\) is odd, then there is exactly one even vertex label in \(H\).
2. If \(G\) contains a component \(K\cong K(m,n)\) with odd \(m\) and \(n\), then \(K\) contains odd number of odd edge labels.

Proof.

(1). By the assumption, \(H\) admits at most one even vertex label. By Lemma 10, there is no even vertex label in \(H\) if \(n\) is even.

(2). Let \((X,Y)\) be the bipartition of a component \(K\). If \(K\) has even vertex labels, then the corresponding vertices lie in \(X\), say. By Lemma 10, there are even number of even vertex labels. Each even vertex label incident with even number of odd edge label as well as each odd vertex label incident with odd number of odd edge label. So, \(K\) contains odd number of odd edge labels.

Theorem 12. The graph \(nK_4\) is edge-prime if and only if \(n \ge 2\).

Proof. (Necessity)

We prove by contrapositive. Suppose there is an edge-prime labeling \(f\) of \(K_4\). By Corollary 11, all 4 vertex labels are odd and distinct. Since each vertex label lies between \(6\) and \(15\). So the set of vertex labels is a subset of \(\{7,9,11,13,15\}\) of size 4. Since the vertex labels are pairwise relatively prime and the sum of all 4 vertex labels is 42 (from (1)), there is no solution. Hence, \(K_4\) is not edge-prime.

(Sufficiency) (a). Suppose \(n=2t\) with \(t\ge 1\). The labeling of the two left-most graphs in Figure \ref{fig:6k4} shows that \(G=2K_4\) is edge-prime with \(d_{xy}\in\{2,4,6,8\}\).

By Remark 1 and Corollary 8, we conclude that \((2t)K_4\) is edge-prime for all \(t\ge 1\).
(b). Suppose \(n=2t+1\) with \(t\ge 1\). If \(t=1\), then the labeling in Figure 1 shows that \(H=3K_4\) is edge-prime and \(d_{xy}\in\{2,4,6,8\}\).

If \(t\ge 2\), then we have just known that \(G=(2t-2)K_4\) is edge-prime. By Remark 1 and Corollary 6, \(G+H=(2t+1)K_4\) is edge-prime.
We now consider cubic graphs with every component of order 6. Each component must be a \(C_3\times K_2\) or a \(K(3,3)\).

Theorem 13. For \(n \ge 1\), \(n(C_3 \times K_2)\) is edge-prime.

Proof. In Figure 3, the labeling of the two top-left graphs shows that \(H=C_3\times K_2\) and \(K=2(C_3\times K_2)\) are edge-prime with \(d_{xy}\in\{2,4,6,8\}\).

For \(n=2t\ge 2\), by Remark 1 and Corollary 8, we have \((2t)(C_3 \times K_2)\) is edge-prime for all \(t\ge 1\). Suppose \(n=2t+1\ge 1\). Since \(H\) is edge-prime, we only need to consider \(2t+1\ge 3\). Since we have already known that \(G=(2t)(C_3\times K_2)\) is edge-prime, by Remark 1 and Corollary 6 \(G+H=(2t+1)(C_3\times K_2)\) is edge-prime.

The next theorem shows that there are infinitely many non-edge-prime cubic graphs.

Theorem 14. For \(n \ge 1\), \(n(K(3,3))\) is edge-prime if and only if \(n \equiv 0\) or \(1 \pmod{4}\).

Proof. (Necessity) Suppose \(h\) is a bijective edge labeling of \(n(K(3,3))\) such that no \(2\) adjacent vertex labels are even. By Corollary 11 each component \(K(3,3)\) contains odd number of odd edge labels. So, \(n(K(3,3))\) satisfies the hypotheses of Theorem 9. Hence if \(n(K(3,3))\) is edge-prime, then \(n\equiv 0\) or \(1\pmod{4}\).
(Sufficiency) (a). Consider \(n=4t\). The labeling in Figure 4 shows that \(G=4K(3,3)\) is edge-prime with \(d_{xy}\in\{1,2,3,4,6,8\}\). Clearly, \(G\) satisfies the hypotheses of Corollary 8. Hence, we have \((4t)K(3,3)\) is edge-prime for each \(t\ge 1\).

(b). Consider \(n=4t+ 1\), \(t\ge 0\). The labeling of the graph in Figure 5 shows that \(H=K(3,3)\) is edge-prime with \(d_{xy}\in\{4,8\}\).

We assume that \(t\ge 1\). Since \(G=(4t)K(3,3)\) is edge-prime, by Remark 1 and Corollary 6 we have \(G+H=(4t+1)K(3,3)\) is edge-prime.

Theorem 15. For \(m,n\ge 1\), \(m(C_3\times K_2)+nK(3,3)\) is edge prime.

Proof. Case 1. \(n=4t\), \(t\ge 1\). From Theorem 14 we know that \((4t)K(3,3)\) is edge-prime (with \(d_{xy}\in\{1,2,3,4,6,8\}\)). From Theorem 13 we know that \(m(C_3\times K_2)\) admits an edge-prime labeling with \(d_{xy}\in\{2,4,6,8\}\). Since the size of \((4t)K(3,3)\) is \(36t\), by Remark 1 and Corollary 6 we have \((4t)K(3,3)+m(C_3\times K_2)\) is edge-prime with \(d_{xy}\in\{1,2,3,4,6,8\}\).
Case 2. \(n=4t+1\), \(t\ge 0\). Combining the labeling of \(K(3,3)\) in Figure 5 and the labeling of \(C_3\times K_2\) in the top-middle of Figure 3 we have an edge-prime labeling of \((C_3\times K_2)+K(3,3)\) (with \(d_{xy}\in\{2,4,6,8\}\)). From Theorem 13, Theorem 14 or Case 1, \((m-1)(C_3\times K_2)+(4t)K(3,3)\) admits an edge-prime labeling \(g\) with \(d_{xy}\in\{1,2,3,4,6,8\}\), \(m\ge 1\) and \(t\ge 0\). Since the size of \(C_3\times K_2\) is 18, \(g\) has Property (C). By Corollary 6 we obtain that \([(C_3\times K_2)+K(3,3)]+[(m-1)(C_3\times K_2)+(4t)K(3,3)]=m(C_3\times K_2)+(4t+1)K(3,3)\) is edge-prime.

Case 3. \(n=4t+2\), \(t\ge 0\). From Case 2 or Theorem 14, there is an edge-prime labeling of \((m-1)(C_3\times K_2)+(4t+1)K(3,3)\) with \(d_{xy}\in\{1,2,3,4,6,8\}\) for \(m\ge 1\) and \(t\ge 0\). By the same argument as in Case 2, we have \([(C_3\times K_2)+K(3,3)]+[(m-1)(C_3\times K_2)+(4t+1)K(3,3)]=m(C_3\times K_2)+(4t+2)K(3,3)\) is edge-prime.
Case 4. \(n=4t+3\), \(t\ge 0\). Consider the edge-prime labeling of \((C_3\times K_2)+3K(3,3)\) as shown in Figure 6.

From Theorem 13, Theorem 14, or Case 1, \((m-1)(C_3\times K_2)+(4t)K(3,3)\) admits an edge-prime labeling \(g\) with \(d_{xy}\in\{1,2,3,4,6,8\}\), \(m\ge 1\) and \(t\ge 0\). Since the size of \((C_3\times K_2)+3K(3,3)\) is 36, by Corollary 6 we get that \(m(C_3\times K_2)+(4t+3)K(3,3)\) are edge-prime. Note that the resulting labeling induces \(d_{xy}\in\{1,2,3,4,6,8,12,16\}\).
It is known that there are exactly 5 connected cubic graphs of order 8. Figure 7 shows edge-prime labelings of these 5 graphs, denoted \(G_k\), \(1\le k\le 5\).

Theorem 16. If every component of \(G\) is a connected cubic graph of order 8, then \(G\) is edge-prime.

Proof. Since each edge labeling shown in the Figure 7 induces \(d_{xy}\in\{2,4,6,8\}\). By Remark 1 and applying Corollary 6 repeatedly, the theorem holds.

3. Cubic graphs with distinct order components

In this section, we completely determine the edge-primality of cubic graphs with components of distinct orders of 4, 6 or 8.

Theorem 17. For \(m,n \ge 1\), \(mK_4 + n(C_3\times K_2)\) is edge-prime.

Proof. Suppose \(m\ge 2\). From Table 1 we know that \(mK_4\) and \(n(C_3\times K_2)\) are edge-prime when \(m\ge 2\) and \(n\ge 1\). Since \(|E(mK_4)|=6m\) and \(d_{xy}\in\{2,4,6,8\}\) under the edge-prime labeling of \(n(C_3\times K_2)\), by Remark 1 and Corollary 6, \(mK_4 + n(C_3\times K_2)\) is edge-prime.
Suppose \(m=1\). Consider odd \(n\). The edge-prime labeling of \(K_4+(C_3\times K_2)\) shown in Figure 8 with \(d_{xy}\in\{2,4,6,8\}\). For \(n\ge 3\), from Table 1 we know that \((n-1)(C_3\times K_2)\) is edge-prime with size \(9(n-1)\). By Corollary 6, \((n-1)(C_3\times K_2) + [K_4+(C_3\times K_2)]=K_4+n(C_3\times K_2)\) is edge-prime with \(d_{xy}\in\{2,4,6,8\}\).

Consider even \(n\). Figure9 shows that \(K_4 + 2(C_3\times K_2)\) is edge-prime. Using Table 1 and by Corollary 6, if needed, we get that \([K_4 + 2(C_3\times K_2)] + (n-2)(C_3\times K_2) = K_4 + n(C_3\times K_2)\) is edge-prime with \(d_{xy}\in\{2,4,6,8\}\).

Theorem 18. For \(m, n \ge 1\), the graph \(m K_4 + n K(3,3)\) is edge-prime.

Proof. (a). Suppose \(n\equiv 0,1\pmod{4}\). Suppose \(m\) is even. By Table 1, we know that \(mK_4\) and \(nK(3,3)\) are edge-prime. Note that \(d_{xy}\in\{1,2,3,4,6,8\}\) for \(xy\in E(nK(3,3))\) and \(|E(mK_4)|=6m\). By Remark 1 and Corollary 6, we get the result.
Suppose \(m\) is odd. From the whole figure and the two top-left graphs of Figure \ref{fig:K44K(33)} we see that \(K_4+K(3,3)\) and \(K_4+4K(3,3)\) are edge-prime with \(d_{xy}\in\{1,2,3,4,6,8\}\).

By Table 1 or the above case, there is an edge-prime labeling of \((m-1)K_4+4(t-1)K(3,3)\) for odd \(m\ge 1\) and \(t\ge 1\). Since \(|E((m-1)K_4+4(t-1)K(3,3))|=6(m-1)+36(t-1)\), the labelings of \(K_4+K(3,3)\) and \(K_4+4K(3,3)\) have Property (C). By Corollary 6 we get that \(mK_4+(4(t-1)+1)K(3,3)\) and \(mK_4+(4t)K(3,3)\) are edge-prime. Note that \(d_{xy}\in\{1,2,3,4,6,8\}\) for the resulting edge labelings above.
(b). Suppose \(n\equiv 2,3\pmod{4}\). Figure \ref{fig:2K(33)K4} shows that \(K_4+2K(3,3)\) is edge-prime.

From the above case, there is an edge-prime labeling \(g\) of \((m-1)K_4+(n-2)K(3,3)\) with \(d_{xy}\in \{1,2,3,4,6,8\}\) for \(m\ge 1\) and \(n\ge 2\). Since \(|E(K_4+2K(3,3))|=24\), \(g\) has Property (C). By Corollary 6 we have \(mK_4+nK(3,3)\) is edge-prime.

The following Table 1 gives a summary of the edge-prime labelings obtained above together with the set of the difference of adjacent vertex labels \(d_{xy}\).

Table 1. Results from Theorems 12 to 18.

Graph \(G\)	Condition(s)	\(\{d_{xy}\;|\; xy\in E(G)\}\)
\(nK_4\)	\(n\ge 2\)	\(\{2,4,6,8\}\)
\(n(C_3\times K_3)\)	\(n\ge 1\)	\(\{2,4,6,8\}\)
\(nK(3,3)\)	\(n\equiv 0,1\pmod{4}\)	\(\{1,2,3,4,6,8\}\)
\(m(C_3\times K_2)+nK(3,3)\)	\(m,n\ge 1\)	\(\{1,2,3,4,6,8,12,16\}\)
\(mK_4 + n(C_3\times K_2)\)	\(m,n\ge 1\)	\(\{2,4,6,8\}\)
\(mK_4 + nK(3,3)\)	\(m,n\ge 1\)	\(\{1,2,3,4,6,8,12,16\}\)
\(\sum_{i=1}^5 n_iG_i\)	\(\sum_{i=1}^5 n_i\ge 1\)	\(\{2,4,6,8\}\)

Theorem 19. For \(m_1, m_2, m_3\ge 1\), the graph \(m_1K_4 + m_2K(3,3)+m_3(C_3\times K_2)\) is edge-prime.

Proof. (a). Suppose \(m_2\) is even. From Table 1, \(m_1K_4+m_2(K(3,3))\) is edge-prime and \(m_3(C_3\times K_2)\) admits an edge-prime labeling with \(d_{xy}\in\{2,4,6,8\}\). Since the size of \(m_1K_4+m_2K(3,3)\) is \(6(m_1+3m_2/2)\), by Corollary 6, we have the theorem.
(b). Suppose \(m_2\) is odd. For even \(m_3\), from the case (a) or Table 1, \(m_1K_4+(m_2-1)K(3,3)+m_3(C_3\times K_2)\) is edge-prime with even size. From Figure 5 there is an edge-prime labeling of \(K(3,3)\) with \(d_{xy}\in\{4,8\}\). By Corollary 6 we have the theorem.
Now, assume that \(m_3\) is odd. If \(m_1=m_2=m_3=1\), then Figure 12 shows an edge-prime labeling of \(K_4+K(3,3)+(C_3\times K_2)\).

So we assume that at least one of \(m_1,m_2, m_3\) is greater than 1. From Case (a) or Table 1, \(m_1K_4+(m_2-1)K(3,3)+(m_3-1)(C_3\times K_2)\) is edge-prime with the size a multiple of 6. From the proof of Theorem 15 there is an edge-prime labeling of \((C_3\times K_2)+K(3,3)\) with \(d_{xy}\in\{2,4,6,8\}\). By Corollary 6 we have the theorem.

Remark 3 The set of \(d_{xy}\) of the edge-prime labeling obtained above is \(\{1,2,3,4,6,8,12,16\}\).

Theorem 20. The graph \(mK_4 + \sum^5_{i=1} n_iG_i\) is edge-prime for all \(m \ge 1\) and \(\sum^5_{i=1} n_i \ge1\).

Proof. Suppose \(m \ge 2\). From Table 1 and by Corollary 6 we get the theorem. Suppose \(m = 1\). For \(1 \le k \le 5\), we choose the smallest \(k\) such that \(n_k > 0\) and label \(K_4 + G_k\) from 1 to 18. The \(K_4\) is labeled by \(1,2,4,5,7,10\) as shown in Figure 9. The labeling of each \(G_k\) by \(\{3,6,8,9\}\cup[11,18]\), if needed, is labeled by the remaining labels shown in Figure 13.

The remaining unlabeled subgraph, if any, admits an edge-prime labeling with \(d_{xy}\in\{2,4,6,8\}\) by Table 1. By Corollary 6 we obtain the theorem.

Theorem 21. The graph \(m(C_3\times K_2) +\sum^5_{i=1} n_iG_i\) is edge-prime for \(m \ge 1\) and \(\sum^5_{i=1} n_i \ge1\).

Proof. From Table 1 we know that there are edge-prime labelings of \(\sum^5_{k=1} n_iG_i\) and \(m(C_3\times K_2)\) with \(d_{xy}\in\{2,4,6,8\}\). Since the size of \(\sum^5_{k=1} n_iG_i\) is 12 multiple, by Corollary 6 we have the theorem.

Theorem 22. The graph \(mK(3,3)+\sum^5_{i=1} n_iG_i\) is edge-prime for \(m \ge 1\) and \(\sum^5_{i=1} n_i \ge1\).

Proof. (a). Suppose \(m\equiv 0,1\pmod{4}\). From Table 1 we know that there are edge-prime labelings of \(\sum^5_{i=1} n_iG_i\) and \(mK(3,3)\) with \(d_{xy}\in\{1,2,3,4,6,8\}\). Since the size of \(\sum^5_{i=1} n_iG_i\) is a 12 multiple, by Corollary 6 we have the theorem. Note that the set of difference adjacent vertex labels of the new edge-prime labeling is \(d_{xy}\in\{1,2,3,4,6,8\}\).
Suppose \(m\equiv 2,3\pmod{4}\). For \(1 \le k \le 5\), we choose the smallest \(k\) such that \(n_k > 0\) and label \(2K(3,3) + G_k\) from 1 to 30. Figure 4 shows an edge labeling of \(2K(3,3)\) labeled by integers in \([1,21]\setminus\{13, 18,20\}\) with \(d_{xy}\in\{2,4,6,8\}\). The labeling of each \(G_k\) by \(\{13,18,20\}\cup [22,30]\), if needed, is shown in Figure 14.

From Case (a) or Table 1 there is an edge-prime labeling of \((m-2)K(3,3)+\sum\limits_{\begin{smallmatrix}1\le i\le 5\\i\ne k\end{smallmatrix}} n_iG_i\), if any, with \(d_{xy}\in\{1,2,3,4,6,8\}\). Since the size of \(2K(3,3) + G_k\) is 30, by Corollary 6 we have the theorem.

Theorem 23. For \(m_1, m_2\ge 1\) and \(\sum^5_{i=1}n_i\ge 1\), the graph \(m_1K_4 + m_2(C_3\times K_2) + \sum^5_{i=1} n_iG_i\) is edge-prime.

Proof. The size of \(\sum^5_{i=1}n_iG_i\) is a multiple of 12. From Table 1 and by Corollary 6 we have the theorem.

Theorem 24. For \(m_1, m_2\ge 1\) and \(\sum^5_{i=1}n_i\ge 1\), the graph \(m_1K_4 + m_2K(3,3) + \sum^5_{i=1} n_iG_i\) is edge-prime.

Proof. The size of \(\sum^5_{i=1}n_iG_i\) is a multiple of 12. From Table 1 and by Corollary 6 we have the theorem.

Theorem 25. \label{v6v8} For \(m_1, m_2\ge 1\) and \(\sum^5_{i=1}n_i\ge 1\), the graph \(m_1(C_3\times K_2)+\) \(m_2K(3,3) +\) \(\sum^5_{i=1} n_iG_i\) is edge-prime.

Proof. The size of \(\sum^5_{i=1}n_iG_i\) is a multiple of 12. From Table 1 and by Corollary 6 we have the theorem.

Theorem 26. Let \(m_1, m_2, m_3, \sum^5_{i=1} n_i\ge 1\), the graph \(m_1 K_4 + m_2 (C_3\times K_2) + m_3 K(3,3) +\) \(\sum^5_{i=1}G_i\) is edge-prime.

Proof. The size of \(\sum^5_{i=1}n_iG_i\) is a multiple of 12. From Remark 3 and by Corollary 6 we have the theorem.

Corollary 27. If \(G\) is a cubic graph with every component of order 4, 6 or 8, then \(G\) is edge-prime if and only if \(G\not\cong K_4\) or \(nK(3,3)\), \(n\equiv2,3\pmod{4}\).

In [7], we proved that (i) a 1-regular graph is edge-prime if and only if it is \(K_2\), (ii) all 2-regular graphs are edge-prime, and (iii) if \(G\) is edge-prime, then \(G+ C_n\) \((n\ge 3)\) and \(G+ K(1,2)\) are edge-prime.

Corollary 28. Let \(G\) be an edge-prime graph as in Sections 2 and 3. For \(\sum m_k\ge 1, n_k\ge 3\), \(G+ \sum m_k C_{n_k}\) and \(G+ K(1,2)\) are edge-prime.

4. Open problems

Problem 29. Determine the edge-primality of the following families of cubic graphs.

1. cylinder graph \(C_n\times K_2\), \(n\ge 5\).
2. Möbius ladder \(M(2n)\), \(n\ge 2\).
3. generalized Petersen graph \(P(n,k)\), \(n\ge 5, k\ge 2\).

It is easy to verify that \(K_n\) is edge-prime for \(n=2,3,5,6,7\) but not \(n=4\), and \(K(n,n)\) is edge-prime for \(n=2,3,4\). We end with the following conjecture.

Conjecture 4.1. For \(n\ge 2\), we have

1. \(K_n\) is edge-prime if and only if \(n\not= 4\).
2. \(K(n,n)\) is edge-prime.
3. All connected cubic graphs, except \(K_4\), are edge-prime.

Author Contributions

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

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

Our aim in this paper is to investigate the behavior of the solution of the following nonlinear difference equation

where the initial conditions \(x_{-4}, x_{-3}, x_{-2}, x_{-1}, x_0\) are arbitrary positive real numbers and \(a, b, c\) and \(d\) are positive constants.

Recently there has been a great interest in studying the qualitative properties of rational difference equations. Some prototypes for the development of the basic theory of the global behavior of nonlinear difference equations of order greater than one come from the results for rational difference equations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

However, there have not been any effective general methods to deal with the global behavior of rational difference equations of order greater than one so far. From the known work, one can see that it is extremely difficult to understand thoroughly the global behaviors of solutions of rational difference equations although they have simple forms (or expressions). One can refer to [16, 17, 18, 19, 20, 21, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] for examples to illustrate this. Therefore, the study of rational difference equations of order greater than one is worth further consideration.

Many researchers have investigated the behavior of the solution of difference equations, for example,

Elsayed et al. [37] has obtained results concerning the dynamics and global attractivity of the rational difference equation \begin{equation*} x_{n+1}=\dfrac{ax_nx_{n-2}}{bx_{n-2}+cx_{n-3}}. \end{equation*} Aloqeili [18] has obtained the solutions of the difference equation \begin{equation*} x_{n+1}=\dfrac{x_{n-1}}{a-x_{n}x_{n-1}}. \end{equation*} Simsek et al. [43] obtained the solution of the difference equation \begin{equation*} x_{n+1}=\dfrac{x_{n-3}}{1+ x_{n-1}} \end{equation*} Cinar [22, 23, 24] got the solutions of the following difference equations \begin{equation*} x_{n+1}=\dfrac{x_{n-1}}{1+ax_{n}x_{n-1}}, x_{n+1}=\dfrac{x_{n-1}}{ -1+ax_{n}x_{n-1}}, x_{n+1}=\dfrac{ax_{n-1}}{1+bx_{n}x_{n-1}}. \end{equation*} In [48], Ibrahim got the form of the solution of the rational difference equation \begin{equation*} x_{n+1}=\dfrac{x_nx_{n-2}}{x_{n-1}(a+bx_{n}x_{n-2})}. \end{equation*} Karatas et al. [46] got the solution of the difference equation \begin{equation*} x_{n+1}=\dfrac{x_{n-5}}{1+x_{n-2}x_{n-5}}. \end{equation*} Here, we recall some notations and results which will be useful in our investigation. Let \(I\) be some interval of real numbers and let \begin{equation*} f : I^{k+1} \rightarrow I, \end{equation*} be a continuously differentiable function. Then for every set of initial conditions \(x_{-k}, x_{-k+1}, x_{-k+2}, ..., x_{0} \in I\), the difference equation has a unique solution \(\{x_n\}_{n=-k}^\infty\).

Definition 1. A point \(\bar x \in I\) is called an \textit{equilibrium point} of Equation (2) if \(\bar x=f(\bar x,\bar x,...,\bar x)\). That is, \(x_n = \bar x\) for \(n \geq 0\), is a solution of Equation (2), or equivalently, \(\bar x\) is a fixed point of \(f\).

Definition 2.

• The equilibrium point \(\bar x\) of Equation (2) is locally stable if for every \(\varepsilon> 0\), there exists \(\delta > 0\) such that for all \(x_{-k}, x_{-k+1}, x_{-k+2}, ..., x_{0} \in I,\) with \begin{equation*} |x_{-k}-\bar x|+|x_{-k+1}-\bar x|+|x_{-k+2}-\bar x|+...+|x_{0}-\bar x| < \delta, \end{equation*} we have \(|x_n - \bar x| < \varepsilon\), for all \(n \geq -k\).
• The equilibrium point \(\bar x\) of Equation (2) is locally asymptotically stable if \(\bar x\) is locally stable solution of Equation (2) and there exists \( \gamma > 0\), such that for all \(x_{-k}, x_{-k+1}, x_{-k+2}, ..., x_{0} \in I, \) with \begin{equation*} |x_{-k}-\bar x|+|x_{-k+1}-\bar x|+|x_{-k+2}-\bar x|+...+|x_{0}-\bar x| < \delta, \end{equation*} we have \(\displaystyle \lim_{n\rightarrow \infty} x_n = \bar x\).
• The equilibrium point \(\bar x\) of Equation (2) is global attractor if for all \(x_{-k}, x_{-k+1}, ... , x_{0} \in I\) we have \begin{equation*} \lim_{n\rightarrow \infty} x_n = \bar x. \end{equation*}
• The equilibrium point \(\bar x\) of Equation (2) is globally asymptotically stable if \(\bar x\) is locally stable, and \(\bar x\) is also a global attractor of Equation (2).
• The equilibrium point \(\bar x\) of Equation (2) is unstable if \(\bar x\) is not locally stable.

The linearized form of Equation (2) about the equilibrium \(\bar{x}\) is the linear difference equation \begin{equation*} y_{n+1}=\sum_{i=0}^{k}\dfrac{\partial f(\bar{x},\bar{x},...,\bar{x})}{ \partial x_{n-i}}y_{n-i}. \end{equation*}

Theorem 3. Assume that \(p, q \in \mathbb{R}\) and \(k \in \{0, 1, 2, ...\}\). Then \(|p| + |q| < 1\) is a sufficient condition for the asymptotic stability of the difference equation \begin{equation*} x_{n+1} + px_n + qx_{n-k} = 0, n = 0, 1, \ldots \end{equation*}

Remark 1. The theorem can be easily extended to a general linear equations of the form

where \(p_1, p_2, ..., p_k \in \mathbb{R}\) and \(k \geq 0\). Then Equation (3) is asymptotically stable provided that \(\displaystyle \sum_{i=0}^k |p_i|< 1\).

Consider the following equation The following theorem will be useful for the proof of our results in this paper.

Theorem 4. Let \([a, b]\) be an interval of real numbers and assume that \begin{equation*} g:[a, b]^3\rightarrow [a, b], \end{equation*} is a continuous function satisfying the following properties:

1. \(g(x,y,z)\) is nondecreasing in \(x\) and \(z\) in \([a, b]\) for each \(y \in[a, b]\), and is nonincreasing in \(y\in[a, b]\) for each \(x\) and \(z\) in \( [a, b]\)
2. if \((m,M)\in [a, b]\times [a, b]\) is a solution of the system \begin{equation*} M=g(M,m,M),\qquad m=g(m,M,m), \end{equation*} then \(m=M\).

Then (4) has a unique equilibrium point \(\bar x \in[a, b]\) and every solution of (4) converges to \(\bar x\).

2. Local stability of equation (1)

In this section we investigate the local stability character of the solutions of Equation (1). Equation (1) has a unique equilibrium point and is given by \begin{equation*} \bar x=a \bar x+\dfrac{b \bar x^2}{ c \bar x+d \bar x}, \end{equation*} or \begin{equation*} \bar x^2(1-a)(c+d)=b \bar x^2, \end{equation*} then if \((1-a)(c+d)\not=b\) , then the unique equilibrium point is \(\bar x=0\).\\ Define the following function \begin{equation*} \begin{array}{llll} f:(0,\infty)^3\rightarrow (0,\infty) & & & \\ f(u,v,w)=a u+\dfrac{b u w}{c v+d w}. & & & \end{array} \end{equation*} It follows that \begin{equation*} f_u(u,v,w)=a +\dfrac{b w}{c v+d w}, \quad f_v(u,v,w)=-\dfrac{bc u w}{(c v+d w)^2},\quad f_w(u,v,w)=\dfrac{b cu v}{(c v+d w)^2}. \end{equation*} Then \begin{equation*} f_u(\bar x,\bar x, \bar x)=a +\dfrac{b }{c +d}, \quad f_v(\bar x,\bar x, \bar x)=-\dfrac{bc}{(c +d )^2},\quad f_w(\bar x,\bar x, \bar x)=\dfrac{bc}{ (c +d )^2}. \end{equation*} The linearized equation of Equation (1) about \(\bar x\) is

Theorem 5. Assume that \begin{equation*} b(d+3c)< (1-a)(c+d)^2. \end{equation*} Then the equilibrium point of Equation (1) is locally asymptotically stable.

Proof. It follows from Theorem 3 that Equation (5) is asymptotically stable if \begin{equation*} \left|a +\dfrac{b }{c +d}\right|+\left|\dfrac{bc}{(c +d )^2}\right|+\left| \dfrac{bc}{(c +d )^2}\right|< 1, \end{equation*} or \begin{equation*} a +\dfrac{b }{c +d}+\dfrac{2bc}{(c +d )^2}< 1, \end{equation*} and so, \begin{equation*} \dfrac{b(d+3c) }{(c +d )^2}< (1-a). \end{equation*} The proof is complete.

3. Global attractor of the equilibrium point of equation (1)

In this section we investigate the global attractivity character of solutions of Equation (1).

Theorem 6. The equilibrium point \(\bar x\) of Equation (1) is global attractor if \(d(1-a)\not=b\)

Proof. Let \(p,q\) be real numbers and assume that \(g:[p, q]^3\rightarrow [p, q]\) is a function defined by \( g(u,v,w)=au+\dfrac{buw}{cv +dw}\), then we can easily see that the function \( g(u,v,w)\) is increasing in \(u,w\) and decreasing in \(v\). Suppose that \((m,M)\) is a solution of the system \begin{equation*} M=g(M,m,M),\qquad m=g(m,M,m). \end{equation*} Then from Equation (1), we see that \begin{equation*} M=aM+\dfrac{bM^2}{cm +dM},\qquad m=am+\dfrac{bm^2}{cM +dm} \end{equation*} or \begin{equation*} M(1-a)=\dfrac{bM^2}{cm +dM},\qquad m(1-a)=\dfrac{bm^2}{cM +dm} \end{equation*} then \begin{equation*} c(1-a)mM+d(1-a)M^2=bM^2,\qquad c(1-a)mM+d(1-a)m^2=bm^2 \end{equation*} subtracting, we obtain \begin{equation*} d(1-a)(M^2-m^2)=b(M^2-m^2). \end{equation*} Since \(d(1-a)\not=b\) therefore \begin{equation*} M=m. \end{equation*} It follows from Theorem 4 that \(\bar x\) is a global attractor of Equation (1), and then the proof is complete.

4. Boundedness of solutions of equation (1)

In this section we study the boundedness of solutions of Equation (1).

Theorem 7. Every solution of Equation (1) is bounded if \(a+\dfrac{b}{d}< 1\).

Proof. Let \(\{x_n\}_{n=-4}^\infty\) be a solution of Equation (1). It follows from Equation (1) that \begin{equation*} x_{n+1}=a x_{n}+\dfrac{b x_{n} x_{n-4}}{c x_{n-3}+dx_{n-4}}\leq a x_{n}+ \dfrac{b x_{n} x_{n-4}}{dx_{n-4}}=\left(a+\dfrac{b}{d}\right)x_{n}. \end{equation*} Then \(x_{n+1}\leq x_{n}, \quad \forall n\geq 0\). Then the sequence \( \{x_n\}_{n=-4}^\infty\) is decreasing and so is bounded from above by \( M=\max\{x_{-4}, x_{-3}, x_{-2}, x_{-1}, x_0\}\).

For explaining the results of this section, we consider numerical example for \(x_{-4}=10, x_{-3}=1, x_{-2}=3, x_{-1}=2, x_0=7\). (See Figure 1). \begin{center}

5. Special cases of equation (1)

Our goal in this section is to find a specific form of the solutions of some special cases of Equation (1) when , \(a,b,c\) and \(d\) are integers and give numerical examples of each case.

5.1. First case: on the difference equation \(x_{n+1}= x_{n}+\dfrac{x_{n} x_{n-4}}{ x_{n-3}+x_{n-4}}\)

In this subsection we study the following special case of Equation (1): where the initial conditions \(x_{-4}, x_{-3}, x_{-2}, x_{-1}, x_0\) are arbitrary nonzero real numbers.

Theorem 8. Let \(\{x_n\}_{n=-4}^\infty\) be a solution of Equation (6). Then for \(n=0,1,2,...\) \begin{equation*} \left\{ \begin{array}{llll} \displaystyle x_{4n}=r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{ (A_ir+2B_ih)}{(B_ir+A_ih)} , & & & \\ \displaystyle x_{4n+1}=r\prod_{i=1}^{n+1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \prod_{i=1}^{n}\dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{ (B_ih+A_ig)}\dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}, & & & \\ \displaystyle x_{4n+2}=r\prod_{i=1}^{n+1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \dfrac{(A_ig+2B_if)}{(B_ig+A_if)} \prod_{i=1}^{n}\dfrac{(A_ih+2B_ig)}{ (B_ih+A_ig)}\dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}, & & & \\ \displaystyle x_{4n+3}=r\prod_{i=1}^{n+1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \prod_{i=1}^{n}\dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)} , & & & \end{array} \right. \end{equation*} where \(x_{-4}=e, x_{-3}=f, x_{-2}=g, x_{-1}=h, x_0=r\), \(\{A_m\}_{m=1}^ \infty=\{1,3,7,17,41,...\}\), \(\{B_m\}_{m=1}^\infty=\{1,2,5,12,29,...\}\) \( A_m=2A_{m-1}+A_{m-2}, B_m=2B_{m-1}+B_{m-2}, m\geq 1, A_{-1}=-1\), \(A_0=1\), \( B_{-1}=1, B_0=0\), or also \(A_m=2B_{m-1}+A_{m-1}, B_m=B_{m-1}+A_{m-1}, m\geq 0 \), and \(\displaystyle \prod_{i=1}^{0}G_i=1\).

Proof. For \(n=0\), the result holds. Now suppose that our assumption holds for \(n-1\) and for \(n-2\). That is \begin{equation*} \left\{ \begin{array}{llll} \displaystyle x_{4n-8}=r\prod_{i=1}^{n-2}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{ (A_ir+2B_ih)}{(B_ir+A_ih)} , & & & \\ \displaystyle x_{4n-7}=r\prod_{i=1}^{n-1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \prod_{i=1}^{n-2}\dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{ (B_ih+A_ig)}\dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}, & & & \\ \displaystyle x_{4n-6}=r\prod_{i=1}^{n-1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \dfrac{(A_ig+2B_if)}{(B_ig+A_if)} \prod_{i=1}^{n-2}\dfrac{(A_ih+2B_ig)}{ (B_ih+A_ig)}\dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}, & & & \\ \displaystyle x_{4n-5}=r\prod_{i=1}^{n-1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \prod_{i=1}^{n-2}\dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}, & & & \end{array} \right. \end{equation*} \begin{equation*} \left\{ \begin{array}{llll} \displaystyle x_{4n-4}=r\prod_{i=1}^{n-1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{ (A_ir+2B_ih)}{(B_ir+A_ih)} , & & & \\ \displaystyle x_{4n-3}=r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \prod_{i=1}^{n-1}\dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{ (B_ih+A_ig)}\dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}, & & & \\ \displaystyle x_{4n-2}=r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \dfrac{(A_ig+2B_if)}{(B_ig+A_if)} \prod_{i=1}^{n-1}\dfrac{(A_ih+2B_ig)}{ (B_ih+A_ig)}\dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}, & & & \\ \displaystyle x_{4n-1}=r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \prod_{i=1}^{n-1}\dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}. & & & \end{array} \right. \end{equation*} Now it follows from Equation (6) that \begin{equation*} \begin{array}{llll} \displaystyle x_{4n} = \displaystyle x_{4n-1}+\dfrac{ x_{4n-1} x_{4n-5}}{ x_{4n-4}+x_{4n-5}}= r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \prod_{i=1}^{n-1} \dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)} \Bigg(1+ & \\ \dfrac{\displaystyle r\prod_{i=1}^{n-1}\dfrac{(A_if+2B_ie)}{ (B_if+A_ie)}\dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig) } \prod_{i=1}^{n-2}\dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}}{\displaystyle r\prod_{i=1}^{n-1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \dfrac{(A_ig+2B_if)}{ (B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{(A_ir+2B_ih)}{ (B_ir+A_ih)} +r\prod_{i=1}^{n-1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \prod_{i=1}^{n-2} \dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}}\Bigg) & \\ = \displaystyle r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \prod_{i=1}^{n-1} \dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}\left(1+ \dfrac{1}{\dfrac{ (A_{n-1}r+2B_{n-1}h)}{(B_{n-1}r+A_{n-1}h)} +1}\right) & \\ = \displaystyle r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \prod_{i=1}^{n-1} \dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}\left(1+ \dfrac{B_{n-1}r+A_{n-1}h}{ (A_{n-1}+B_{n-1})r+(2B_{n-1}+A_{n-1})h} \right) & \\ = \displaystyle r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \prod_{i=1}^{n-1} \dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}\; \dfrac{ (A_{n-1}+2B_{n-1})r+(2B_{n-1}+2A_{n-1})h}{ (A_{n-1}+B_{n-1})r+(2B_{n-1}+A_{n-1})h} & \\ = \displaystyle r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \prod_{i=1}^{n-1} \dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}\; \dfrac{A_{n}r+2B_{n}h}{B_{n}r+A_{n}h} & \\ = \displaystyle r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{ (A_ir+2B_ih)}{(B_ir+A_ih)}. & \end{array} \end{equation*} Similarly, \begin{equation*} \begin{array}{llll} \displaystyle x_{4n+1} = \displaystyle x_{4n}+\dfrac{ x_{4n} x_{4n-4}}{ x_{4n-3}+x_{4n-4}}= r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{ (A_ir+2B_ih)}{(B_ir+A_ih)}\end{array} \end{equation*}} {\small \begin{equation*} \begin{array}{llll} \Bigg(1+ \\ \dfrac{\displaystyle r\prod_{i=1}^{n-1}\dfrac{(A_if+2B_ie)}{ (B_if+A_ie)}\dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig) } \dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}}{\displaystyle r\prod_{i=1}^{n}\dfrac{ (A_if+2B_ie)}{(B_if+A_ie)}\prod_{i=1}^{n-1} \dfrac{(A_ig+2B_if)}{(B_ig+A_if)} \dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)} +r\prod_{i=1}^{n-1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{(A_ig+2B_if)}{ (B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{(A_ir+2B_ih)}{ (B_ir+A_ih)}}\Bigg) \\ \end{array} \end{equation*} \begin{equation*} \begin{array}{llll} = \displaystyle r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{ (A_ir+2B_ih)}{(B_ir+A_ih)}\left(1+ \dfrac{1}{\dfrac{(A_{n}r+2B_{n}h)}{ (B_{n}r+A_{n}h)} +1}\right) \\ = \displaystyle r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{ (A_ir+2B_ih)}{(B_ir+A_ih)}\left(1+ \dfrac{B_{n}r+A_{n}h}{ (A_{n}+B_{n-1})r+(2B_{n}+A_{n})h} \right) \\ = \displaystyle r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{ (A_ir+2B_ih)}{(B_ir+A_ih)}\; \dfrac{(A_{n}+2B_{n})r+(2B_{n}+2A_{n})h}{ (A_{n}+B_{n})r+(2B_{n}+A_{n})h} \\ = \displaystyle r\prod_{i=1}^{n}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)}\dfrac{ (A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{(B_ih+A_ig)} \dfrac{ (A_ir+2B_ih)}{(B_ir+A_ih)}\; \dfrac{A_{n+1}r+2B_{n+1}h}{B_{n+1}r+A_{n+1}h} \\ = \displaystyle r\prod_{i=1}^{n+1}\dfrac{(A_if+2B_ie)}{(B_if+A_ie)} \prod_{i=1}^{n}\dfrac{(A_ig+2B_if)}{(B_ig+A_if)}\dfrac{(A_ih+2B_ig)}{ (B_ih+A_ig)} \dfrac{(A_ir+2B_ih)}{(B_ir+A_ih)}. \end{array} \end{equation*} Similarly, one can easily obtain the other relations. Thus, the proof is completed.

For explaining the results of this section, we consider numerical example for \(x_{-4}=10, x_{-3}=1, x_{-2}=3, x_{-1}=2, x_0=7\), (See Figure 2).

5.2. Second case: on the difference equation \(x_{n+1}= x_{n}+\dfrac{ x_{n} x_{n-4}}{ x_{n-3}-x_{n-4}}\)

In this subsection we study the following special case of Equation (1): where the initial conditions \(x_{-4}, x_{-3}, x_{-2}, x_{-1}, x_0\) are arbitrary nonzero real numbers.

Theorem 9. Let \(\{x_n\}_{n=-4}^\infty\) be a solution of Equation (7). Then for \(n=0,1,2,...\) \begin{equation*} \left\{ \begin{array}{llll} \displaystyle x_{8n-4}=\dfrac{r^{n}}{e^{n-1}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n-3}=f\dfrac{r^{n}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n-2}=g\dfrac{r^{n}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n-1}=h\dfrac{r^{n}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n}=\dfrac{r^{n+1}}{e^n}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n+1}=-f\dfrac{r^{n+1}}{e^n}\dfrac{(rfgh)^{n}}{ (-f+e)^{n+1}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n+2}=fg\dfrac{r^{n+1}}{e^n}\dfrac{(rfgh)^{n}}{ (-f+e)^{n+1}(-g+f)^{n+1}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n+3}=-fgh\dfrac{r^{n+1}}{e^n}\dfrac{(rfgh)^{n}}{ (-f+e)^{n+1}(-g+f)^{n+1}(-h+g)^{n+1}(-r+h)^{n}}. & & & \end{array} \right. \end{equation*}

Proof. For \(n=0\), the result holds. Now suppose that our assumption holds for \(n-1\). That is \begin{equation*} \left\{ \begin{array}{llll} \displaystyle x_{8n-12}=\dfrac{r^{n-1}}{e^{n-2}}\dfrac{(rfgh)^{n-1}}{ (-f+e)^{n-1}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-11}=f\dfrac{r^{n-1}}{e^{n-1}}\dfrac{(rfgh)^{n-1}}{ (-f+e)^{n-1}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-10}=g\dfrac{r^{n-1}}{e^{n-1}}\dfrac{(rfgh)^{n-1}}{ (-f+e)^{n-1}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-9}=h\dfrac{r^{n-1}}{e^{n-1}}\dfrac{(rfgh)^{n-1}}{ (-f+e)^{n-1}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-8}=\dfrac{r^{n}}{e^{n-1}}\dfrac{(rfgh)^{n-1}}{ (-f+e)^{n-1}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-7}=-f\dfrac{r^{n}}{e^{n-1}}\dfrac{(rfgh)^{n-1}}{ (-f+e)^{n}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-6}=fg\dfrac{r^{n}}{e^{n-1}}\dfrac{(rfgh)^{n-1}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-5}=-fgh\dfrac{r^{n}}{e^{n-1}}\dfrac{(rfgh)^{n-1}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n-1}}. & & & \end{array} \right. \end{equation*} Now it follows from Equation (7) that \begin{equation*} \begin{array}{llll} \displaystyle x_{8n}=\displaystyle x_{8n-1}+\dfrac{ x_{8n-1} x_{8n-5}}{ x_{8n-4}-x_{8n-5}}= h\dfrac{r^{n}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} & & & \\ \left(1+\dfrac{-fgh\dfrac{r^{n}}{e^{n-1}}\dfrac{(rfgh)^{n-1}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n-1}}}{\dfrac{r^{n}}{e^{n-1}}\dfrac{ (rfgh)^{n}}{(-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}+fgh\dfrac{r^{n}}{ e^{n-1}}\dfrac{(rfgh)^{n-1}}{(-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n-1}}} \right) & & & \\ =\displaystyle h\dfrac{r^{n}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+\dfrac{-1}{\dfrac{r}{(-r+h) }+1}\right) \\ =\displaystyle h\dfrac{r^{n}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+\dfrac{r-h}{h}\right) & & & \\ =\displaystyle h\dfrac{r^{n}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \dfrac{r}{h}\\ =\displaystyle \dfrac{ r^{n+1}}{e^{n}}\dfrac{(rfgh)^{n}}{(-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} & & & \end{array} \end{equation*} Similarly, \begin{equation*} \begin{array}{llll} \displaystyle x_{8n+1}=\displaystyle x_{8n}+\dfrac{ x_{8n} x_{8n-4}}{ x_{8n-3}-x_{8n-4}}\\ = \dfrac{r^{n+1}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} & & & \\ \left(1+\dfrac{\dfrac{r^{n}}{e^{n-1}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}}{f\dfrac{r^{n}}{e^{n}}\dfrac{ (rfgh)^{n}}{(-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}-\dfrac{r^{n}}{e^{n-1}} \dfrac{(rfgh)^{n}}{(-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}}\right) & & & \\ =\displaystyle \dfrac{r^{n+1}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+\dfrac{1}{\dfrac{f}{e}-1} \right)\\ =\displaystyle \dfrac{r^{n+1}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+\dfrac{e}{f-e}\right) & & & \\ =\displaystyle \dfrac{r^{n+1}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \dfrac{f}{f-e}\\ =\displaystyle -f \dfrac{r^{n+1}}{e^{n}}\dfrac{(rfgh)^{n}}{ (-f+e)^{n+1}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} & & & \end{array} \end{equation*} Similarly, one can easily obtain the other relations. Thus, the proof is completed.

Consider numerical examples which represent different types of solutions to Equation (7).
See Figure 3, since \(x_{-4}=20, x_{-3}=10, x_{-2}=30, x_{-1}=2, x_0=10\). The solution is bounded and converges to \(\bar x=0\).
Now, if we take \(x_{-4}=1, x_{-3}=3, x_{-2}=1, x_{-1}=40, x_0=10\), the solution is unbounded (see Figure 4).

In this subsection we study the following special case of Equation (1): where the initial conditions \(x_{-4}, x_{-3}, x_{-2}, x_{-1}, x_0\) are arbitrary nonzero real numbers.

Theorem 10. Let \(\{x_n\}_{n=-4}^\infty\) be a solution of Equation (8). Then for \(n=0,1,2,...\) \begin{equation*} \left\{ \begin{array}{llll} \displaystyle x_{4n} & = & \dfrac{ r\;(rfgh)^n}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g)(ir+h)}, & \\ \displaystyle x_{4n+1} & = & \dfrac{ rf\;(rfgh)^n}{\displaystyle \prod_{i=1}^{n+1}(if+e)\prod_{i=1}^{n}(ig+f)(ih+g)(ir+h)}, & \\ \displaystyle x_{4n+2} & = & \dfrac{ rfg\;(rfgh)^n}{\displaystyle \prod_{i=1}^{n+1}(if+e)(ig+f)\prod_{i=1}^{n}(ih+g)(ir+h)}, & \\ \displaystyle x_{4n+3} & = & \dfrac{ (rfgh)^{n+1}}{\displaystyle \prod_{i=1}^{n+1}(if+e)(ig+f)(ih+g)\prod_{i=1}^{n}(ir+h)}. & \end{array} \right. \end{equation*}

Proof. For \(n=0\), the result holds. Now suppose that our assumption holds for \(n-1\) and for \(n-2\). That is \begin{equation*} \left\{ \begin{array}{llll} \displaystyle x_{4n-4} = \dfrac{ r\;(rfgh)^{n-1}}{\displaystyle \prod_{i=1}^{n-1}(if+e)(ig+f)(ih+g)(ir+h)}, & \\ \displaystyle x_{4n-3} = \dfrac{ rf\;(rfgh)^{n-1}}{\displaystyle \prod_{i=1}^{n}(if+e)\prod_{i=1}^{n-1}(ig+f)(ih+g)(ir+h)}, & \\ \displaystyle x_{4n-2} = \dfrac{ rfg\;(rfgh)^{n-1}}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)\prod_{i=1}^{n-1}(ih+g)(ir+h)}, & \\ \displaystyle x_{4n-1} = \dfrac{ (rfgh)^{n}}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g)\prod_{i=1}^{n-1}(ir+h)}, & \end{array} \right. \left\{ \begin{array}{llll} \displaystyle x_{4n-8} = \dfrac{ r\;(rfgh)^{n-2}}{\displaystyle \prod_{i=1}^{n-2}(if+e)(ig+f)(ih+g)(ir+h)}, & \\ \displaystyle x_{4n-7} = \dfrac{ rf\;(rfgh)^{n-2}}{\displaystyle \prod_{i=1}^{n-1}(if+e)\prod_{i=1}^{n-2}(ig+f)(ih+g)(ir+h)}, & \\ \displaystyle x_{4n-6} = \dfrac{ rfg\;(rfgh)^{n-2}}{\displaystyle \prod_{i=1}^{n-1}(if+e)(ig+f)\prod_{i=1}^{n-2}(ih+g)(ir+h)}, & \\ \displaystyle x_{4n-5} = \dfrac{ (rfgh)^{n-1}}{\displaystyle \prod_{i=1}^{n-1}(if+e)(ig+f)(ih+g)\prod_{i=1}^{n-2}(ir+h)}. & \end{array} \right. \end{equation*} Now it follows from Equation (8) that \begin{equation*} \begin{array}{llll} \displaystyle x_{4n}=x_{4n-1}-\dfrac{ x_{4n-1} x_{4n-5}}{ x_{4n-4}+x_{4n-5}} = \dfrac{ (rfgh)^{n}}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g) \prod_{i=1}^{n-1}(ir+h)} & & & \\ \left(1-\dfrac{\dfrac{ (rfgh)^{n-1}}{\displaystyle \prod_{i=1}^{n-1}(if+e)(ig+f)(ih+g)\prod_{i=1}^{n-2}(ir+h)}}{\dfrac{ r\;(rfgh)^{n-1}}{\displaystyle \prod_{i=1}^{n-1}(if+e)(ig+f)(ih+g)(ir+h)} + \dfrac{ (rfgh)^{n-1}}{\displaystyle \prod_{i=1}^{n-1}(if+e)(ig+f)(ih+g) \prod_{i=1}^{n-2}(ir+h)}} \right) & & & \\ \end{array} \end{equation*} \begin{equation*} \begin{array}{llll}\displaystyle = \dfrac{ (rfgh)^{n}}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g)\prod_{i=1}^{n-1}(ir+h)} \left(1-\dfrac{1}{ \dfrac{ r}{(n-1)r+h} + 1} \right)\\ \displaystyle = \dfrac{ (rfgh)^{n}}{ \displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g)\prod_{i=1}^{n-1}(ir+h)} \left(1-\dfrac{(n-1)r+h}{nr+h} \right) & & & \\ \displaystyle = \dfrac{ (rfgh)^{n}}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g)\prod_{i=1}^{n-1}(ir+h)} \left(\dfrac{r}{ nr+h} \right) \\ \displaystyle = \dfrac{ r(rfgh)^{n}}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g)(ir+h)} . & & & \\ \end{array} \end{equation*} Similarly, \begin{equation*} \begin{array}{llll} \displaystyle x_{4n+1}=x_{4n}-\dfrac{ x_{4n} x_{4n-4}}{ x_{4n-3}+x_{4n-4}}= \dfrac{ r(rfgh)^{n}}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g)(ir+h)} & & & \\ \left(1-\dfrac{\dfrac{ r(rfgh)^{n-1}}{\displaystyle \prod_{i=1}^{n-1}(if+e)(ig+f)(ih+g)(ir+h)}}{\dfrac{ rf\;(rfgh)^{n-1}}{ \displaystyle \prod_{i=1}^{n}(if+e)\prod_{i=1}^{n-1}(ig+f)(ih+g)(ir+h)} + \dfrac{ r(rfgh)^{n-1}}{\displaystyle \prod_{i=1}^{n-1}(if+e)(ig+f)(ih+g)(ir+h)}} \right) & & & \\ \end{array} \end{equation*} \begin{equation*} \begin{array}{llll} \displaystyle = \dfrac{r (rfgh)^{n}}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g)(ir+h)} \left(1-\dfrac{1}{\dfrac{ f}{nf+e} + 1} \right) \\ \displaystyle = \dfrac{ r(rfgh)^{n}}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g)(ir+h)} \left(1-\dfrac{nf+e}{(n+1)f+e} \right) & & & \\ \displaystyle = \dfrac{ r(rfgh)^{n}}{\displaystyle \prod_{i=1}^{n}(if+e)(ig+f)(ih+g)(ir+h)} \left(\dfrac{f}{(n+1)f+e} \right) \\ \displaystyle = \dfrac{ r(rfgh)^{n}}{\displaystyle \prod_{i=1}^{n+1}(if+e) \prod_{i=1}^{n}(ig+f)(ih+g)(ir+h)} . & & & \end{array} \end{equation*} Similarly, one can easily obtain the other relations. Thus, the proof is completed.

Consider numerical examples which represent different types of solutions to Equation (8).
Assume that \(x_{-4}=100, x_{-3}=30, x_{-2}=80, x_{-1}=1, x_0=3\) (See Figure \ref{fig5}).
Now for \(x_{-4}=0.1, x_{-3}=0.5, x_{-2}=30, x_{-1}=50, x_0=300\) (See Figure \ref{fig6}).

5.3. Fourth case: on the difference equation \(x_{n+1}= x_{n}+\dfrac{ x_{n} x_{n-4}}{ -x_{n-3}+x_{n-4}}\)

In this subsection we study the following special case of Equation (1): where the initial conditions \(x_{-4}, x_{-3}, x_{-2}, x_{-1}, x_0\) are arbitrary nonzero real numbers.

Theorem 11. Let \(\{x_n\}_{n=-4}^\infty\) be a solution of Equation (9). Then for \(n=0,1,2,...\) \begin{equation*} \left\{ \begin{array}{llll} \displaystyle x_{8n-4}=\dfrac{r^{n}}{e^{n-1}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n-3}=f\dfrac{r^{n}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n-2}=g\dfrac{r^{n}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n-1}=h\dfrac{r^{n}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n}=\dfrac{r^{n+1}}{e^n}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} , & & & \\ \displaystyle x_{8n+1}=\dfrac{r^{n+1}}{e^n}\dfrac{ (-f+2e)^{n+1}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n+1}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n+2}=\dfrac{r^{n+1}}{e^n}\dfrac{ (-f+2e)^{n+1}(-g+2f)^{n+1}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n+1}(-g+f)^{n+1}(-h+g)^{n}(-r+h)^{n}}, & & & \\ \displaystyle x_{8n+3}=\dfrac{r^{n+1}}{e^n}\dfrac{ (-f+2e)^{n+1}(-g+2f)^{n+1}(-h+2g)^{n+1}(-r+2h)^{n}}{ (-f+e)^{n+1}(-g+f)^{n+1}(-h+g)^{n+1}(-r+h)^{n}}. & & & \end{array} \right. \end{equation*}

Proof. For \(n=0\), the result holds. Now suppose that our assumption holds for \(n-1\). That is \begin{equation*} \left\{ \begin{array}{llll} \displaystyle x_{8n-12}=\dfrac{r^{n-1}}{e^{n-2}}\dfrac{ (-f+2e)^{n-1}(-g+2f)^{n-1}(-h+2g)^{n-1}(-r+2h)^{n-1}}{ (-f+e)^{n-1}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-11}=f\dfrac{r^{n-1}}{e^{n-1}}\dfrac{ (-f+2e)^{n-1}(-g+2f)^{n-1}(-h+2g)^{n-1}(-r+2h)^{n-1}}{ (-f+e)^{n-1}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-10}=g\dfrac{r^{n-1}}{e^{n-1}}\dfrac{ (-f+2e)^{n-1}(-g+2f)^{n-1}(-h+2g)^{n-1}(-r+2h)^{n-1}}{ (-f+e)^{n-1}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-9}=h\dfrac{r^{n-1}}{e^{n-1}}\dfrac{ (-f+2e)^{n-1}(-g+2f)^{n-1}(-h+2g)^{n-1}(-r+2h)^{n-1}}{ (-f+e)^{n-1}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-8}=\dfrac{r^{n}}{e^{n-1}}\dfrac{ (-f+2e)^{n-1}(-g+2f)^{n-1}(-h+2g)^{n-1}(-r+2h)^{n-1}}{ (-f+e)^{n-1}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}} , & & & \\ \displaystyle x_{8n-7}=\dfrac{r^{n}}{e^{n-1}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n-1}(-h+2g)^{n-1}(-r+2h)^{n-1}}{ (-f+e)^{n}(-g+f)^{n-1}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-6}=\dfrac{r^{n}}{e^{n-1}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n-1}(-r+2h)^{n-1}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n-1}(-r+h)^{n-1}}, & & & \\ \displaystyle x_{8n-5}=\dfrac{r^{n}}{e^{n-1}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n-1}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n-1}}. & & & \end{array} \right. \end{equation*} Now it follows from Equation (9) that \begin{equation*} \begin{array}{llll} \displaystyle x_{8n}=x_{8n-1}+\dfrac{ x_{8n-1} x_{8n-5}}{-x_{8n-4}+x_{8n-5}} & & & \\ = h\dfrac{r^{n}}{e^{n}}\dfrac{(-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} & & & \\ \left(1+ \dfrac{\dfrac{r^{n}}{e^{n-1}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n-1}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n-1}}}{-\dfrac{r^{n}}{e^{n-1}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}+\dfrac{r^{n}}{e^{n-1}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n-1}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n-1}} }\right ) \\ \displaystyle = h \dfrac{r^{n}}{e^{n}}\dfrac{(-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+ \dfrac{1}{-\dfrac{(-r+2h) }{(-r+h)}+1 }\right ) \\ \displaystyle = h\dfrac{r^{n}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+ \dfrac{(-r+h)}{-h}\right )\\ \displaystyle = h\dfrac{r^{n}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(\dfrac{r}{h}\right ) \\ \displaystyle = \dfrac{r^{n+1}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}. \\ & & & \\ \displaystyle = h\dfrac{r^{n}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+ \dfrac{1}{-\dfrac{(-r+2h) }{(-r+h)}+1 }\right ) & & & \\ \displaystyle = h\dfrac{r^{n}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+ \dfrac{(-r+h)}{-h}\right )\\ \displaystyle = h\dfrac{r^{n}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(\dfrac{r}{h}\right ) & & & \\ \displaystyle = \dfrac{r^{n+1}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}. & & & \end{array} \end{equation*} Similarly, \begin{equation*} \begin{array}{llll} \displaystyle x_{8n+1}=x_{8n}+\dfrac{ x_{8n} x_{8n-4}}{-x_{8n-3}+x_{8n-4}} = \dfrac{r^{n+1}}{e^{n}}\dfrac{(-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} & & & \\ \end{array} \end{equation*} \begin{equation*} \begin{array}{llll} \left(1+ \dfrac{\dfrac{r^{n}}{e^{n-1}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}}{-f\dfrac{r^{n}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}+\dfrac{r^{n}}{e^{n-1}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} }\right ) \\ \displaystyle = \dfrac{ r^{n+1}}{e^{n}}\dfrac{(-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+ \dfrac{1}{-\dfrac{f}{e}+1 }\right ) \\ \displaystyle = \dfrac{r^{n+1}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+ \dfrac{e}{-f+e}\right ) \\ \displaystyle = \dfrac{r^{n+1}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(\dfrac{-f+2e}{-f+e}\right ) \\ \displaystyle = \dfrac{r^{n+1}}{e^{n}}\dfrac{ (-f+2e)^{n+1}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n+1}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}\\ \displaystyle = \dfrac{r^{n+1}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+ \dfrac{1}{-\dfrac{f}{e}+1 }\right )\\ \displaystyle = \dfrac{r^{n+1}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(1+ \dfrac{e}{-f+e}\right ) & & & \\ \displaystyle = \dfrac{r^{n+1}}{e^{n}}\dfrac{ (-f+2e)^{n}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}} \left(\dfrac{-f+2e}{-f+e}\right ) \\ \displaystyle = \dfrac{r^{n+1}}{e^{n}}\dfrac{ (-f+2e)^{n+1}(-g+2f)^{n}(-h+2g)^{n}(-r+2h)^{n}}{ (-f+e)^{n+1}(-g+f)^{n}(-h+g)^{n}(-r+h)^{n}}. & & & \end{array} \end{equation*} Similarly, one can easily obtain the other relations. Thus, the proof is completed.

Consider numerical examples which represent different types of solutions to Equation (9).
The solution is unbounded since we choose \(x_{-4}=20, x_{-3}=13, x_{-2}=3, x_{-1}=2, x_0=1\) (see Figure \ref{fig7}).
However the solution converges to \(\bar x=0\) by choosing \(x_{-4}=100, x_{-3}=30, x_{-2}=10, x_{-1}=1, x_0=3\) (see Figure \ref{fig8}).

6. Conclusion

This paper discussed global stability, boundedness, and the solutions of some special cases of Equation (1). In Section 2 we proved that if \( b(d+3c)< (1-a)(c+d)^{2}\) then the equilibrium point of Equation (1) is locally asymptotically stable. In Section 3 we showed that the unique equilibrium of Equation (1) is globally asymptotically stable if \(d(1-a)\not=b\). In Section 4 we proved that the solution of Equation (1) is bounded if \(a+\dfrac{b}{d}< 1\). In Section 5 we gave the form of the solution of four special cases of Equation (1) and gave numerical examples of each case.

Author Contributions

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

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

Let \(G=(V,E)\) be a simple connected graph. The length of the shortest \(x-y\) path in \(G\) is the distance between \(x\) and \(y\) denoted by \(d(x,y)\) and \(max \{d(x,y): x,y \in V(G)\}\) is the diameter of \(G\) denoted by \(diam(G)\). Let \(P_n\), \(C_n\), \(K_n\), \(W_n\) and \(S_n\) denote the path of length \(n\), cycle of length \(n\), complete graph of length \(n\), wheel of length \(n\) and star graph of length \(n\) respectively. A subset \(D \subset V(G)\) is a dominating set (DS), of a graph \(G\) if every vertex of \(V(G) \setminus D\) has a neighbor in \(D\). The domination number of a graph \(G\), denoted by \(\gamma (G)\), is the minimum cardinality of a dominating set of \(G\), see [1]. A vertex \(v \in V(G)\) dominates an edge \(e \in E(G)\) if \(v\) is incident with \(e\) or with an edge adjacent to \(e\). The vertex-edge dominating set of \(G\) was introduced in [2] as the subset \(D \subset V(G)\) if every edge of \(G\) is vertex-edge dominated by a vertex in \(D\). The vertex-edge domination number of a graph \(G\) is the minimum cardinality of a vertex-edge dominating set of \(G\) and it is denoted by \(\gamma_{ve}(G)\). Further studies of vertex-edge domination in graphs can be found in [3, 4, 5].

A subset \(D\) of \(V(G)\) is an outer-connected dominating set of a graph \(G\) if \(D\) is a dominating set and \(G \setminus D\) is connected. In [6], the outer-connected domination number of a graph \(G\) was given as the outer-connected dominating set of \(G\) with minimum cardinality and is denoted by \(\gamma_c(G)\). Vizing [7] posed a conjecture concerning the domination number of the Cartesian product graphs. He showed that \(\gamma(G)\gamma(H) \leq \gamma(G \times H)\). For a survey of domination in Cartesian products, see [8]. In [9], the notion of outer-connected vertex edge dominating set was introduced. A subset \(D\) of \(V(G)\) is an outer-connected vertex-edge dominating set (OCVEDS) of \(G\) if \(D\) is a vertex edge dominating set of \(G\) and the graph \(G \setminus D\) is connected. The outer connected vertex-edge domination number of a graph \(G\), denoted by \(\gamma_{ve}^{oc}(G) \), is the minimum cardinality of an outer connected vertex edge dominating set of \(G\). The outer connected vertex edge domination number in Cartesian product of graphs has been given in [10].

The rest of this paper is organized as follows. In next Section 2, we give definition of lexicographic product and some known results. In Section 3, we give the outer-connected vertex edge domination number of the product \(P_m \ \circ \ H\) and \(C_m \ \circ \ H\) and the outer-connected vertex edge domination number of the product \(G \ \circ \ H\) where \(\gamma_{ve}^{oc}(G)=1\) and Section 4 contains concluding remarks.

2. Preliminaries

Definition 1. The lexicographic product \(G\circ H\) of two graphs \(G\) and \(H\) is a graph with vertex set \(V(G)\times V(H)\) in which \((x_1,y_1)\) and \((x_2,y_2)\) are adjacent if one of the following condition holds:

1. \(\{x_1,x_2\}\in E(G)\).
2. \(x_1= x_2\) and \(\{y_1,y_2\}\in E(H)\).

The graphs \(G\) and \(H\) are known as the factors of \(G\circ H\).

Suppose we are dealing with \(m\)-copies of a graph \(G\), we denote these \(m\)-copies of \(G\) by \(G^i\), where \(i=1,2,3,...,m\). The Lexicographic product \(G\circ H\), of graphs \(G\) and \(H\), can also be viewed as the graph obtained by replacing each vertex of \(G\) by a copy of \(H\) and every edge of \(G\) by the complete bipartite graph \(K_{[H],[H]}\). Henceforth, for any vertex \(i \in V(G)\), we refer the copy of \(H\), denoted by \(H^i\), in \(G \ \circ \ H\) corresponding to the vertex \(i\) as the \(i^{th}\) copy of \(H\) in \(G \ \circ \ H\). The following is the outer-connected vertex edge domination number of some standard graphs that would be used in this paper.

1. [9] \( \gamma_{ve}^{oc}(P_n)= \left\{\begin{array}{cc} 1 ,& \; if \ n=2 \ or \ 3 \\ \ \\ 2,& \; if \ n=4 \ or \ 5 \\ \ \\ n-3 ,& \; otherwise \\ \end{array}\right. \)
2. [9] \( \gamma_{ve}^{oc}(C_n)= \left\{\begin{array}{cc} 1 ,& \; if \ n=3 \ or \ 4 \\ \ \\ n-3 ,& \; otherwise \\ \end{array}\right. \)
3. [9] \( \gamma_{ve}^{oc}(K_n)=1\), for \(n \geq 1\)
4. \( \gamma_{ve}^{oc}(S_n)=1\), for \(n \geq 1\)
5. \( \gamma_{ve}^{oc}(W_n)=1\), for \(n \geq 1\)

3. Main results

In this section, we give our main results.

3.1. Outer-connected vertex edge domination number in \(P_m \ \circ \ H\) and \(C_m \ \circ \ H\)

Theorem 2. Let \(G\) be a cycle or path of order \(m\) and \(H\) be a path with maximum length four, then \(\gamma_{ve}^{oc}(G \ \circ \ H)= \lfloor \frac{m+2}{3} \rfloor\).

Proof. Let \(D=\{x_1,x_2,x_3,...,x_j\}\), \( 1 \leq j \leq \lfloor \frac{m+2}{3} \rfloor \in \gamma_{ve}^{oc}(G \ \circ \ P_n)\). The graph \(G \ \circ \ P_n\) has \(m\) copies of \(P_n\). Next we consider the following three cases.
Case 1: whenever \(m \equiv 0 (mod \ 3)\)
Let each element of \(D\) belong to any vertex of copies \(P_n^2,P_n^5,P_n^8,...,P_n^{m-1}\) respectively.
Case 2: whenever \(m \equiv 1 (mod \ 3)\)
Let each element of \(D\) belong to any vertex of copies \(P_n^2,P_n^5,P_n^8,...,P_n^{m-2}\) and \(P_n^m\) respectively.
Case 3: whenever \(m \equiv 2 (mod \ 3)\)
Again, let each element of \(D\) belong to any vertex of copies \(P_n^2,P_n^5,P_n^8,...,P_n^m\) respectively.
Clearly, for each of the cases above, each edge in \(G \ \circ \ P_n\), \(2 \leq n \leq 5\) is dominated by a vertex in \(D\) and \((G \ \circ \ P_n)\setminus D\) is connected. Hence the proof.

Next, we have the following corollary which is immediate from Theorem 2.

Corollary 3. Let \(G\) be a cycle or path of order \(m\), suppose we have a graph \(H\) such that \(\gamma_{ve}^{oc}(H)=1\), then \(\gamma_{ve}^{oc}(G \ \circ \ H)=\lfloor \frac{m+2}{3} \rfloor \).

Proof. The proof of this Corollary is same as that of Theorem 2.

Remark 1. Since the Wheel graph, Star graph, Complete graph and the Cycle of order \(3 \ or \ 4\) has their outer connected vertex edge domination number to be equal to \textit{one}, then the bound mentioned in Corollary 3 covers them.

Lemma 4. Let \(G\) be a cycle or path of order three and \(H=P_n\), \( n \geq 6\) then \(\gamma_{ve}^{oc}(G \ \circ \ P_n)=2\).

Proof. Consider the copies \(P_n^2\) and \( P_n^3\) in \(G \ \circ \ P_n\). Assume that \(D=\{x_1,x_2\} \in \{P_n^2 \cup P_n^3 \}\), where \(x_1 \in V(P_n^2)\) and \(x_2 \in V(P_n^3)\). It is easy to see that each edge in \(G \ \circ \ P_n\) is dominated by a vertex in \(D\). Also, \((G \ \circ \ P_n)\setminus D\) is connected. Therefore \(\gamma_{ve}^{oc}(G \ \circ \ P_n)=2\), and this completes Hence the proof.

Theorem 5. Let \(G\) be a cycle or path of order \(m\) and \(H=P_n, \ n\geq 6\), then
\(\gamma_{ve}^{oc}(G \ \circ \ P_n)=\left\{\begin{array}{cc} \frac{m+1}{2} ,& \; if \ m \ is \ odd \\ \ \\ \frac{m}{2},& \; if \ m \equiv 0 \ (mod \ 4) \\ \ \\ \frac{m+2}{2} ,& \; otherwise \\ \end{array}\right. \)

Proof. Let \(D\) be an OCVEDS of \(G \ \circ \ P_n\). We now consider the following three cases.
Case 1: whenever \(m \equiv \ 0 \ or \ 3 \ (mod \ 4)\)
Applying Lemma 4 to the first three copies of \(P_n\) we have \(|D|=2\). Next, for \(4 \leq q \leq m\), consider the \(q^{th}\) inner copies of \(P_n\) in the following way. Let \(D'=\{y_1,y_2\}\) such that \(y_1 \in V(P_n^q)\) if \(q \equiv \ 2 \ (mod \ 4)\) and \(y_2 \in V(P_n^q)\) if \(q \equiv \ 3 \ (mod \ 4)\).
Case 2: whenever \(m \equiv \ 1 \ (mod \ 4)\).
Here also, we apply Lemma 4 to the first three copies of \(P_n\). This gives \(|D|=2\). Now, for \(4 \leq q \leq m-1\), consider the \(q^{th}\) inner copies of \(P_n\) as follows. Let \(D'=x \cup \{y_1,y_2\}\) such that \(x \in V(P_n^{m-1})\) and \(y_1 \in V(P_n^q)\) if \(q \equiv \ 2 \ (mod \ 4)\) and \(y_2 \in V(P_n^q)\) if \(q \equiv \ 3 \ (mod \ 4)\).
Case 3: whenever \(m \equiv \ 2 \ (mod \ 4)\)
Applying Lemma 4 to the first three and last three copies of \(P_n\) we have \(|D|=4\). Next, for \(4 \leq q \leq m-3\), consider the \(q^{th}\) inner copies of \(P_n\) in the following way. Let \(D'=\{y_1,y_2\}\) such that \(y_1 \in V(P_n^q)\) if \(q \equiv \ 2 \ (mod \ 4)\) and \(y_2 \in V(P_n^q)\) if \(q \equiv \ 3 \ (mod \ 4)\).
Furthermore, notice that for each of these cases, every edge in the graph \(G \ \circ \ P_n\) is dominated by a vertex in \((D \cup D' )\) and \((D \cup D')\) is minimum. Also, \((G \ \circ \ P_n)\setminus (D \cup D')\) is connected. This completes the proof.

We now have the following corollary which is immediate from Theorem 5.

Corollary 6. Let \(G\) be a cycle or path of order \(m\) and \(H\) be a cycle of order \(n\) such that \( \ n\geq 5\), then
\(\gamma_{ve}^{oc}(G \ \circ \ C_n)=\left\{\begin{array}{cc} \frac{m+1}{2} ,& \; if \ m \ is \ odd \\ \ \\ \frac{m}{2},& \; if \ m \equiv 0 \ (mod \ 4) \\ \ \\ \frac{m+2}{2} ,& \; otherwise \\ \end{array}\right. \)

Proof. By applying the proof of Theorem 5 to the graph \(G \ \circ \ C_n\), it is not hard to see that the edges of the graph \(G \ \circ \ C_n\) is dominated by \((D \cup D')\) and \((D \cup D')\) is minimum. Also, \((G \ \circ \ C_n)\setminus (D \cup D')\) is not disconnected. Hence the proof.

3.2. Outer-connected vertex edge domination number in \(G \ \circ \ H\)

In this subsection, we provide the outer connected vertex edge domination number of the product \(G \ \circ \ H\) when \(\gamma_{ve}^{oc}(G)=1\).

Theorem 7. Let \(H\) be a simple connected graph and \(n>1\), then
\(\gamma_{ve}^{oc}(K_n \ \circ \ H)=\left\{\begin{array}{cc} 1 ,& \; if \ \gamma_{ve}^{oc}(H)=1 \\ \ \\ 2,& \; otherwise \\ \end{array}\right. \)

Proof. The graph \(K_n \ \circ \ H\) has \(n\) copies of graph \(H\) and the distance between any two pair of vertices of \(K_n \ \circ \ H\) is equal to two. We now split the proof in two cases.
Case 1: whenever \(\gamma_{ve}^{oc}(H)=1\)
Let the vertex \(x \in D\), where \(D\) is the OCVEDS of \(K_n \ \circ \ H\), be a vertex in any of the copy of \(H\) in \(K_n \ \circ \ H\). Clearly, \(D\) dominates the graph \(K_n \ \circ \ H\) and \((K_n \ \circ \ H) \setminus D\) is connected. Hence \(\gamma_{ve}^{oc} (K_n \ \circ \ H)=1\).
Case 2: whenever \(\gamma_{ve}^{oc}(H) \ne 1\)
Let the vertices \(x,y \in D\), where \(D\) is the OCVEDS of \(K_n \ \circ \ H\), be such that \(d_G(x,y)=1\), i.e., \(x\) and \(y\) are not in the same copy of \(K_n\) in \(K_n \ \circ \ H\). Thus \(D\) dominates the graph \(K_n \ \circ \ H\) and \((K_n \ \circ \ H) \setminus D\) is connected. Hence \(\gamma_{ve}^{oc} (K_n \ \circ \ H)=2\). The proof is complete.

Corollary 8. Suppose we have the graph \(G \ \circ \ H\), such that \(\gamma_{ve}^{oc}(G)=1\). If \(\gamma_{ve}^{oc}(H)=\gamma_{ve}^{oc}(G)\), then \(\gamma_{ve}^{oc}(G \ \circ \ H)=1\) and whenever \(\gamma_{ve}^{oc}(H)>1\), \(\gamma_{ve}^{oc}(G \ \circ \ H)=2\).

Proof. The proof of this Corollary can be drawn in the same way as that of Theorem 7.

4. Conclusion

We begin our conclusion with the following remark.

Remark Although the lexicographic product \(G \ \circ \ H\) is commutative, it is seen that the \(\gamma_{ve}^{oc}(G \ \circ \ H)\) may not necessarily be equal to \(\gamma_{ve}^{oc}(H \ \circ \ G)\). For instance, from Corollary 3 and Theorem 7, \(\gamma_{ve}^{oc}(C_m \ \circ \ K_n) \ne \gamma_{ve}^{oc}(K_n \ \circ \ C_m)\), whenever \(m>3\).

So far in this paper, we have provided the outer connected vertex edge domination number of the lexicographic product of two graphs. For some of the products mentioned, the outer connected vertex edge domination number of the graph \(G \ \circ \ H\) depends on either the \(\gamma_{ve}^{oc}(G)\), \(\gamma_{ve}^{oc}(H)\) or both.

Author Contributions

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

Competing Interests

The authors declare that they have no competing interests.

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.