Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring Zprqs

1. Introduction

For a positive integer n, ${\mathbb{Z}}_n$ denotes the ring of integers modulo n. In 1990, the unit graph was first introduced by Grimaldi [1] for the ring ${\mathbb{Z}}_n$ as follows: the unit graph $G({\mathbb{Z}}_n)$ is the graph obtained by setting all the elements of ${\mathbb{Z}}_n$ to be vertices and defining distinct vertices x and y to be adjacent if and only if $x+y\in U({\mathbb{Z}}_n)$. He discussed certain basic properties of the structure of the unit graph $G({\mathbb{Z}}_n)$ and studied the degree of a vertex, covering number, independence number, Hamilton cycles, and chromatic polynomial of the graph $G({\mathbb{Z}}_n)$. More about the unit graph $G({\mathbb{Z}}_n)$ can be seen in [2,3,4]. Later, Ashrafi et al. [5] generalized the unit graph from $G({\mathbb{Z}}_n)$ to $G(R)$ for an arbitrary ring R. They studied the chromatic index, diameter, girth, and planarity of $G(R)$. Some of the work associated with the unit graph on the rings can be found in [6,7,8,9].

In recent years, many researchers have studied the Laplacian spectrum and vertex connectivity of graphs associated with algebraic structures. In 2020, Chattopadhyay et al. [10] studied the Laplacian spectrum of the zero divisor graph $\Gamma ({\mathbb{Z}}_n)$ of the ring ${\mathbb{Z}}_n$. They discussed the Laplacian integrality, algebraic connectivity, vertex connectivity, and Laplacian spectral radius of $\Gamma ({\mathbb{Z}}_n)$. For other related works on the Laplacian spectrum and vertex connectivity of graphs associated to the ring ${\mathbb{Z}}_n$, one may refer to [11,12]. Shen et al. [3] determined the Laplacian spectrum of the unit graphs of the ring ${\mathbb{Z}}_n$ for $n=p^m$, where p is an odd prime and m is a positive integer. They proved that the algebraic connectivity and vertex connectivity of $G({\mathbb{Z}}_n)$ coincide if and only if $n=p^m$.

In this paper, we investigate the structure of $G({\mathbb{Z}}_n)$. Based on this structure, we study the Laplacian spectrum and vertex connectivity of $G({\mathbb{Z}}_n)$ for various n. The paper is arranged as follows: In Section 2, we provide the preliminary concepts and results that are used throughout the paper. In Section 3, we examine the structure of $G({\mathbb{Z}}_n)$ for $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, where $p_1,p_2,...,p_k$ are distinct primes and $k,r_1,r_2,...,r_k$ are positive integers such that at least one of the $r_i$ must be greater than 1. We prove that the graph $G({\mathbb{Z}}_n)$ is a generalized join of certain complete graphs and null graphs. In Section 4, we study the Laplacian spectrum of $G({\mathbb{Z}}_n)$, we prove that $G({\mathbb{Z}}_n)$ is Laplacian integral, and we deduce the algebraic connectivity and Laplacian spectral radius of $G({\mathbb{Z}}_n)$. In Section 5, we investigate the vertex connectivity of $G({\mathbb{Z}}_{pq})$ and $G({\mathbb{Z}}_{p^r{q}^s})$, where $2\ne p<q$ are primes and r and s are positive integers, based on their structure and Menger’s theorem. Moreover, we present the following conjectures:

Conjecture I: Let $n=p_1{p}_2...p_k$, where $2\ne p_1<p_2<...<p_k$ are primes. Then, the vertex connectivity of $G({\mathbb{Z}}_n)$ is ${\displaystyle (p_1-2)\prod _{2\le i\le k}{p}_i.}$

Conjecture II: Let $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, where $2\ne p_1<p_2<...<p_k$ are primes and $k,r_1,r_2,...,r_k$ are positive integers such that at least one of the $r_i$ must be greater than 1. Then, the vertex connectivity of $G({\mathbb{Z}}_n)$ is ${\displaystyle (p_1-2)p_1^{(r_1-1)}\prod _{2\le i\le k}{p}_i^{r_i}.}$

2. Preliminaries

In this section, we will present preliminary definitions and theorems that will be necessary for the following sections. Let G be a graph with vertex set $V(G)=\{v_1,v_2,...,v_n\}$ and edge set $E(G)=\{e_1,e_2,...,e_m\}$. For $1\le i,j\le n$, two vertices $v_i$ and $v_j$ in G are adjacent (or neighbors) in G if $v_i$ and $v_j$ are endpoints of an edge e of G, and we write $v_i\sim v_j$ if $v_i$ is adjacent to $v_j$ in G. For $v\in V(G)$, we denote by $N_G(v)$ the set of all neighbors of v in G. The degree of a vertex v in G, denoted by $deg(v)$, is the number of edges incident with it. A path in a graph is a sequence of distinct vertices with the property that each vertex in the sequence is adjacent to the next vertex of it. The graph G is said to be connected if G contains a path between every pair of vertices. A complete graph is a graph in which each pair of distinct vertices is joined by an edge. We denote the complete graph with n vertices by $K_n$. The complement of $K_n$ is a null graph and is denoted by ${\overline{K}}_n$. A clique of a graph G is a complete subgraph of G. A coclique in a graph G is a set of pairwise nonadjacent vertices. An isomorphism of graphs $G_1$ and $G_2$, $G_1\cong G_2$, is a bijection $\varphi$ between the vertex sets of $G_1$ and $G_2$ such that for any two vertices x and y of $G_1$, x and y are adjacent in $G_1$ if and only if $\varphi (x)$ and $\varphi (y)$ are adjacent in $G_2$. For two graphs $G_1$ and $G_2$ with disjoint vertex sets, the join $G_1\vee G_2$ of $G_1$ and $G_2$ is the graph obtained from the union of $G_1$ and $G_2$ by adding new edges from each vertex of $G_1$ to every vertex of $G_2$.

Let R be a ring with unity and $U(R)$ be the set of units of R. The unit graph $G(R)$ of R is the graph whose vertices are all the elements of R, defining distinct vertices x and y to be adjacent if and only if $x+y$ is a unit in R. Let R be a commutative ring with unity. An element $r\in R$ is nilpotent if there exists an integer $k>1$ such that $r^k=0$. The nilradical of R, denoted $nil(R)$, is the set of all nilpotent elements of R. An ideal $N\ne R$ in R is a prime ideal if $ab\in N$ implies that either $a\in N$ or $b\in N$ for $a,$$b\in R$. The nilradical of R is the intersection of prime ideals of R. A maximal ideal of R is an ideal m different from R such that there is no proper ideal I of R properly containing m. The Jacobson radical $J(R)$ of R is the intersection of maximal ideals of R. Every maximal ideal in R is a prime ideal. So, $J(R)=nil(R)$. Recall that, the sum of a unit and a nilpotent element is a unit. The following two results give some properties of the unit graph $G(R)$.

Lemma 1.

[5] Let R be a commutative ring and suppose that $J(R)$ denotes the Jacobson radical of R. If $x,y\in R$, then the following statements hold:

1. 

If $x+J(R)$ and $y+J(R)$ are adjacent in $G(R/J(R))$, then every element of $x+J(R)$ is adjacent to every element of $y+J(R)$.

2. 

If $2x\notin U(R)$, then $x+J(R)$ is a coclique in $G(R)$.

3. 

If $2x\in U(R)$, then $x+J(R)$ is a clique in $G(R)$.

Proposition 1.

[5] Let R be a finite ring. Then, the following statements hold for the unit graph of R:

1. 

If $2\notin U(R)$, then the unit graph $G(R)$ is a $|U(R)|$-regular graph.

2. 

If $2\in U(R)$, then for every $x\in U(R)$ we have $deg(x)=|U(R)|-1$, and for every $x\in R\setminus U(R)$ we have $deg(x)=|U(R)|$.

For a finite simple undirected graph G, the adjacency matrix $A(G)$ is defined as the $n\times n$ matrix whose $(i,j)th$ entry is 1 if $v_i\sim v_j$ and 0 otherwise. The Laplacian matrix $L(G)$ of G is defined by $L(G):=D(G)-A(G)$, where $D(G)=Diag(d_1,d_2,...,d_n)$ is the diagonal matrix such that $d_i$ are degrees of vertices of G. The matrix $L(G)$ is a real, symmetric, and positive semidefinite so that its eigenvalues are real and nonnegative. Since the sum of each row of $L(G)$ is zero, the smallest eigenvalue of $L(G)$ is 0. The largest eigenvalue of $L(G)$ is known as the Laplacian spectral radius $\lambda (G)$ of G, and the second smallest eigenvalue of $L(G)$ is known as the algebraic connectivity $\mu (G)$ of G and $\mu (G)>0$ if and only if G is connected. A graph G is called Laplacian integral if all the eigenvalues of $L(G)$ are integers. More literature about the Laplacian matrix of graphs can be seen in [13,14].

The spectrum of a square matrix C, denoted by $\sigma (C)$, is the multiset of all the eigenvalues of C. If ${\eta }_1,{\eta }_2,...,{\eta }_c$ are distinct eigenvalues of B with respective multiplicities ${\gamma }_1$, ${\gamma }_2$$,...,{\gamma }_c$, then we shall denote the spectrum of C by

$$\sigma (C)=\left\{\begin{array}{cccc}{\eta }_1& {\eta }_2& ...& {\eta }_c\\ {\gamma }_1& {\gamma }_2& ...& {\gamma }_c\end{array}\right\}.$$

For a graph G, the Laplacian spectrum of G is the spectrum of $L(G)$, we write $\sigma (L(G))$ as ${\sigma }_L(G)$. For example,

$${\sigma }_L(K_n)=\left\{\begin{array}{cc}0& n\\ 1& n-1\end{array}\right\}\mathrm{and}{\sigma }_L({\overline{K}}_n)=\left\{\begin{array}{c}0\\ n\end{array}\right\}.$$

(1)

Let G be a graph on k vertices with $V(G)=\{v_1,v_2,...,v_k\}$ and let $H_1,H_2,...,H_k$ be k pairwise disjoint graphs. The G-generalized join graph $G[H_1,H_2,...,H_k]$ of $H_1,H_2,...,H_k$ is the graph formed by replacing each vertex $v_i$ of G by the graph $H_i$ and then joining each vertex of $H_i$ to every vertex of $H_j$ whenever $v_i\sim v_j$ in G [15]. The following result is useful in the sequel.

Theorem 1.

[16] Let G be a graph on k vertices with $V(G)=\{v_1,v_2,...,v_k\}$ and let $H_1,H_2,...,H_k$ be k pairwise disjoint graphs on $n_1,n_2,...,n_k$ vertices, respectively. Then, the Laplacian spectrum of $G[H_1,H_2,...,H_k]$ is given by

$${\sigma }_L(G[H_1,H_2,...,H_k])=\left(\begin{array}{c}{\displaystyle \bigcup _{j=1}^k(M_j+({\sigma }_L(H_j))\setminus \left\{0\right\})}\end{array}\right)\bigcup \sigma (\mathit{L}(G)),$$

(2)

where

$$M_j=\left\{\begin{array}{cc}{\displaystyle \sum _{v_i\sim v_j}{n}_i}& if{N}_G(v_j)\ne \varnothing ;\hfill \\ \\ 0& otherwise,\hfill \end{array}\right.$$

$$\mathit{L}(G)=\left[\begin{array}{cccc}{M}_1& -s_{1,2}& ...& -s_{1,k}\\ -s_{1,2}& M_2& ...& -s_{2,k}\\ ...& ...& ...& ...\\ -s_{1,k}& -s_{2,k}& ...& M_k\end{array}\right],$$

and

$$s_{i,j}=\left\{\begin{array}{cc}\sqrt{n_i{n}_j}& if{v}_i\sim v_{j}inG;\hfill \\ 0& otherwise.\hfill \end{array}\right.$$

In (2), ${\sigma }_L(H_j)\setminus \left\{0\right\}$ means that one copy of the eigenvalue 0 is removed from the multiset ${\sigma }_L(H_j)$, and $(M_j+({\sigma }_L(H_j))\setminus \left\{0\right\})$ means that $M_j$ is added to each element of ${\sigma }_L(H_j)\setminus \left\{0\right\}$.

Let n be a positive integer. Euler’s totient function, denoted by $\phi (n)$, is the number of positive integers less than or equal to n that are relatively prime to n. Let $n=p_1{p}_2...p_k$ and $p_i$ be distinct primes for $i=1,2,...,k$. Note that, $\phi (p_i)=p_i-1$ and $\phi (p_1{p}_2...p_k)=\phi (p_1)\phi (p_2)...\phi (p_k)$, so $\phi (n)={\prod }_{i=1}^k\phi (p_i)$. Fakieh et al. studied the Laplacian spectrum of the unit graphs associated to the ring ${\mathbb{Z}}_{p_1{p}_2...p_k}$, where $p_i$ are distinct primes and $i=1,2,...,k$ [4]. This result is the main tool to prove Theorem 4.

Theorem 2.

[4] Let $p_i\ne 2$ be distinct primes and k be a positive integer, $1\le i,j\le k$. Then:

1. 

If $n=p_1{p}_2...p_k$, then the Laplacian spectrum of $G({\mathbb{Z}}_n)$ is

$$\begin{array}{cc}\hfill {\sigma }_L(G({\mathbb{Z}}_n))& =\left\{\begin{array}{ccccc}0& [\phi (p_i)\pm 1]\prod _{i\ne j}\phi (p_j)& [\phi (p_i)\phi (p_j)\pm 1]\prod _{h\ne i,j}\phi (p_h)& ...& \\ \\ 1& {\displaystyle \frac{\phi (p_i)}{2}}& {\displaystyle \frac{\phi (p_i)\phi (p_j)}{2}}& ...\end{array}\right.\hfill \\ \\ \hfill & \left.\begin{array}{ccc}[\phi (p_1)\phi (p_2)...\phi (p_{k-1})\pm 1]\phi (p_k)& \left[\prod _{1\le i\le k}\phi (p_i)\right]\pm 1& \\ \\ {\displaystyle \frac{{\prod }_{1\le i\le k-1}\phi (p_i)}{2}}& {\displaystyle \frac{{\prod }_{1\le i\le k}\phi (p_i)}{2}}\end{array}\right\}.\hfill \end{array}$$

2. 

If $n=2p_1{p}_2...p_k$, then the Laplacian spectrum of $G({\mathbb{Z}}_n)$ is

$$\begin{array}{cc}\hfill {\sigma }_L(G({\mathbb{Z}}_n))& =\left\{\begin{array}{ccccc}0& [\phi (p_i)\pm 1]\prod _{i\ne j}\phi (p_j)& [\phi (p_i)\phi (p_j)\pm 1]\prod _{h\ne i,j}\phi (p_h)& ...& \\ \\ 1& \phi (p_i)& \phi (p_i)\phi (p_j)& ...\end{array}\right.\hfill \\ \\ \hfill & \left.\begin{array}{ccc}\left[\phi (p_1)\phi (p_2)...\phi (p_{k-1})\pm 1\right]\phi (p_k)& \left[\prod _{1\le i\le k}\phi (p_i)\right]\pm 1& 2\prod _{1\le i\le k}\phi (p_i)\\ \\ \prod _{1\le i\le k-1}\phi (p_i)& \prod _{1\le i\le k}\phi (p_i)& 1\end{array}\right\}.\hfill \end{array}$$

The vertex connectivity $\kappa (G)$ of a graph G is the minimum number of vertices whose removal from G leaves a disconnected or trivial graph. A family of two or more paths in a graph G is said to be internally disjoint if no vertex of G is an internal vertex of more than one path in the family. There is a strong result for the vertex connectivity $\kappa (G)$ by Menger’s theorem. Menger’s theorem says that the maximum number of internally disjoint $uv$-paths in G is equal to the minimum number of vertices whose deletion destroys all $uv$-paths, where u and v are nonadjacent in G [17]. So that

$$\begin{array}{cc}\hfill \kappa (G)& =min\{\rho (u,v):\rho (u,v)\mathrm{is}\mathrm{the}\mathrm{maximum}\mathrm{number}\mathrm{of}\mathrm{internally}\mathrm{disjoint}uv\text{-}\mathrm{paths}\mathrm{in}G,\hfill \\ \hfill & \mathrm{where}u\nsim v\}.\hfill \end{array}$$

This paper uses Menger’s theorem to examine the vertex connectivity of $G({\mathbb{Z}}_n)$.

3. $G({\mathbb{Z}}_n)$ as a Generalized Join Graph

For a positive integer n, ${\mathbb{Z}}_n$ denotes the ring of integers modulo n. The elements of the ring ${\mathbb{Z}}_n$ are referred to as $0,1,2,$ and $n-1$. A nonzero element $x\in {\mathbb{Z}}_n$ is a unit in ${\mathbb{Z}}_n$ if x is relatively prime with n, $(x,n)=1$. Through this section, we use $p_i$ as a prime number. Also, an integer n can be written in the form $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, where $p_1,p_2,...,p_k$ are distinct primes and $k,r_1,r_2,...,r_k$ are positive integers such that at least one of the $r_i$ must be greater than 1. In this section, we prove that $G({\mathbb{Z}}_n)$ is a generalized join graph of some complete graphs and null graphs. To this end, first, we study the structure of $G({\mathbb{Z}}_n)$. Denote the maximal ideal of ${\mathbb{Z}}_n$ by ${\mathrm{m}}_i$, ${\mathrm{m}}_i$ is an ideal generated by the prime divisors $p_i$ of n, that is, ${\mathrm{m}}_i=〈p_i〉$. So, $J({\mathbb{Z}}_n)=\cap {\mathrm{m}}_i=〈{\prod }_{i=1}^k{p}_i〉\ne \left\{0\right\}$. Put $\mathbf{p}={\prod }_{i=1}^k{p}_i$, then

$${\mathbb{Z}}_n/J({\mathbb{Z}}_n)=\{i+J({\mathbb{Z}}_n):i\in \{0,1,...,\mathbf{p}-1\}\}\mathrm{and}{\mathbb{Z}}_n/J({\mathbb{Z}}_n)\cong {\mathbb{Z}}_{\mathbf{p}}.$$

Let S be the set of distinct representatives of ${\mathbb{Z}}_n/J({\mathbb{Z}}_n)$. For $i\in S$, we denote

$$B_i=i+J({\mathbb{Z}}_n).$$

Note that the sets $B_0,B_1,...,B_{\mathbf{p}-1}$ form a partition of the vertex set of $G({\mathbb{Z}}_n)$. Thus,

$${\displaystyle V(G({\mathbb{Z}}_n))=\bigcup _{i=0}^{\mathbf{p}-1}{B}_i.}$$

(3)

Let $B_0=J({\mathbb{Z}}_n)$. Then, for $j\in S$

$$|B_j|=|B_0|=|J({\mathbb{Z}}_n)|=p_1^{r_1-1}{p}_2^{r_2-1}...p_k^{r_k-1}.$$

The following result describes the adjacency criterion of vertices in $G({\mathbb{Z}}_n)$, where $V(G({\mathbb{Z}}_n))$ is described in Equation (3).

Lemma 2.

For $i,j\in S$, every vertex of $B_i$ is adjacent to every vertex of $B_j$ in $G({\mathbb{Z}}_n)$ if and only if $(i+j,n)=1$.

Proof.$(\Rightarrow )$ Clearly.

$(\Leftarrow )$ Suppose that $(i+j,n)=1$. Let $a\in B_i$ and $b\in B_j$, which can be written as $a=i+a_i$ and $b=j+b_j$, where $a_i,b_j\in J({\mathbb{Z}}_n)$. Now, $a+b=(i+j)+(a_i+b_j)$. Here $i+j$ is a unit by assumption, and $a_i+b_j$ is nilpotent. So, $a+b$ is a unit, and hence every vertex of $B_i$ is adjacent to every vertex of $B_j$ in $G({\mathbb{Z}}_n)$. □

By using Lemmas 1 [(2),(3)] and 2, the following is evident.

Corollary 1.

The following statements hold:

1. 

For $i\in S$, the induced subgraph $G(B_i)$ of $G({\mathbb{Z}}_n)$ on the vertex set $B_i$ is either the complete graph $K_{|J({\mathbb{Z}}_n)|}$ or its complement graph ${\overline{K}}_{|J({\mathbb{Z}}_n)|}$. Indeed, $G(B_i)$ is $K_{|J({\mathbb{Z}}_n)|}$ if and only if $2i\in U({\mathbb{Z}}_n)$.

2. 

For $i,j\in S$ with $i\ne j$, a vertex of $G(B_i)$ is adjacent to either all or none of the vertices of $G(B_j)$ in $G({\mathbb{Z}}_n)$.

The above corollary implies that the partition $B_0\cup B_1\cup ...\cup B_{\mathbf{p}-1}$ of the vertex set $V(G({\mathbb{Z}}_n))$ of $G({\mathbb{Z}}_n)$ is an equitable partition in such a way that every vertex of the $B_i$ has equal number of neighbors in $B_j$ for all $i,j\in S$.

We define $G_n$ by the simple graph whose vertices are the distinct representatives of ${\mathbb{Z}}_n/J({\mathbb{Z}}_n)$, that is, the set of vertices is S, and in which two distinct vertices i and j are adjacent if and only if $(i+j,n)=1$. The graph $G_n$ will play an important role in the rest of the paper.

Lemma 3.

$G({\mathbb{Z}}_{\mathbf{p}})\cong G_n$.

Proof. 

We define a map $\varphi :V(G({\mathbb{Z}}_{\mathbf{p}}))\to V(G_n)$ such that $\varphi (B_i)=i$. Clearly, $\varphi$ is well-defined and bijection. From Lemma 2, the adjacency relationships are preserved by $\varphi$. Hence, the result follows. □

The following lemma states that $G({\mathbb{Z}}_n)$ can be expressed as a generalized join of certain complete graphs and null graphs.

Lemma 4.

Let $G(B_i)$ be the induced subgraph of $G({\mathbb{Z}}_n)$ on the vertex set $B_i$ for $0\le i\le \mathbf{p}-1$. Then,

$$G({\mathbb{Z}}_n)=G_n[G(B_0),G(B_1),\dots ,G(B_{\mathbf{p}-1})].$$

Proof. 

Replace the vertex i of $G_n$ by $G(B_i)$ for $0\le i\le \mathbf{p}-1$. Thus, the result follows from Lemma 2 and Corollary 1. □

Example 1.

The unit graph $G({\mathbb{Z}}_{25})$ is shown in Figure 1. By Lemma 4, we have

$$G({\mathbb{Z}}_{25})=G_{25}[G(B_0),G(B_1),G(B_2),G(B_3),G(B_4)],$$

where $G_{25}$ is shown in Figure 2, $G(B_0)={\overline{K}}_5$, and $G(B_i)=K_5$ for $1\le i\le 4$. In Figure 1, the lines between two squares mean that each vertex in one square is adjacent to every vertex in the other square.

4. Laplacian Spectrum of $G({\mathbb{Z}}_n)$

In this section, we investigate the Laplacian spectrum of $G({\mathbb{Z}}_n)$ for $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, where $2\ne p_1<p_2<...<p_k$ are primes. For $0\le i\le \mathbf{p}-1$, we give the weight $|J({\mathbb{Z}}_n)|=t$ to the vertex i of the graph $G_n$. Define the integer

$$M_j=t{b}_j,where{b}_j=deg(j)in{G}_n$$

for $0\le j\le \mathbf{p}-1$. The $\mathbf{p}\times \mathbf{p}$ vertex weighted Laplacian matrix $\mathbf{L}(G_n)$ of $G_n$ defined in Theorem 1 is given by

$$\mathbf{L}(G_n)=\left[\begin{array}{cccc}{M}_0& -h_{0,1}& ...& -h_{0,\mathbf{p}-1}\\ -h_{0,1}& M_1& ...& -h_{1,\mathbf{p}-1}\\ ...& ...& ...& ...\\ -h_{0,\mathbf{p}-1}& -h_{1,\mathbf{p}-1}& ...& M_{\mathbf{p}-1}\end{array}\right],$$

(4)

where

$$h_{i,j}=\left\{\begin{array}{cc}t& ifi\sim jin{G}_n;\hfill \\ 0& otherwise\hfill \end{array}\right.$$

for $0\le i\ne j\le \mathbf{p}-1$.

The following remark is an immediate result of Proposition 1.

Remark 1.

The following statements hold:

1. 

If $2\notin U({\mathbb{Z}}_{\mathbf{p}})$, then $M_j=t\phi (\mathbf{p})$.

2. 

If $2\in U({\mathbb{Z}}_{\mathbf{p}})$, then

$$M_j=\left\{\begin{array}{cc}t(\phi (\mathbf{p})-1)& ifj\in U({\mathbb{Z}}_{\mathbf{p}});\hfill \\ \\ t\phi (\mathbf{p})& ifj\notin U({\mathbb{Z}}_{\mathbf{p}}).\hfill \end{array}\right.$$

Lemma 5.

$\mathit{L}(G_n)=t(L(G_n))$

Proof. 

The proof is direct from definition of $\mathit{L}(G_n)$ in Equation (4). □

The following theorem describes the Laplacian spectrum of $G({\mathbb{Z}}_n)$.

Theorem 3.

Let $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, where $p_1<p_2<...<p_k$ are primes, $r_i$ and k are positive integers, and $1\le i\le k$. The Laplacian spectrum of $G({\mathbb{Z}}_n)$ is given by

$${\sigma }_L(G({\mathbb{Z}}_n))=\left(\begin{array}{c}{\displaystyle \bigcup _{j=0}^{\mathbf{p}-1}(M_j+({\sigma }_L(G(B_j))\setminus \left\{0\right\}))}\end{array}\right)\bigcup t{\sigma }_L(G_n),$$

where $j\in S$ and $M_j+({\sigma }_L(G(B_j)\setminus \left\{0\right\})$ means that $M_j$ is added to each element of the multiset ${\sigma }_L(G(B_j)\setminus \left\{0\right\}$.

Proof. 

By Lemma 4, we have

$$G({\mathbb{Z}}_n)=G_n[G(B_0),G(B_1),\dots ,G(B_{\mathbf{p}-1})].$$

Consequently, the result can be obtained by using Theorem 1 and Lemma 5. □

By Corollary 1, $G(B_i)$ is either $K_t$ or ${\overline{K}}_t$ for $1\le i\le k$. By Theorem 3, out of the n number of Laplacian eigenvalues of $G({\mathbb{Z}}_n)$, $n-\mathbf{p}$ of them are known to be nonzero integer values. The remaining $\mathbf{p}$ Laplacian eigenvalues of $G({\mathbb{Z}}_n)$ will come from the Laplacian eigenvalues of $G({\mathbb{Z}}_{\mathbf{p}})$.

Corollary 2.

Let $2\ne p_1<p_2<...<p_k$ be distinct primes and $r,$$r_i$, $s_i$, k be positive integers, where $1\le i\le k$. Then, the Laplacian spectrum of $G({\mathbb{Z}}_n)$ is given by

1. 

If $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, then

$$\begin{array}{cc}\hfill {\sigma }_L(G({\mathbb{Z}}_n))& =(\underset{\phi (\mathbf{p})-\mathrm{times}}{\underbrace{t(\phi (\mathbf{p})-1)+({\sigma }_L(K_t)\setminus \left\{0\right\})}})\bigcup (\underset{[\mathbf{p}-\phi (\mathbf{p})]-\mathrm{times}}{\underbrace{t(\phi (\mathbf{p})+({\sigma }_L({\overline{K}}_t)\setminus \left\{0\right\})}})\bigcup t{\sigma }_L(G_n).\hfill \end{array}$$

2. 

If $n=2^r{p}_1^{s_1}{p}_2^{s_2}...p_k^{s_k}$, then

$$\begin{array}{cc}\hfill {\sigma }_L(G({\mathbb{Z}}_n))& =(\underset{2\mathbf{p}-\mathrm{times}}{\underbrace{t(\phi (\mathbf{p})+({\sigma }_L({\overline{K}}_t)\setminus \left\{0\right\})}})\bigcup t{\sigma }_L(G_n).\hfill \end{array}$$

Proof. 

By the above argument and Remark 1, the result holds. □

The following result gives the Laplacian spectrum of $G({\mathbb{Z}}_n)$ for $n=p^r{q}^s$, where $p<q$ are primes and r, s are positive integers.

Theorem 4.

Let $2\ne p<q$ be primes and $r,s$ be positive integers. Then:

1. 

If $n=p^r{q}^s$, then the Laplacian spectrum of $G({\mathbb{Z}}_n)$ consists of

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cccc}0& p^{r-1}{q}^{s-1}[\phi (p)-1]\phi (q)& p^{r-1}{q}^{s-1}[\phi (q)-1]\phi (p)& p^{r-1}{q}^{s-1}[\phi (p)\phi (q)-1]\\ \\ 1& {\displaystyle \frac{\phi (p)}{2}}& {\displaystyle \frac{\phi (q)}{2}}& {\displaystyle \frac{\phi (p)\phi (q)}{2}}\end{array}\right.\hfill \\ \\ \hfill & \left.\begin{array}{cccc}{p}^{r-1}{q}^{s-1}[\phi (p)\phi (q)+1]& p^{r-1}{q}^s\phi (p)& p^r{q}^{s-1}\phi (q)& p^{r-1}{q}^{s-1}\phi (p)\phi (q)\\ \\ {\displaystyle \frac{\phi (p)\phi (q)}{2}}& {\displaystyle \frac{\phi (q)}{2}}& {\displaystyle \frac{\phi (p)}{2}}& p^r{q}^s-pq\end{array}\right\}.\hfill \end{array}$$

(5)

2. 

If $n=2^r{q}^s$, then the Laplacian spectrum of $G({\mathbb{Z}}_n)$ is

$$\begin{array}{c}\hfill {\sigma }_L(G({\mathbb{Z}}_n))=\left\{\begin{array}{ccccc}0& 2^{r-1}{p}^{s-1}[\phi (p)-1]& 2^{r-1}{p}^s& 2^r{p}^{s-1}\phi (p)& 2^{r-1}{p}^{s-1}\phi (p)\\ \\ 1& \phi (p)& \phi (p)& 1& 2^r{p}^s-2p\end{array}\right\}.\end{array}$$

Proof. 

1. Let $n=p^r{q}^s$, where $2<p<q$ are primes and r, s are positive integers. So, the Jacobson radical of ${\mathbb{Z}}_n$ is $〈pq〉$ and the set of distinct representatives of ${\mathbb{Z}}_n/〈pq〉$ is $\{0,1,...,pq-1\}$. Thus,

$$G({\mathbb{Z}}_{p^r{q}^s})=G_{p^r{q}^s}[\underset{\phi (pq)-\mathrm{times}}{\underbrace{K_{p^{r-1}{q}^{s-1}},\dots ,K_{p^{r-1}{q}^{s-1}}}},\underset{[pq-\phi (pq)]-\mathrm{times}}{\underbrace{{\overline{K}}_{p^{r-1}{q}^{s-1}},\dots ,{\overline{K}}_{p^{r-1}{q}^{s-1}}}}].$$

By Corollary 2, the Laplacian spectrum of $G({\mathbb{Z}}_n)$ is given by

$$\begin{array}{cc}\hfill {\sigma }_L(G({\mathbb{Z}}_n))& =(\underset{\phi (pq)-\mathrm{times}}{\underbrace{t(\phi (pq)-1)+({\sigma }_L(K_{p^{r-1}{q}^{s-1}})\setminus \left\{0\right\})}})\bigcup (\underset{[pq-\phi (pq)]-\mathrm{times}}{\underbrace{t(\phi (pq)+({\sigma }_L({\overline{K}}_t)\setminus \left\{0\right\})}})\hfill \\ \hfill & \bigcup p^{r-1}{q}^{s-1}{\sigma }_L(G_{p^r{q}^s}).\hfill \end{array}$$

By Eq (1), the Laplacian spectrum of $K_{p^{r-1}{q}^{s-1}}$ and ${\overline{K}}_{p^{r-1}{q}^{s-1}}$ are

$${\sigma }_L(K_{p^{r-1}{q}^{s-1}})=\left\{\begin{array}{cc}0& p^{r-1}{q}^{s-1}\\ \\ 1& p^{r-1}{q}^{s-1}-1\end{array}\right\}\mathrm{and}{\sigma }_L({\overline{K}}_{p^{r-1}{q}^{s-1}})=\left\{\begin{array}{c}0\\ \\ p^{r-1}{q}^{s-1}\end{array}\right\}.$$

Then,

$$\begin{array}{cc}\hfill {\sigma }_L(G({\mathbb{Z}}_n))& =\left\{\begin{array}{c}{p}^{r-1}{q}^{s-1}[\phi (pq)-1]+p^{r-1}{q}^{s-1}\\ \\ \phi (pq)(p^{r-1}{q}^{s-1}-1)\end{array}\right\}\bigcup \left\{\begin{array}{c}{p}^{r-1}{q}^{s-1}\phi (pq)\\ \\ pq-\phi (pq)(p^{r-1}{q}^{s-1}-1)\end{array}\right\}\hfill \\ \\ \hfill & \bigcup \left\{p^{r-1}{q}^{s-1}{\sigma }_L(G_{p^r{q}^s})\right\}.\hfill \end{array}$$

By using Lemma 3, $G_{p^r{q}^s}$ is isomorphic to $G({\mathbb{Z}}_{pq})$, and hence ${\sigma }_L(G_{p^r{q}^s})={\sigma }_L(G({\mathbb{Z}}_{pq}))$. So, by Theorem 2, we have

$$\begin{array}{cc}\hfill {\sigma }_L(G_{p^r{q}^s})& =\left\{\begin{array}{ccccc}0& [\phi (p)-1]\phi (q)& [\phi (q)-1]\phi (p)& \phi (p)\phi (q)-1& \phi (p)\phi (q)+1\\ \\ 1& {\displaystyle \frac{\phi (p)}{2}}& {\displaystyle \frac{\phi (q)}{2}}& {\displaystyle \frac{\phi (p)\phi (q)}{2}}& {\displaystyle \frac{\phi (p)\phi (q)}{2}}\end{array}\right.\hfill \\ \\ \hfill & \left.\begin{array}{cc}\phi (p)[\phi (q)+1]& \phi (q)[\phi (p)+1]\\ \\ {\displaystyle \frac{\phi (q)}{2}}& {\displaystyle \frac{\phi (p)}{2}}\end{array}\right\}.\hfill \end{array}$$

So,

$$\begin{array}{cc}\hfill {\sigma }_L(G({\mathbb{Z}}_{p^r{q}^s}))& =\left\{\begin{array}{c}{p}^{r-1}{q}^{s-1}\phi (pq)\\ \\ p^r{q}^s-pq\end{array}\right\}\bigcup \left\{\begin{array}{ccc}0& p^{r-1}{q}^{s-1}[\phi (p)-1]\phi (q)& p^{r-1}{q}^{s-1}[\phi (q)-1]\phi (p)\\ \\ 1& {\displaystyle \frac{\phi (p)}{2}}& {\displaystyle \frac{\phi (q)}{2}}\end{array}\right.\hfill \\ \\ \hfill & \left.\begin{array}{cccc}{p}^{r-1}{q}^{s-1}[\phi (p)\phi (q)-1]& p^{r-1}{q}^{s-1}[\phi (p)\phi (q)+1]& p^{r-1}{q}^s\phi (p)& p^r{q}^{s-1}\phi (q)\\ \\ {\displaystyle \frac{\phi (p)\phi (q)}{2}}& {\displaystyle \frac{\phi (p)\phi (q)}{2}}& {\displaystyle \frac{\phi (q)}{2}}& {\displaystyle \frac{\phi (p)}{2}}\end{array}\right\}.\hfill \end{array}$$

Hence, the Laplacian spectrum of $G({\mathbb{Z}}_{p^r{q}^s})$ is as in Eq (5).

2. Let $n=2^r{p}^s$, where $p\ne 2$ is a prime and $r,$s are positive integers. Note that, $S=\{0,1,...,2p-1\}$ is the vertex set of the graph $G_{2^r{p}^s}$. Thus,

$$G({\mathbb{Z}}_{2^r{p}^s})=G_{2^r{p}^s}[\underset{2p-\mathrm{times}}{\underbrace{{\overline{K}}_{2^{r-1}{p}^{s-1}},{\overline{K}}_{2^{r-1}{p}^{s-1}},\dots ,{\overline{K}}_{2^{r-1}{p}^{s-1}}}}].$$

By Corollary 2, the Laplacian spectrum of $G({\mathbb{Z}}_n)$ is given by

$$\begin{array}{cc}\hfill {\sigma }_L(G({\mathbb{Z}}_{2^r{p}^s}))& =(\underset{2p-\mathrm{times}}{\underbrace{2^{r-1}{p}^{s-1}(\phi (p)+({\sigma }_L({\overline{K}}_{2^{r-1}{p}^{s-1}})\setminus \left\{0\right\})}})\bigcup 2^{r-1}{p}^{s-1}{\sigma }_L(G_{2^r{p}^s})\hfill \\ \\ \hfill & =\left\{\begin{array}{c}{2}^{r-1}{p}^{s-1}\phi (p)\\ \\ 2p(2^{r-1}{p}^{s-1}-1)\end{array}\right\}\bigcup \left\{2^{r-1}{p}^{s-1}{\sigma }_L(G_{2^r{p}^s})\right\}.\hfill \end{array}$$

By using Lemma 3, $G_{2^r{p}^s}$ is isomorphic to $G({\mathbb{Z}}_{2p})$, and hence ${\sigma }_L(G_{2^r{p}^s})={\sigma }_L(G({\mathbb{Z}}_{2p}))$. So, by Theorem 2, we have

$${\sigma }_L(G_{2^r{p}^s})=\left\{\begin{array}{cccc}0& \phi (p)-1& \phi (p)+1& 2\phi (p)\\ \\ 1& \phi (p)& \phi (p)& 1\end{array}\right\}.$$

Hence, the Laplacian spectrum of $G({\mathbb{Z}}_n)$ is

$$\begin{array}{c}\hfill {\sigma }_L(G({\mathbb{Z}}_n))=\left\{\begin{array}{ccccc}0& 2^{r-1}{p}^{s-1}[\phi (p)-1]& 2^{r-1}{p}^s& 2^r{p}^{s-1}\phi (p)& 2^{r-1}{p}^{s-1}\phi (p)\\ \\ 1& \phi (p)& \phi (p)& 1& 2^r{p}^s-2p\end{array}\right\}.\end{array}$$

□

Now, we find the Laplacian spectrum of $G({\mathbb{Z}}_n)$ for $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, where $p_1<p_2<...<p_k$ are primes, $r_i,k$ are positive integers, and $1\le i\le k$. The following theorem can be obtained by arguments similar to those used in the proof of Theorem 4, and therefore the proof is omitted.

Theorem 5.

Let $2\ne p_1<p_2<...<p_k$ be primes and $r,r_i,s_i,k$ be positive integers, $1\le i\le k$. Then:

1. 

If $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, then the Laplacian spectrum of $G({\mathbb{Z}}_n)$ consists of

$$\begin{array}{cc}\hfill & \left\{\begin{array}{ccccc}0& {\displaystyle [\phi (p_i)\pm 1]\prod _{1\le i\le k}{p}_i^{(r_i-1)}\prod _{i\ne j}\phi (p_j)}& {\displaystyle [\phi (p_i)\phi (p_j)\pm 1]\prod _{1\le i\le k}{p}_i^{(r_i-1)}\prod _{h\ne i,j}\phi (p_h)}& ...& \\ \\ 1& {\displaystyle \frac{\phi (p_i)}{2}}& {\displaystyle \frac{\phi (p_i)\phi (p_j)}{2}}& ...\end{array}\right.\hfill \\ \hfill & \begin{array}{ccc}{\displaystyle [\phi (p_1)\phi (p_2)...\phi (p_{k-1})\pm 1]\phi (p_k)\prod _{1\le i\le k}{p}_i^{(r_i-1)}}& {\displaystyle \left[\left[\prod _{1\le i\le k}\phi (p_i)\right]\pm 1\right]\prod _{1\le i\le k}{p}_i^{(r_i-1)}}& \\ \\ {\displaystyle \frac{{\displaystyle \prod _{1\le i\le k-1}\phi (p_i)}}{2}}& {\displaystyle \frac{{\displaystyle \prod _{1\le i\le k}\phi (p_i)}}{2}}\end{array}\hfill \\ \\ \hfill & \left.\begin{array}{c}{\displaystyle \prod _{1\le i\le k}{p}_i^{(r_i-1)}\phi (p_i)}\\ \\ n-p_1{p}_2...p_k\end{array}\right\}.\hfill \end{array}$$

2. 

If $n=2^r{p}_1^{s_1}{p}_2^{s_2}...p_k^{s_k}$, then the Laplacian spectrum of $G({\mathbb{Z}}_n)$ consists of

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cccc}0& {\displaystyle 2^{r-1}[\phi (p_i)\pm 1]\prod _{1\le i\le k}{p}_i^{(s_i-1)}\prod _{i\ne j}\phi (p_j)}& {\displaystyle 2^{r-1}[\phi (p_i)\phi (p_j)\pm 1]\prod _{1\le i\le k}{p}_i^{(s_i-1)}\prod _{h\ne i,j}\phi (p_h)}& ...\\ \\ 1& \phi (p_i)& \phi (p_i)\phi (p_j)& ...\end{array}\right.\hfill \\ \\ \hfill & \begin{array}{ccc}{\displaystyle 2^{r-1}[\phi (p_1)\phi (p_2)...\phi (p_{k-1})\pm 1]\phi (p_k)\prod _{1\le i\le k}{p}_i^{(s_i-1)}}& {\displaystyle 2^{r-1}\left[\left[\prod _{1\le i\le k}\phi (p_i)\right]\pm 1\right]\prod _{1\le i\le k}{p}_i^{(s_i-1)}}& \\ \\ {\displaystyle \prod _{1\le i\le k-1}\phi (p_i)}& {\displaystyle \prod _{1\le i\le k}\phi (p_i)}\end{array}\hfill \\ \\ \hfill & \left.\begin{array}{cc}{\displaystyle 2^r\prod _{1\le i\le k}{p}_i^{(s_i-1)}\phi (p_i)}& {\displaystyle 2^{(r-1)}\prod _{1\le i\le k}{p}_i^{(s_i-1)}\phi (p_i)}\\ \\ 1& n-2p_1{p}_2...p_k\end{array}\right\}.\hfill \end{array}$$

As an immediate consequence of Theorem 5, we have the following results.

Corollary 3.

$G({\mathbb{Z}}_n$) is Laplacian integral for all n.

Corollary 4.

The Laplacian spectral radius of $G({\mathbb{Z}}_n)$ is

$$\lambda (G({\mathbb{Z}}_n))=\left\{\begin{array}{cc}{\displaystyle [\phi (p_1)+1]\prod _{1\le i\le k}{p}_i^{(r_i-1)}\prod _{j\ne 1}\phi (p_j)}\hfill & \mathrm{if}n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k};\hfill \\ \\ {\displaystyle 2^r\prod _{1\le i\le k}{p}_i^{(s_i-1)}\phi (p_i)}\hfill & \mathrm{if}n=2^r{p}_1^{s_1}{p}_2^{s_2}...p_k^{s_k},\hfill \end{array}\right.$$

where $2\ne p_1<p_2<...<p_k$ are primes, $r,r_i,s_i,k$ are positive integers, and $1\le i\le k$.

Corollary 5.

The algebraic connectivity of $G({\mathbb{Z}}_n)$ is

$$\mu (G({\mathbb{Z}}_n))=\left\{\begin{array}{cc}{\displaystyle [\phi (p_1)-1]\prod _{1\le i\le k}{p}_i^{(r_i-1)}\prod _{j\ne 1}\phi (p_j)}\hfill & \mathrm{if}n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k};\hfill \\ \\ {\displaystyle 2^{r-1}[\phi (p_1)-1]\prod _{1\le i\le k}{p}_i^{(s_i-1)}\prod _{j\ne 1}\phi (p_j)}\hfill & \mathrm{if}n=2^r{p}_1^{s_1}{p}_2^{s_2}...p_k^{s_k},\hfill \end{array}\right.$$

where $2\ne p_1<p_2<...<p_k$ are primes, $r,r_i,s_i,k$ are positive integers, and $1\le i\le k$.

5. Vertex Connectivity of $G({\mathbb{Z}}_n)$

In this section, we obtain the vertex connectivity of $G({\mathbb{Z}}_n)$ when $n=pq$ and $n=p^r{q}^s$, where $2\ne p<q$ are primes and r and s are positive integers. To achieve this goal, we calculate the number of internally disjoint paths between any two nonadjacent vertices in $G({\mathbb{Z}}_{pq})$, which allow us to be ready to explore the vertex connectivity of $G({\mathbb{Z}}_{pq})$ by using Menger’s theorem. We end this section by presenting conjectures about the vertex connectivity of $G({\mathbb{Z}}_n)$ when $n=p_1{p}_2...p_k$ and $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, where $p_i$ are distinct primes, $r_i$ are positive integers, and $1\le i\le k$.

5.1. Structure of $G({\mathbb{Z}}_{pq})$

The following result will be used in the sequel.

Lemma 6.

[18] Let $n=pq$, where p and q are distinct odd primes. Then the following statements hold:

1. 

Let $i\in \{1,2,...,p-1\}$. The induced subgraph $G(i+〈p〉)$ of $G({\mathbb{Z}}_n)$ is isomorphic to $K_1\vee CP(q-1)$1. If $i=0$, then $G(i+〈p〉)$ is ${\overline{K}}_q$.

2. 

Let $i,j\in \{0,1,...,p-1\}$ and $i\ne j$. If $(i+j,p)=1$, then every vertex of $G(i+〈p〉)$ is adjacent to $(q-1)$ vertices of $G(j+〈p〉)$.

3. 

Let $i,j\in \{1,2,...,p-1\}$ and $i\ne j$. If $(i+j,p)\ne 1$, then every vertex of $G(i+〈p〉)$ is nonadjacent to any vertex of $G(j+〈p〉)$.

Now, we study the structure of $G({\mathbb{Z}}_{pq})$, where $2\ne p<q$ are primes, analogously to Section 3. In this case, we choose the maximal ideal $〈p〉$. Let $S^{\prime }$ be the set of distinct representatives of ${\mathbb{Z}}_{pq}/〈p〉$. For $i\in S^{\prime }$, we denote

$$B_i^{\prime }=i+〈p〉.$$

Note that the sets $B_0^{\prime },B_1^{\prime },...,B_{p-1}^{\prime }$ form a partition of the vertex set of $G({\mathbb{Z}}_{pq})$. Thus,

$${\displaystyle V(G({\mathbb{Z}}_n))=\bigcup _{i=0}^{p-1}{B}_i^{\prime }.}$$

This implies that any two vertices x and y that belong to the above union are adjacent if and only if $(x+y,pq)=1$. Let $B_0^{\prime }=〈p〉$. Then, $|B_i^{\prime }|=|B_0^{\prime }|=|〈p〉|=q$ for $i\in S^{\prime }$.

Note that, Lemma 6 implies that the partition ${\cup }_{i=0}^{p-1}G(B_i^{\prime })$ of $V(G({\mathbb{Z}}_{pq}))$ is an almost equitable partition in such a way that every vertex of $G(B_i^{\prime })$ has an equal number of neighbors in $G(B_j^{\prime })$ where $i\ne j$ and $i,j\in S^{\prime }$. Also, $G(B_0^{\prime })$ is isomorphic to ${\overline{K}}_q$ and $G(B_i^{\prime })$ is isomorphic to $K_1\vee CP(q-1)$, where $i\in S^{\prime }-\left\{0\right\}$.

Let $G_{pq}^{\prime }$ be defined as the simple graph whose vertices are $G(B_i^{\prime })$, where $i\in S^{\prime }$, so that $|V(G_{pq}^{\prime })|=p$. Two distinct vertices $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$ are adjacent if and only if $(i+j,p)=1$, which is equivalent to each vertex in $G(B_i^{\prime })$ being adjacent to $(q-1)$ vertices in $G(B_j^{\prime })$.

Example 2.

The vertex set of $G({\mathbb{Z}}_{15})$, denoted as $V(G_{15}^{\prime })$, can be expressed as the union $V(G_{15}^{\prime })=G(B_0^{\prime })\cup G(B_1^{\prime })\cup G(B_2^{\prime })$, where $G(B_0^{\prime })=G(0+〈3〉)$, $G(B_1^{\prime })=G(1+〈3〉)$, and $G(B_2^{\prime })=G(2+〈3〉)$. From Figure 3 bellow, we observe the following:

1. 

$G(B_1^{\prime })$ is nonadjacent to $G(B_2^{\prime })$ since $(1+2,3)\ne 1$.

2. 

Each vertex in $G(B_1^{\prime })$ and $G(B_2^{\prime })$ is adjacent to 4 vertices in $G(B_0^{\prime })$.

3. 

$G(B_0^{\prime })$ is isomorphic to ${\overline{K}}_5$ and $G(B_1^{\prime })$ and $G(B_2^{\prime })$ are isomorphic to $K_1\vee CP(4)$. The red vertices 0, 5, and 10 represent $K_1$ in $G(B_0^{\prime })$, $G(B_2^{\prime })$, and $G(B_1^{\prime })$, respectively. Note that these vertices are multiples of 5.

The following two results determine the neighbors and the number of common neighbors of the vertices in $G_{pq}^{\prime }$, which help us to calculate the number of internally disjoint paths between any two nonadjacent vertices in ${\cup }_{i=0}^{p-1}G(B_i^{\prime })$.

Lemma 7.

Let $i\in S^{\prime }$. If $i\ne 0$, then there are $(p-2)$ neighbors of $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. On the other hand, $G(B_0^{\prime })$ has $(p-1)$ neighbors in $G_{pq}^{\prime }$.

Proof. 

Let $i\ne 0$. For $j\ne i$, $(i+j,p)=1$ for all $j\in S^{\prime }$ except $j=p-i$. Consequently, $G(B_i^{\prime })$ is adjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$ when $j\ne i,p-i$. Then, there are $(p-2)$ neighbors of $G(B_i)$ in $G_{pq}^{\prime }$. If $i=0$, then $(0+j,p)=1$ for all $j\ne i$. Thus, $G(B_0^{\prime })$ is adjacent to all vertices in $G_{pq}^{\prime }$. Therefore, there are $(p-1)$ neighbors of $G(B_0^{\prime })$ in $G_{pq}^{\prime }$. □

Lemma 8.

If $i,j\in S^{\prime }$, then the following statements hold:

1. 

If $G(B_i^{\prime })$ and $G(B_j^{\prime })$ are nonadjacent in $G_{pq}^{\prime }$, then the number of common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$ is $(p-2)$.

1. 

If $G(B_i^{\prime })$ and $G(B_j^{\prime })$ are adjacent in $G_{pq}^{\prime }$, where $i,j\ne 0$, then the number of common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$ is $(p-4)$.

3. 

The number of common neighbors between $G(B_0^{\prime })$ and $G(B_i^{\prime })$ in $G_{pq}^{\prime }$ is $(p-3)$.

Proof. 

Let $i,j\in S^{\prime }$.

• Let $G(B_i^{\prime })$ and $G(B_j^{\prime })$ be nonadjacent in $G_{pq}^{\prime }$. By Lemma 7, $j=p-i$ and $i=p-j$. Hence, $G(B_i^{\prime })$ is adjacent to all vertices in $G_{pq}^{\prime }$ except $G(B_{p-i}^{\prime })$. Similarly, $G(B_j^{\prime })$ is adjacent to all vertices in $G_{pq}^{\prime }$ except $G(B_{p-j}^{\prime })$. So, $G(B_i^{\prime })$ and $G(B_j^{\prime })$ are adjacent to all vertices in $G_{pq}^{\prime }$ except $G(B_j^{\prime })$ and $G(B_i^{\prime })$. Therefore, there are $(p-2)$ common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$.
• Let $G(B_i^{\prime })$ and $G(B_j^{\prime })$ be adjacent in $G_{pq}^{\prime }$, where $i,j\ne 0$. According to Lemma 7, $G(B_i^{\prime })$ and $G(B_j^{\prime })$ are nonadjacent to $G(B_{p-i}^{\prime })$ and $G(B_{p-j}^{\prime })$ in $G_{pq}^{\prime }$, respectively. So, $G(B_i^{\prime })$ and $G(B_j^{\prime })$ are adjacent to all vertices in $G_{pq}^{\prime }$ except $G(B_{p-i}^{\prime })$ and $G(B_{p-j}^{\prime })$, respectively. Then, the set of common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$ is

  $$V(G_{pq}^{\prime })-\{G(B_i^{\prime }),G(B_j^{\prime }),G(B_{p-i}^{\prime }),G(B_{p-j}^{\prime })\}.$$
  Thus, there are $(p-4)$ common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$.
• By Lemma 7, $G(B_0^{\prime })$ is adjacent to all vertices in $G_{pq}^{\prime }$. Also, $G(B_i^{\prime })$ is adjacent to all vertices in $G_{pq}^{\prime }$ except $G(B_{p-i}^{\prime })$. So, the set of common neighbors between $G(B_0^{\prime })$ and $G(B_i^{\prime })$ in $G_{pq}^{\prime }$ is

  $$V(G_{pq}^{\prime })-\{G(B_0^{\prime }),G(B_i^{\prime }),G(B_{p-i}^{\prime })\}.$$
  Then, there are $(p-3)$ common neighbors between $G(B_0^{\prime })$ and $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. □

The following result determines the number of common neighbors between nonadjacent vertices $x,y\in G(B_i^{\prime })$ through $G(B_z^{\prime })$, where $G(B_z^{\prime })$ is a neighbor of $G(B_i^{\prime })$ in $G_{pq}^{\prime }$.

Lemma 9.

Let $x,y\in G(B_i^{\prime })$ be nonadjacent and $G(B_z^{\prime })$ be a neighbor of $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. Then, x and y have $(q-2)$ common neighbors in $G(B_z^{\prime })$.

Proof. 

By Part 2 of Lemma 6, both x and y are adjacent to $(q-1)$ vertices in $G(B_z^{\prime })$. Suppose that x and y have the same neighbors in $G(B_z^{\prime })$. Then, x and y are adjacent to all vertices in $G(B_z^{\prime })$ except $z^{\prime }$. This implies that $z^{\prime }$ is adjacent to $(q-2)$ vertices in $G(B_i^{\prime })$, a contradiction with Part 2 of Lemma 6. Then, x and y are adjacent to all vertices in $G(B_z^{\prime })$ except $a_z$ and $b_z$, respectively. So, the number of common neighbors between x and y in $G(B_z^{\prime })$ is $(q-2)$. □

The following result determines the number of common neighbors between nonadjacent vertices $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$ through $G(B_z^{\prime })$, where $G(B_z^{\prime })$ is a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$.

Lemma 10.

Let $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$ be nonadjacent and $G(B_z^{\prime })$ be a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Then:

1. 

If x and y have the same neighbors in $G(B_z^{\prime })$, then the number of common neighbors in $G(B_z^{\prime })$ between x and y is $(q-1)$.

2. 

If x and y do not have the same neighbors in $G(B_z^{\prime })$, then the number of common neighbors in $G(B_z^{\prime })$ between x and y is $(q-2)$.

Proof. 

Let $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$ be nonadjacent and $G(B_z^{\prime })$ be a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$.

1. 

The proof is direct from Part 2 of Lemma 6.

2. 

Let x and y do not have the same neighbors in $G(B_z^{\prime })$. By Part 2 of Lemma 6, both x and y have $(q-1)$ neighbors in $G(B_z^{\prime })$. That is, x and y are adjacent to all vertices in $G(B_z^{\prime })$ except $a_z$ and $b_z$, respectively. So, the number of the common neighbors in $G(B_z^{\prime })$ between x and y is $(q-2)$. □

From now to the rest of this section, we denote x, which is a multiple of q in $G(B_i^{\prime })$, by $x^{*}$ (see Example 2). The following proposition characterizes the nonadjacent vertices of $G({\mathbb{Z}}_{pq})$ for which the relation in Part 1 of Lemma 10 holds when $G(B_i^{\prime })$ is adjacent to $G(B^{\prime }j)$ in $G^{\prime }pq$.

Proposition 2.

Let $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$ be nonadjacent, $G(B_i^{\prime })$ be adjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$, and $G(B_z^{\prime })$ be a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Then, x and y have the same neighbors in $G(B_z^{\prime })$ if and only if $x=x^{*}$ and $y=y^{*}$.

Proof.$(\Rightarrow )$ Suppose that x and y have the same neighbors in $G(B_z^{\prime })$. Then, x and y are adjacent to all vertices in $G(B_z^{\prime })$ except $z^{\prime }$ by Part 2 of Lemma 6. We assume that $x\ne x^{*}$ and $y\ne y^{*}$. Suppose that $x=i+a^{\prime }p$, $y=j+b^{\prime }p$, and $z^{\prime }=z+c^{\prime }p$, where $i,j,z\in S^{\prime }$ and $a^{\prime },b^{\prime },c^{\prime }\in \{0,1,...,q-1\}$. Since $G(B_i^{\prime })$ is adjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$ and x is nonadjacent to y, then $(i+j,p)=1$ and $((i+j)+(a^{\prime }+b^{\prime })p,pq)\ne 1$, this implies that $((i+j)+(a^{\prime }+b^{\prime })p,q)\ne 1$. Similarly, since $G(B_z^{\prime })$ is adjacent to $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$, then $((i+z^{\prime })+(a^{\prime }+c^{\prime })p,q)\ne 1$ and $((j+z^{\prime })+(b^{\prime }+c^{\prime })p,q)\ne 1$. So,

$$\begin{array}{c}\hfill x+y=(i+j)+(a^{\prime }+b^{\prime })p=a_{1}q,\\ \hfill x+z^{\prime }=(i+z)+(a^{\prime }+c^{\prime })p=a_{2}q,\\ \hfill y+z^{\prime }=(j+z)+(b^{\prime }+c^{\prime })p=a_{3}q.\end{array}$$

So, $2z^{\prime }=(a_2+a_3-a_1)q$; this implies that 2 divide $(a_2+a_3-a_1)$, and hence $z^{\prime }$ is multiple of q, in this case $z^{\prime }=z^{*}$. But $z^{*}$ is adjacent to all vertices in $G(B_i^{\prime })$ and $G(B_j^{\prime })$ except $x=x^{*}$ and $y=y^{*}$, respectively. This is a contradiction with $x\ne x^{*}$ and $y\ne y^{*}$. So, $x=x^{*}$ and $y=y^{*}$.

$(\Leftarrow )$ Assume that $x=x^{*}$ and $y=y^{*}$. Then, x is adjacent to $(q-1)$ vertices of $G(B_z^{\prime })$ by Part 2 of Lemma 6, that is x is adjacent to all vertices of $G(B_z^{\prime })$ except $z^{*}$, where $z^{*}=cq$ and $c\in \{0,1,...,q-1\}$. Similarly, y is adjacent to all vertices of $G(B_z^{\prime })$ except $z^{*}$. Then, the result is obtained. □

5.2. Number of Internally Disjoint Paths between Nonadjacent Vertices in $G({\mathbb{Z}}_{pq})$

The following two lemmas calculate the number of internally disjoint paths between nonadjacent vertices x and y in $G(B_i^{\prime })$.

Lemma 11.

Let $x,y\in G(B_i^{\prime })$ be nonadjacent. Then, there are $(p-1)(q-2)$ internally disjoint paths of length 2 between x and y.

Proof. 

If $i\ne 0$, then $G(B_i^{\prime })$ is isomorphic to $K_1\vee CP(q-1)$ by Part 1 of Lemma 6. Since $x^{*}$ represents $K_1$, then $x^{*}$ adjacent to all vertices in $G(B_i^{\prime })$. Since x is nonadjacent to y, then both x and y are not $x^{*}$ in $G(B_i^{\prime })$. So, there are $(q-2)$ common neighbors between x and y, and hence there are $(q-2)$ internally disjoint paths of length 2 between x and y in $G(B_i^{\prime })$. Let $G(B_z^{\prime })$ be a neighbor of $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. By Lemma 9, there are $(q-2)$ common neighbors between x and y in $G(B_z^{\prime })$. Hence, there are $(q-2)$ internally disjoint paths of length 2 through $G(B_z^{\prime })$. By Lemma 7, there are $(p-2)$ neighbors of $G(B_i^{\prime })$ in $G_{pq}^{\prime }$, and hence there are $(p-2)(q-2)$ internally disjoint paths of length 2 between x and y through all neighbors of $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. Thus, the total number of internally disjoint paths of length 2 between x and y is

$$(q-2)+(p-2)(q-2)=(p-1)(q-2).$$

If $i=0$, then $G(B_0^{\prime })$ is isomorphic to ${\overline{K}}_q$, by Part 1 of Lemma 6. So, there is no path between x and y in $G(B_0^{\prime })$. By Lemma 9, there are $(q-2)$ common neighbors between x and y in $G(B_z^{\prime })$, where $G(B_z^{\prime })$ is a neighbor of $G(B_0^{\prime })$ in $G_{pq}^{\prime }$. Hence, there are $(q-2)$ internally disjoint paths of length 2 through $G(B_z^{\prime })$. By Lemma 7, there are $(p-1)$ neighbors of $G(B_0^{\prime })$ in $G_{pq}^{\prime }$, and hence there are $(p-1)(q-2)$ internally disjoint paths of length 2 between x and y. □

Lemma 12.

Let $x,y\in G(B_i^{\prime })$ be nonadjacent. Then:

1. 

If $i\ne 0$, then there are $(p-2)$ internally disjoint paths of length 4 between x and y.

2. 

If $i=0$, then there are $(p-1)$ internally disjoint paths of length 4 between x and y.

Proof. 

Let $x,y\in G(B_i^{\prime })$ be nonadjacent and $G(B_z^{\prime })$ be a neighbor of $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. By proof of Lemma 9, x and y are adjacent to all vertices of $G(B_z^{\prime })$ except $a_z$ and $b_z$, respectively.

• Let $i\ne 0$. By Part 2 of Lemma 6, $a_z$ and $b_z$ in $G(B_z^{\prime })$ are adjacent to $(q-1)$ vertices of $G(B_{p-i}^{\prime })$. Since we investigate the internally disjoint paths between x and y through $a_z$ and $b_z$, we can choose a vertex w from $G(B_{p-i}^{\prime })$ that is adjacent to both $a_z$ and $b_z$. This path will be of length 4, as illustrated in Figure 4 (a). Similarly, for each neighbor $G(B_z^{\prime })$ of $G(B_i^{\prime })$ in $G_{pq}^{\prime }$ there is one internally disjoint path of length 4 between x and y. By Lemma 7, there are $(p-2)$ neighbors of $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. Therefore, there are $(p-2)$ internally disjoint paths of length 4 between x and y through all neighbors of $G(B_i^{\prime })$.
• Let $i=0$. By Part 2 of Lemma 6, $a_z$ and $b_z$ in $G(B_z^{\prime })$ are adjacent to $(q-1)$ vertices of $G(B_0^{\prime })$. Approaching the proof in a similar manner as with Part 1, there is a path of length 4, as shown in Figure 4 (b). By Lemma 7, the number of neighbors of $G(B_0^{\prime })$ in $G_{pq}^{\prime }$ is $(p-1)$. So, there are $(p-1)$ internally disjoint paths of length 4 between x and y through all neighbors of $G(B_0^{\prime })$. □

The following two lemmas determine the number of internally disjoint paths between nonadjacent vertices $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$, where $G(B_i^{\prime })$ is nonadjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$.

Lemma 13.

Let $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$ be nonadjacent, $G(B_i^{\prime })$ be nonadjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$, and $G(B_z^{\prime })$ be a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Then:

1. 

If x and y have the same neighbors in $G(B_z^{\prime })$, then there are $(p-2)(q-1)$ internally disjoint paths of length 2 between x and y.

2. 

If x and y do not have the same neighbors in $G(B_z^{\prime })$, then there are $(p-2)(q-2)$ internally disjoint paths of length 2 between x and y.

Proof. 

Let $G(B_i^{\prime })$ be nonadjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. By Part 1 of Lemma 8, there are $(p-2)$ common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Then:

• If x and y have the same neighbors in $G(B_z^{\prime })$, then there are $(q-1)$ common neighbors between x and y in $G(B_z^{\prime })$ by Lemma 10. Thus, there are $(q-1)$ internally disjoint paths of length 2 between x and y through $G(B_z^{\prime })$. Hence, there are $(p-2)(q-1)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$.
• If x and y do not have the same neighbors in $G(B_z^{\prime })$, then there are $(q-2)$ common neighbors between x and y in $G(B_z^{\prime })$ by Lemma 10. So, there are $(q-2)$ internally disjoint paths of length 2 between x and y through $G(B_z^{\prime })$. Therefore, there are $(p-2)(q-2)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. □

Lemma 14.

Let $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$ be nonadjacent, $G(B_i^{\prime })$ be nonadjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$, and $G(B_z^{\prime })$ be a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Then:

1. 

If x and y have the same neighbors in $G(B_z^{\prime })$, then there are $(p-2)$ internally disjoint paths of length 4 between x and y.

2. 

If x and y do not have the same neighbors in $G(B_z^{\prime })$, then there are $2(p-2)$ internally disjoint paths of length 3 between x and y.

Proof. 

Let $G(B_i^{\prime })$ be nonadjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. By Part 1 of Lemma 8, there are $(p-2)$ common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$.

• Let x and y have the same neighbors in $G(B_z^{\prime })$. By Lemma 6, $x\in G(B_i^{\prime })$ has at least $(q-2)$ neighbors in $G(B_i^{\prime })$ and each of them is adjacent to $(q-1)$ vertices of $G(B_z^{\prime })$. Then, we can choose a neighbor, say $x^{\prime }$, of x in $G(B_i^{\prime })$ such that $x^{\prime }\sim z^{\prime }$, where $z^{\prime }\in G(B_z^{\prime })$ is nonadjacent to both x and y. Similarly, we can choose a neighbor $y^{\prime }$ of y in $G(B_j^{\prime })$ such that $y^{\prime }\sim z^{\prime }$. Therefore, there exists a path of length 4 of the form $x-x^{\prime }-z^{\prime }-y^{\prime }-y$, see Figure 5 (a). So, there are $(p-2)$ internally disjoint paths of length 4 between x and y through all common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$.
• Let x and y do not have the same neighbors in $G(B_z^{\prime })$. By proof of Lemma 10, x and y are adjacent to all vertices in $G(B_z^{\prime })$ except $a_z$ and $b_z$, respectively. Approaching the proof similarly as Part 1, we can choose a neighbor $x^{\prime }$ of x in $G(B_i^{\prime })$ and a neighbor $y^{\prime }$ of y in $G(B_j^{\prime })$, where $x^{\prime }\sim a_z$ and $y^{\prime }\sim b_z$. Since $a_z\sim y$ and $b_z\sim x$, two internally disjoint paths of length 3 exist. These paths are described in Figure 5 (b). Hence, there are $2(p-2)$ internally disjoint paths of length 3 between x and y through all common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. □

The following results find the number of internally disjoint paths between nonadjacent vertices $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$, where $G(B_i^{\prime })$ is adjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$.

Lemma 15.

Let $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$ be nonadjacent and $G(B_i^{\prime })$ be adjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. One of the following cases holds:

1. 

There are $(p-2)(q-1)$ internally disjoint paths of length 2 between x and y.

2. 

There are $(p-2)(q-2)$ internally disjoint paths of length 2 between x and y.

Proof. 

By Lemma 8, there are $(p-4)$ common neighbors between the adjacent vertices $G(B_i^{\prime })$ and $G(B_j^{\prime })$, where $i,j\ne 0$, in $G_{pq}^{\prime }$ and there are $(p-3)$ common neighbors between $G(B_0^{\prime })$ and $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. Let $G(B_z^{\prime })$ be a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Since x and y are nonadjacent, then x and y have the following probabilities:

• Assume that x and y have the same neighbors in $G(B_z^{\prime })$. By Proposition 2 and Lemma 10, $x=x^{*}$ and $y=y^{*}$ and there are $(q-1)$ common neighbors between x and y through $G(B_z^{\prime })$. So, there are $(q-1)$ internally disjoint paths of length 2 between x and y through $G(B_z^{\prime })$. If $i,j\ne 0$, then there are $(p-4)(q-1)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Further, x (resp. y) is adjacent to all vertices in $G(B_i^{\prime })$ (resp. $G(B_j^{\prime })$). Also, x (resp. y) is adjacent to all vertices in $G(B_j^{\prime })$ (resp. $G(B_i^{\prime })$) except y (resp. x). So, the set of common neighbors between x and y in $G(B_i^{\prime })$ and $G(B_j^{\prime })$ is $G(B_i^{\prime })-\left\{x\right\}$ union $G(B_j^{\prime })-\left\{y\right\}$. Thus, there are $2(q-1)$ common neighbors between x and y in $G(B_i^{\prime })$ and $G(B_j^{\prime })$. Consequently, there are $2(q-1)$ internally disjoint paths of length 2 between x and y in $G(B_i^{\prime })$ and $G(B_j^{\prime })$. Therefore, the total number of internally disjoint paths of length 2 between x and y is

  $$(p-4)(q-1)+2(q-1)=(p-2)(q-1).$$
  If $j=0$, then there are $(p-3)(q-1)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G(B_0^{\prime })$ and $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. Further, x is adjacent to all vertices in $G(B_i^{\prime })$ and y is nonadjacent to any vertex in $G(B_0^{\prime })$. Also, x (resp. y) is adjacent to all vertices in $G(B_0^{\prime })$ (resp. $G(B_i^{\prime })$) except y (resp. x). So, the set of common neighbors between x and y in $G(B_0^{\prime })$ and $G(B_i^{\prime })$ are $G(B_i^{\prime })-\left\{x\right\}$. Thus, there are $(q-1)$ common neighbors between x and y in $G(B_0^{\prime })$ and $G(B_i^{\prime })$. Then, there are $(q-1)$ internally disjoint paths of length 2 between x and y in $G(B_0^{\prime })$ and $G(B_i^{\prime })$. Thus, the total number of internally disjoint paths of length 2 between x and y is

  $$(p-3)(q-1)+(q-1)=(p-2)(q-1).$$
• Assume that x and y do not have the same neighbors in $G(B_z^{\prime })$. By Proposition 2 and Lemma 10, $x\ne x^{*}$ and $y\ne y^{*}$ and there are $(q-2)$ common neighbors between x and y in $G(B_z^{\prime })$. Thus, there are $(q-2)$ internally disjoint paths of length 2 between x and y through $G(B_z^{\prime })$. If $i,j\ne 0$, then there are $(p-4)(q-2)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Furthermore, x (resp. y) is adjacent to all vertices except only one vertex $x^{\prime }$ (resp. $y^{\prime }$) in $G(B_i^{\prime })$ (resp. $G(B_j^{\prime })$). Also, x (resp. y) is adjacent to all vertices in $G(B_j^{\prime })$ (resp. $G(B_i^{\prime })$) except y (resp. x). Thus, the set of common neighbors between x and y in $G(B_i^{\prime })$ and $G(B_j^{\prime })$ is $G(B_i^{\prime })-\{x,x^{\prime }\}$ union $G(B_j^{\prime })-\{y,y^{\prime }\}$. So, there are $2(q-2)$ common neighbors between x and y in $G(B_i^{\prime })$ and $G(B_j^{\prime })$. As a result, there are $2(q-2)$ internally disjoint paths of length 2 between x and y in $G(B_i^{\prime })$ and $G(B_j^{\prime })$. Therefore, the total number of internally disjoint paths of length 2 between x and y is

  $$(p-4)(q-2)+2(q-2)=(p-2)(q-2).$$
  If $j=0$, then there are $(p-3)(q-2)$ internally disjoint paths of length 2 between x and y through all common neighbors between $G(B_0^{\prime })$ and $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. Further, x is adjacent to all vertices except only one vertex $x^{\prime }$ in $G(B_i^{\prime })$, and y is nonadjacent to any vertex in $G(B_0^{\prime })$. Also, x (resp. y) is adjacent to all vertices in $G(B_0^{\prime })$ (resp. $G(B_i^{\prime })$) except y (resp. x). Thus, the set of common neighbors between x and y in $G(B_0^{\prime })$ and $G(B_i^{\prime })$ is $G(B_i^{\prime })-\{x,x^{\prime }\}$. So, there are $(q-2)$ common neighbors between x and y in $G(B_0^{\prime })$ and $G(B_i^{\prime })$. Then, there are $(q-2)$ internally disjoint paths of length 2 between x and y in $G(B_0^{\prime })$ and $G(B_i^{\prime })$. Thus, the total number of internally disjoint paths of length 2 between x and y is

  $$(p-3)(q-2)+(q-2)=(p-2)(q-2).$$

□

Lemma 16.

Let $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$ be nonadjacent, where $G(B_i^{\prime })$ is adjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. One of the following cases holds:

1. 

There are $(q-1)$ internally disjoint paths of length 3 between x and y.

2. 

There are $(p+q-5)$ internally disjoint paths of length 3 between x and y.

3. 

There are $(p+q-4)$ internally disjoint paths of length 3 between x and y.

Proof. 

We need to examine whether any $xy$-path passes through $G(B_{p-j}^{\prime })$ and $G(B_{p-i}^{\prime })$ because we are sure that $G(B_{p-j}^{\prime })$ (resp. $G(B_{p-i}^{\prime })$) is adjacent to $G(B_i^{\prime })$ (resp. $G(B_j^{\prime })$) and nonadjacent to $G(B_j^{\prime })$ (resp. $G(B_i^{\prime })$) in $G_{pq}^{\prime }$. Let $G(B_z^{\prime })$ be a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Since x and y are nonadjacent, then x and y have the following cases:

• Assume that x and y have the same neighbors in $G(B_z^{\prime })$. Indeed, $x=x^{*}$ and $y=y^{*}$ by Proposition 2. If $i,j\ne 0$, there are $(q-1)$ neighbors of x in $G(B_{p-j}^{\prime })$, denote these neighbors by $x_k^{\prime }$ such that $k=1,2,...,q-1$, and each of them is adjacent to $(q-1)$ vertices of $G(B_{p-i}^{\prime })$ by Part 2 of Lemma 6. Similarly, there are $(q-1)$ neighbors of y in $G(B_{p-i}^{\prime })$, and each of them is adjacent to $(q-1)$ vertices of $G(B_{p-j}^{\prime })$. To get the internally disjoint paths of length 3 between x and y, we choose one of the neighbors of $x_k^{\prime }$, say $y_k^{\prime }$, in $G(B_{p-i}^{\prime })$ such that $y_k^{\prime }$ is a neighbor of y. Indeed, for each $x_k^{\prime }$ in $G(B_{p-j}^{\prime })$ there is one internally disjoint path between x and y through $G(B_{p-j}^{\prime })$ and $G(B_{p-i}^{\prime })$. Therefore, the total number of internally disjoint paths of length 3 between x and y through $G(B_{p-j}^{\prime })$ and $G(B_{p-i}^{\prime })$ together is equal to the number of neighbors of x in $G(B_{p-j}^{\prime })$, which is $(q-1)$. Now let $j=0$. There are $(q-1)$ neighbors of x in $G(B_0^{\prime })$ and each of them is adjacent to $(q-1)$ vertices of $G(B_{p-i}^{\prime })$. Since there are $(q-1)$ neighbors of y in $G(B_{p-i}^{\prime })$, so there are more than $(q-1)$ paths of length 3 between x and y through $G(B_0^{\prime })$ and $G(B_{p-i}^{\prime })$ together. By applying the same method in the case where $i,j\ne 0$, there are $(q-1)$ internally disjoint paths of length 3 between x and y.
• Assume that x and y do not have the same neighbors in $G(B_z^{\prime })$. So, $x\ne x^{*}$ and $y\ne y^{*}$ by Proposition 2. Suppose that $G(B_{z_k}^{\prime })$ is a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. By proof of Lemma 10, x and y are adjacent to all vertices in $G(B_{z_k}^{\prime })$ except $a_{z_k}$ and $b_{z_k}$, respectively. Let $i,j\ne 0$. Since x has $(q-1)$ neighbors in $G(B_{p-j}^{\prime })$ and each of these neighbors is adjacent to $(q-1)$ vertices of $G(B_{z_k}^{\prime })$, then we can choose a neighbor $x_k^{\prime }$ of x in $G(B_{p-j}^{\prime })$ such that $x_k^{\prime }\sim a_{z_K}$. Similarly, we can choose a neighbor $y_k^{\prime }$ of y in $G(B_{p-i}^{\prime })$ such that $y_k^{\prime }\sim b_{z_K}$. Since $a_{z_k}\sim y$ and $b_{z_k}\sim x$, there exist two internally disjoint paths of length 3 between x and y, as illustrated in Figure 6 (a). By Part 2 of Lemma 8, there are $(p-4)$ common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Then, there are $2(p-4)$ internally disjoint paths of length 3 between x and y through all common neighbors between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. After removing all $x_k^{\prime }$ and $y_k^{\prime }$ from $G(B_{p-j}^{\prime })$ and $G(B_{p-i}^{\prime })$, respectively, then the number of remaining neighbors of x and y in $G(B_{p-j}^{\prime })$ and $G(B_{p-i}^{\prime })$, respectively, is $(q-1)-(p-4)=q-p+3$. So, there are $(q-p+3)$ internally disjoint paths length 3 between x and y that pass through the remaining of neighbors of x and y in $G(B_{p-j}^{\prime })$ and $G(B_{p-i}^{\prime })$, respectively, together. So, the total number of internally disjoint paths of length 3 between x and y is

  $$2(p-4)+(q-p+3)=p+q-5.$$
  Let $j=0$. Since x has $(q-1)$ neighbors in $G(B_0^{\prime })$ and each of these neighbors is adjacent to $(q-1)$ vertices of $G(B_{z_k}^{\prime })$, then we can choose a neighbor $x_k^{\prime }$ of x in $G(B_0^{\prime })$ such that $x_k^{\prime }\sim a_{z_k}$. Similarly, we can choose a neighbor $y_k^{\prime }$ of y in $G(B_{p-i}^{\prime })$ such that $y_k^{\prime }\sim b_{z_k}$. Since $a_{z_k}\sim y$ and $b_{z_k}\sim x$, there exist two internally disjoint path of length 3 between x and y, as illustrated in Figure 6 (b). By Part 3 of Lemma 8, there are $(p-3)$ common neighbors between $G(B_0^{\prime })$ and $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. Consequently, there are $2(p-3)$ internally disjoint paths of length 3 between x and y through all common neighbors between $G(B_0^{\prime })$ and $G(B_i^{\prime })$ in $G_{pq}^{\prime }$. After removing all $x_k^{\prime }$ and $y_k^{\prime }$ from $G(B_0^{\prime })$ and $G(B_{p-i}^{\prime })$, respectively, then the number of remaining neighbors of x and y in $G(B_0^{\prime })$ and $G(B_{p-i}^{\prime })$, respectively, is $(q-1)-(p-3)=q-p+2$. So, there are $(q-p+2)$ internally disjoint paths length 3 between x and y that pass through the rest of neighbors of x and y in $G(B_0^{\prime })$ and $G(B_{p-i}^{\prime })$, respectively, together. So, the total number of internally disjoint paths of length 3 between x and y is

  $$2(p-3)+(q-p+2)=p+q-4.$$

□

5.3. Vertex Connectivity of $G({\mathbb{Z}}_{p^r{q}^s})$

The following result is of crucial importance to our study in this section.

Theorem 6.

Let $2\ne p<q$ be distinct primes. The vertex connectivity of $G({\mathbb{Z}}_{pq})$ is

$$\kappa (G({\mathbb{Z}}_{pq}))=(p-2)q.$$

Proof. 

Let x and y be nonadjacent in $G({\mathbb{Z}}_{pq})$. In this proof, we will calculate the maximum number of internally disjoint paths between any two nonadjacent vertices. There are several cases for x and y, as follows:

Case 1: Let $x,y\in G(B_i^{\prime })$. By Lemma 11, then there are $(p-1)(q-2)$ internally disjoint paths of length 2 between x and y. In addition, there are other internally disjoint paths depending on the following cases for i:

(a)

Let $i\ne 0$. By lemma 12, there are $(p-2)$ internally disjoint paths of length 4 between x and y. So, the maximum number of internally disjoint paths between x and y is

$$(p-1)(q-2)+(p-2)=pq-p-q.$$

(b)

Let $i=0$. By Lemma 12, there are $(p-1)$ internally disjoint paths of length 4 between x and y. So, the maximum number of internally disjoint paths between x and y is

$$(p-1)(q-2)+(p-1)=(p-1)(q-1).$$

Case 2: Let $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$, where $G(B_i^{\prime })$ is nonadjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Let $G(B_z^{\prime })$ be a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. The following cases arise for x and y:

(a)

If y has the same neighbors as x in $G(B_z^{\prime })$, then there are $(p-2)(q-1)$ internally disjoint paths of length 2 between x and y by Lemma 13. According to Lemma 14, there are $(p-2)$ internally disjoint paths of length 4 between x and y. Hence, the maximum number of internally disjoint paths between x and y is

$$(p-2)(q-1)+(p-2)=(p-2)q.$$

(b)

If x and y do not have the same neighbors in $G(B_z^{\prime })$, there are $(p-2)(q-2)$ internally disjoint paths of length 2 between x and y by Lemma 13. According to Lemma 14, there are $2(p-2)$ internally disjoint paths of length 3 between x and y. Hence, the maximum number of internally disjoint paths between x and y is

$$(p-2)(q-2)+2(p-2)=(p-2)q.$$

Case 3: Let $x\in G(B_i^{\prime })$ and $y\in G(B_j^{\prime })$, where $G(B_i^{\prime })$ is adjacent to $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. Let $G(B_z^{\prime })$ be a common neighbor between $G(B_i^{\prime })$ and $G(B_j^{\prime })$ in $G_{pq}^{\prime }$. There are the following cases for x and y:

(a)

If y has the same neighbors as x in $G(B_z^{\prime })$, then there are $(p-2)(q-1)$ internally disjoint paths of length 2 between x and y by proof of Lemma 15. According to proof of Lemma 16, there are $(q-1)$ internally disjoint paths of length 3 between x and y. Hence, the maximum number of internally disjoint paths between x and y is

$$(p-2)(q-1)+(q-1)=(p-1)(q-1).$$

(b)

If x and y do not have the same neighbors in $G(B_z^{\prime })$, then there are $(p-2)(q-2)$ internally disjoint paths of length 2 between x and y by proof of Lemma 15. In addition, there are other internally disjoint paths depending on the following cases for i and j:

(1)

Let $i,j\ne 0$. According to proof of Lemma 16, there are $(p+q-5)$ internally disjoint paths of length 3 between x and y. Therefore, the maximum number of internally disjoint paths between x and y is

$$(p-2)(q-2)+p+q-5=pq-p-q-1.$$

(2)

Let $j=0$. According to proof of Lemma 16, there are $(p+q-4)$ internally disjoint paths of length 3 between x and y. So, the maximum number of internally disjoint paths between x and y is

$$(p-2)(q-2)+(p+q-4)=pq-p-q.$$

From the above cases and by Menger’s theorem, we have

$$\begin{array}{cc}\hfill \kappa (G({\mathbb{Z}}_{pq}))& =min\{pq-p-q,(p-1)(q-1),(p-2)q,pq-p-q-1\}\hfill \\ \hfill & =(p-2)q.\hfill \end{array}$$

□

Now, let us explore the vertex connectivity of $G({\mathbb{Z}}_n)$ if $n=p^r{q}^s$, where $2\ne p<q$ are primes and $r,s$ are positive integers such that at least one of $r,s$ must be greater than 1.

Theorem 7.

Let $n=p^r{q}^s$, where $2\ne p<q$ are distinct primes, r and s are positive integers. Then, the vertex connectivity of $G({\mathbb{Z}}_n)$ is given by

$$\kappa (G({\mathbb{Z}}_n))=(p-2)p^{(r-1)}{q}^s.$$

Proof. 

By Lemma 4, the unit graph $G({\mathbb{Z}}_n)$ is

$$G({\mathbb{Z}}_n)=G_{p^r{q}^s}[G(B_0),G(B_1),\dots ,G(B_{pq-1})].$$

According to Lemma 3, $G_{p^r{q}^s}$ is isomorphic to $G({\mathbb{Z}}_{pq})$. Hence, by Theorem 6, we get

$$\kappa (G_{p^r{q}^s})=(p-2)q.$$

Note that, for every vertex i of $G_{p^r{q}^s}$, we have $|G(B_i)|$ vertices in $G({\mathbb{Z}}_n)$. Since $|G(B_i)|=p^{r-1}{q}^{s-1}$ for $0\le i\le pq-1$, then the vertex connectivity of $G({\mathbb{Z}}_n)$ is

$$\begin{array}{cc}\hfill \kappa (G({\mathbb{Z}}_n))& =p^{r-1}{q}^{s-1}\kappa (G_{p^r{q}^s})\hfill \\ \hfill & =(p-2)p^{(r-1)}{q}^s.\hfill \end{array}$$

□

Based on our results for the vertex connectivity of the unit graph, in Theorems 6 and 7, we state the following conjectures:

Conjecture I: Let $n=p_1{p}_2...p_k$, where $2\ne p_1<p_2<...<p_k$ are distinct primes. The vertex connectivity of $G({\mathbb{Z}}_n)$ is

$${\displaystyle \kappa (G({\mathbb{Z}}_n))=(p_1-2)\prod _{2\le i\le k}{p}_i.}$$

Conjecture II: Let $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, where $2\ne p_1<p_2<...<p_k$ are distinct primes, $r_i$ and k are positive integers, and $1\le i\le k$. Then, the vertex connectivity of $G({\mathbb{Z}}_n)$ is given by

$${\displaystyle \kappa (G({\mathbb{Z}}_n))=(p_1-2)p_1^{(r_1-1)}\prod _{2\le i\le k}{p}_i^{r_i}.}$$

6. Conclusions

In this paper, we have investigated the structure of $G({\mathbb{Z}}_n)$. Based on this structure, the Laplacian spectrum and vertex connectivity of $G({\mathbb{Z}}_n)$ have been determined for various n. First, we study the structure of $G({\mathbb{Z}}_n)$ for $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, where $p_1,p_2,...,p_k$ are distinct primes and $k,r_1,r_2,...,r_k$ are positive integers such that at least one of the $r_i$ must be greater than 1, and we prove that the graph $G({\mathbb{Z}}_n)$ is a generalized join of certain complete graphs and null graphs. Then, we determine the Laplacian spectrum of $G({\mathbb{Z}}_n)$, we prove that $G({\mathbb{Z}}_n)$ is Laplacian integral, and we deduce the algebraic connectivity and Laplacian spectral radius of $G({\mathbb{Z}}_n)$. Furthermore, we examine the vertex connectivity of $G({\mathbb{Z}}_{pq})$ and $G({\mathbb{Z}}_{p^r{q}^s})$, where $2\ne p<q$ are primes and r and s are positive integers by using their structure and Menger’s theorem. Finally, we present conjectures about the vertex connectivity of $G({\mathbb{Z}}_n)$ when $n=p_1{p}_2...p_k$ and $n=p_1^{r_1}{p}_2^{r_2}...p_k^{r_k}$, where $p_i$ are distinct primes, $r_i$ are positive integers, and $1\le i\le k$. Our results are precise and dependable, as verified by Python programming (see Appendix A).