Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)

Abstract

In this paper, we examine the interplay between the structural and spectral properties of the unit graph $G\left({\mathbb{Z}}_n\right)$ for $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, where $p_1,p_2,\dots ,p_k$ are distinct primes and $k,r_1,r_2,\dots ,r_k$ are positive integers such that at least one of the $r_i$ must be greater than 1. We first analyze the structure of the unit graph of ${\mathbb{Z}}_n$, treating it as what we will define as a ‘generalized join graph’ under these conditions. We then determine the Laplacian spectrum of $G\left({\mathbb{Z}}_n\right)$ and deduce that it is integral for all n. Consequently, we obtain the Laplacian spectral radius and algebraic connectivity of $G\left({\mathbb{Z}}_n\right)$. We also prove that the vertex connectivity of $G\left({\mathbb{Z}}_{pq}\right)$ is $(p-2)q$, where $2\ne p<q$. We deduce the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ when $n=p^r{q}^s$, where $2\ne p<q$ are primes and $r,s$ are positive integers. Finally, we present conjectures regarding the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ when $n=p_1{p}_2\dots p_k$ and $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, where $p_i$ are distinct primes, $r_i$ are positive integers, and $1\le i\le k$.

1. Introduction

For any positive integer n, ${\mathbb{Z}}_n$ represents the ring of integers modulo n. The elements of this ring are denoted as $0,1,2,\dots ,n-1$. A nonzero element $x\in {\mathbb{Z}}_n$ is considered a unit if x is coprime with n, meaning $\gcd (x,n)=1$. In 1990, Grimaldi [1] introduced the concept of the unit graph for the ring ${\mathbb{Z}}_n$. The unit graph $G\left({\mathbb{Z}}_n\right)$ is constructed by taking all elements of ${\mathbb{Z}}_n$ as vertices, with two distinct vertices x and y being adjacent if and only if $x+y$ is a unit in ${\mathbb{Z}}_n$. Grimaldi analyzed the fundamental properties of $G\left({\mathbb{Z}}_n\right)$, including the vertex degrees, covering numbers, independence numbers, Hamiltonian cycles, and chromatic polynomials. Additional studies on $G\left({\mathbb{Z}}_n\right)$ can be found in [2,3,4]. Ashrafi et al. subsequently extended the notion of unit graphs from $G\left({\mathbb{Z}}_n\right)$ to $G\left(R\right)$, where R is an arbitrary ring [5]. Their work examined features such as the diameter, planarity, chromatic index, and girth of $G\left(R\right)$. Other research contributions related to unit graphs over rings are documented in [6,7,8,9].

In recent years, the Laplacian spectrum and vertex connectivity of graphs related to algebraic structures have garnered significant attention from researchers. In 2020, Chattopadhyay et al. [10] investigated the Laplacian spectrum and vertex connectivity of the zero divisor graph $\Gamma \left({\mathbb{Z}}_n\right)$ associated with the ring ${\mathbb{Z}}_n$. Further studies on the Laplacian spectrum and vertex connectivity of graphs connected to ${\mathbb{Z}}_n$ can be found in [11,12]. Shen et al. [3] examined the Laplacian spectrum of unit graphs $G\left({\mathbb{Z}}_n\right)$ when $n=q^{\alpha }$, where q is an odd prime and $\alpha$ is a positive integer. They demonstrated that the vertex connectivity and algebraic connectivity of $G\left({\mathbb{Z}}_n\right)$ coincide if and only if $n=q^{\alpha }$. Moreover, they explored properties such as the Laplacian spectral radius and Laplacian integrality of $G\left({\mathbb{Z}}_{q^{\alpha }}\right)$.

In this paper, we explore the structure of $G\left({\mathbb{Z}}_n\right)$ and use it as a foundation to analyze the Laplacian spectrum and vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ for various values of n. This paper is arranged as follows: In Section 2, we provide the preliminary concepts and results that are used throughout this paper. In Section 3, we examine the structure of $G\left({\mathbb{Z}}_n\right)$ for $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, where $p_1,p_2,\dots ,p_k$ are distinct primes and $k,r_1,r_2,\dots ,r_k$ are positive integers such that at least one of the $r_i$ must be greater than 1. We prove that $G\left({\mathbb{Z}}_n\right)$ is a generalized join graph. In Section 4, we investigate the Laplacian spectrum of $G\left({\mathbb{Z}}_n\right)$, establishing that it is the Laplacian integral. Additionally, we derive the algebraic connectivity and the Laplacian spectral radius of $G\left({\mathbb{Z}}_n\right)$. In Section 5, we investigate the vertex connectivity of $G\left({\mathbb{Z}}_{pq}\right)$ and $G\left({\mathbb{Z}}_{p^r{q}^s}\right)$, where $2\ne p<q$ are primes and r and s are positive integers, based on their structure and Menger’s theorem. Lastly, we conjecture formulas for the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ for odd n.

2. Preliminaries

In this section, we present fundamental definitions and results that are necessary for the following sections. Consider a graph G with a vertex set $V\left(G\right)=\{v_1,v_2,\dots ,v_n\}$ and an edge set $E\left(G\right)=\{e_1,e_2,\dots ,e_m\}$. For $1\le i,j\le n$, two vertices $v_i$ and $v_j$ in G are said to be adjacent (or neighbors) if they are connected by an edge e in G. This adjacency is denoted by $v_i\sim v_j$. For $v\in V\left(G\right)$, let $N_G\left(v\right)$ represent the set of all vertices in G that are adjacent to v. The degree of a vertex v, denoted as $\mathrm{deg}\left(v\right)$, is defined as the number of edges incident to v. A path in a graph is a sequence of distinct vertices where each vertex in the sequence is connected to the subsequent vertex by an edge. A graph G is considered connected if there exists a path between any two vertices in G. A complete graph is a graph where every pair of distinct vertices is connected by an edge. The complete graph containing n vertices is denoted by $K_n$. A null graph ${\overline{K}}_n$ is a graph consisting of n vertices and no edges. An isomorphism between two graphs, $G_1$ and $G_2$, denoted by $G_1\cong G_2$, is defined as a bijective mapping $\varphi$ from $V\left(G_1\right)$ to $V\left(G_2\right)$, such that $x\sim y$ in $G_1$ if and only if $\varphi \left(x\right)\sim \varphi \left(y\right)$ in $G_2$. For two graphs, $G_1$ and $G_2$, with disjoint vertex sets, the join $G_1\vee G_2$ is constructed by combining $G_1$ and $G_2$ and adding edges connecting every vertex in $G_1$ to each vertex in $G_2$.

Let R be a ring with unity, and let $U\left(R\right)$ denote the set of all units in R. The unit graph $G\left(R\right)$ of a ring R is defined as the graph where the vertices correspond to all elements of R. Two distinct vertices, x and y, are adjacent if and only if their sum, $x+y$, is a unit in R. Let R be a commutative ring with unity. An element $r\in R$ is called nilpotent if there exists an integer $k>1$ such that $r^k=0$. The nilradical of R, denoted as $\mathrm{nil}\left(R\right)$, is defined as the set of all nilpotent elements in R. Alternatively, it can also be characterized as the intersection of all prime ideals in R. A maximal ideal of R is an ideal $m\ne R$ such that no proper ideal I in R exists with $m⊊I$. The Jacobson radical $J\left(R\right)$ of a ring R is the set obtained by taking the intersection of all maximal ideals contained within R. Every maximal ideal in R is also a prime ideal. So, $J\left(R\right)=nil\left(R\right)$. Recall that, the sum of a nilpotent element and a unit is always a unit.

For a finite, simple, undirected graph G, the adjacency matrix $A\left(G\right)$ is an $n\times n$ matrix where the $(i,j)$-th entry is 1 if $v_i\sim v_j$, and 0 otherwise. The Laplacian matrix $L\left(G\right)$ of a graph G is defined as $L\left(G\right)=D\left(G\right)-A\left(G\right)$, where $D\left(G\right)$ is a diagonal matrix $\mathrm{Diag}(d_1,d_2,\dots ,d_n)$, with $d_i$ representing the degrees of the vertices in G. The second smallest eigenvalue of $L\left(G\right)$ is called the algebraic connectivity, represented by $\mu \left(G\right)$. The value $\mu \left(G\right)$ is greater than zero if and only if the graph G is connected. The largest eigenvalue of $L\left(G\right)$ is referred to as the Laplacian spectral radius, denoted by $\lambda \left(G\right)$. A graph G is said to be the Laplacian integral if all the eigenvalues of its Laplacian matrix $L\left(G\right)$ are integers. Additional information on the Laplacian matrix of graphs is available in [13,14].

The spectrum of an $n\times n$ matrix C, denoted as $\sigma \left(C\right)$, represents the multiset of all its eigenvalues. If the distinct eigenvalues of C are ${\eta }_1,{\eta }_2,\dots ,{\eta }_c$ with respective multiplicities ${\gamma }_1,{\gamma }_2,\dots ,{\gamma }_c$, the spectrum of C is expressed as follows:   

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

For a graph G, the Laplacian spectrum refers to the spectrum of its Laplacian matrix $L\left(G\right)$, commonly denoted as ${\sigma }_L\left(G\right)$. For example:

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

(1)

Consider a graph G with k vertices, $V\left(G\right)=\{v_1,v_2,\dots ,v_k\}$, and let $W_1,W_2,\dots ,W_k$ be k pairwise disjoint graphs. The G-generalized join graph, denoted as $G[W_1,W_2,\dots ,W_k]$, is constructed by replacing each vertex $v_i$ of G with the graph $W_i$, and then adding edges between every vertex in $W_i$ and every vertex of $W_j$ whenever $v_i\sim v_j$ in G [15]. This result is instrumental for subsequent analysis.

Theorem 1

([16]). Consider a graph G with k vertices, $V\left(G\right)=\{v_1,v_2,\dots ,v_k\}$, and let $W_1,W_2,\dots ,$$W_k$ be k pairwise disjoint graphs containing $n_1,n_2,\dots ,n_k$ vertices, respectively. Then we have the following:

$${\sigma }_L\left(G[W_1,W_2,\dots ,W_k]\right)=\left(\begin{array}{c}{\displaystyle \bigcup _{j=1}^k(M_j+({\sigma }_L\left(W_j\right)\setminus \left\{0\right\}))}\end{array}\right)\bigcup \sigma \left(\mathit{L}\left(G\right)\right),$$

(2)

where

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

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

and

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

In (2), ${\sigma }_L\left(W_j\right)\setminus \left\{0\right\}$ indicates the multiset ${\sigma }_L\left(W_j\right)$ with a single occurrence of the eigenvalue 0 removed. Additionally, $M_j+({\sigma }_L\left(W_j\right)\setminus \left\{0\right\})$ denotes the operation of adding $M_j$ to each element in ${\sigma }_L\left(W_j\right)\setminus \left\{0\right\}.$

In this paper, $q,p,p_1,p_2,\dots ,p_k$ are used to denote distinct prime numbers, while $r,r_1,r_2,\dots ,r_k,s,s_1,s_2,\dots ,s_k$ represent positive integers. Let n be a positive integer. Euler’s totient function, $\phi \left(n\right)$, calculates the number of integers from 1 to n that are coprime to n. Recall that $\phi \left(p_i\right)=p_i-1$ and $\phi (p_1{p}_2\cdots p_k)=\phi \left(p_1\right)\phi \left(p_2\right)\cdots \phi \left(p_k\right)={\prod }_{i=1}^k\phi \left(p_i\right)$. Fakieh et al. examined the Laplacian spectrum of unit graphs corresponding to the ring ${\mathbb{Z}}_{p_1{p}_2\dots p_k}$ [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, $1\le i,j\le k$.

1. 

If $n=p_1{p}_2\dots p_k$, then we have the following:

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

2. 

If $n=2p_1{p}_2\dots p_k$, then we have the following:

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

The vertex connectivity $\kappa \left(G\right)$ of a graph G is defined as the minimum number of vertices that need to be removed from G to produce either a disconnected graph or a trivial graph. Fiedler presented a famous result that establishes a relation between vertex connectivity and algebraic connectivity as in [17]. A set of two or more paths in G is considered internally disjoint if no vertex in G serves as an internal vertex for more than one path in the set. Menger’s theorem provides a fundamental result related to vertex connectivity $\kappa \left(G\right)$. It states that the minimum number of vertices that must be deleted to eliminate all $xy$-paths in G is equal to the maximum number of internally disjoint $xy$-paths, where x and y are nonadjacent vertices [18]. In other words, we have the following:

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

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

3. $\mathit{G}\left({\mathbb{Z}}_{\mathit{n}}\right)$ as a Generalized Join Graph

Any integer $n\ge 2$ can be written in the following form: $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$. In this section and the next, we assume that at least one of the $r_i>1$. In this case, note that $J\left({\mathbb{Z}}_n\right)$ is nontrivial. We prove that $G\left({\mathbb{Z}}_n\right)$ is a generalized join graph. To this end, we first study the structure of $G\left({\mathbb{Z}}_n\right)$. Denote the maximal ideal of ${\mathbb{Z}}_n$ by ${\mathrm{m}}_i$, where ${\mathrm{m}}_i$ is the ideal generated by the prime divisor $p_i$ of n, that is, ${\mathrm{m}}_i=⟨p_i⟩$. So, $J\left({\mathbb{Z}}_n\right)=\cap {\mathrm{m}}_i=⟨{\prod }_{i=1}^k{p}_i⟩\ne \left\{0\right\}$. Put $\mathbf{p}={\prod }_{i=1}^k{p}_i$, then we have the following:

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

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

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

The cosets $B_0,B_1,\dots ,B_{\mathbf{p}-1}$ of $J\left({\mathbb{Z}}_n\right)$ form a partition of the vertex set of $G\left({\mathbb{Z}}_n\right)$. Thus, we have the following:

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

(3)

Let $B_0=J\left({\mathbb{Z}}_n\right)$. Then, for $j\in S$, we have the following:

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

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

Lemma 1.

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

Proof. 

$(\Rightarrow )$ Clear.

$(\Leftarrow )$ Suppose that $\gcd (i+j,n)=1$. 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 $a_i,b_j\in J\left({\mathbb{Z}}_n\right)$. 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 $G\left({\mathbb{Z}}_n\right)$.   □

By using parts (b) and (c) of Lemma 2.7 in [5] and Lemma 1, the following is evident:

Corollary 1.

The following statements hold:

1. 

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

2. 

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

The above result implies that the partition $B_0\cup B_1\cup \dots \cup B_{\mathbf{p}-1}$ of $V\left(G\left({\mathbb{Z}}_n\right)\right)$ of $G\left({\mathbb{Z}}_n\right)$ forms an equitable partition. This means that every vertex in $B_i$ has the same 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 ${\mathbb{Z}}_n/J\left({\mathbb{Z}}_n\right)$, that is, the set of vertices is S, and in which two distinct vertices i and j are adjacent if and only if $\gcd (i+j,n)=1$. The graph $G_n$ serves a pivotal role throughout the remainder of this paper.

Lemma 2.

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

Proof. 

We define a map $\varphi :V\left(G\left({\mathbb{Z}}_{\mathbf{p}}\right)\right)\to V\left(G_n\right)$ such that $\varphi \left(B_i\right)=i$. Clearly, $\varphi$ is well-defined and bijection. From Lemma 1, the adjacency relationships are preserved by $\varphi$. As a result, the conclusion is validated.   □

The following lemma states that $G\left({\mathbb{Z}}_n\right)$ can be expressed as a generalized join graph.

Lemma 3.

Let $G\left(B_i\right)$ be the induced subgraph of $G\left({\mathbb{Z}}_n\right)$ for $0\le i\le \mathbf{p}-1$. Then, we have the following:

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

Proof. 

Replace each vertex i of $G_n$ with $G\left(B_i\right)$ for $0\le i\le \mathbf{p}-1$. Consequently, the result is derived from Lemma 1 and Corollary 1.   □

Example 1.

The unit graph $G\left({\mathbb{Z}}_{25}\right)$, as illustrated in Figure 1, can be expressed using Lemma 3 as follows:

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

where $G_{25}$ is depicted in Figure 2, $G\left(B_0\right)={\overline{K}}_5$, and $G\left(B_i\right)=K_5$ for $1\le i\le 4$. In Figure 1, the lines connecting two squares indicate that every vertex in one square is adjacent to all vertices in the other square.

Figure 1. The graph $G\left({\mathbb{Z}}_{25}\right)$.

Figure 2. The graph $G_{25}$.

4. Laplacian Spectrum of $\mathit{G}\left({\mathbb{Z}}_{\mathit{n}}\right)$

In this section, we explore the Laplacian spectrum of $G\left({\mathbb{Z}}_n\right)$ for $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$. For $0\le i\le \mathbf{p}-1$, we give the weight $\left|J\right({\mathbb{Z}}_n\left)\right|=t$ to the vertex i of the graph $G_n$. We define the integer

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

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

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

(4)

where   

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

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

The next remark is an immediate result of Proposition 2.4 in [5].

Remark 1.

The following statements hold:

1. 

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

2. 

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

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

Lemma 4.

$\mathbf{L}\left(G_n\right)=t\left(L\left(G_n\right)\right)$.

Proof. 

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

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

Theorem 3.

Let $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$. Then, we have the following:

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

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

Proof. 

By Lemma 3, we have the following:

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

As a result, the outcome follows directly from Lemma 4 and Theorem 1.   □

By Corollary 1, $G\left(B_i\right)$ 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 $G\left({\mathbb{Z}}_n\right)$, $n-\mathbf{p}$ are known to be nonzero integer values. The remaining $\mathbf{p}$ Laplacian eigenvalues of $G\left({\mathbb{Z}}_n\right)$ will come from the Laplacian eigenvalues of $G\left({\mathbb{Z}}_{\mathbf{p}}\right)$.

Corollary 2.

Let $2\ne p_1<p_2<\dots <p_k$. The Laplacian spectrum of $G\left({\mathbb{Z}}_n\right)$ is as follows:

1. 

If $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, then we have the following:

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

2. 

If $n=2^r{p}_1^{s_1}{p}_2^{s_2}\dots p_k^{s_k}$, then we have the following:

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

Proof. 

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

The following Theorem presents the Laplacian spectrum of $G\left({\mathbb{Z}}_n\right)$ for $n=p^r{q}^s$.

Theorem 4.

Let $2\ne p<q$.

1. 

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

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

(5)

2. 

If $n=2^r{q}^s$, then

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

Proof. 

1.

Let $n=p^r{q}^s$, where $2\ne p<q$. So, $J\left({\mathbb{Z}}_n\right)=⟨pq⟩$, and the set of distinct representatives of ${\mathbb{Z}}_n/⟨pq⟩$ is $\{0,1,\dots ,pq-1\}$. Thus, we have the following:

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

By Corollary 2,

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

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

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

Then, we have the following:

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

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

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

So,

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

Thus, the Laplacian spectrum of $G\left({\mathbb{Z}}_{p^r{q}^s}\right)$ is given by Equation (5).

2.

Let $n=2^r{p}^s$, where $p\ne 2$. Note that $S=\{0,1,\dots ,2p-1\}$ is the vertex set of the graph $G_{2^r{p}^s}$. Thus, we have the following:

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

By Corollary 2,

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

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

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

Hence,

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

   □

Now, we find the Laplacian spectrum of $G\left({\mathbb{Z}}_n\right)$ for $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$. The following theorem can be derived using reasoning analogous to that employed in the proof of Theorem 4, and thus its proof is omitted.

Theorem 5.

Let $2\ne p_1<p_2<\dots <p_k$.

1. 

If $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, then the Laplacian spectrum of $G\left({\mathbb{Z}}_n\right)$ is as follows:

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

2. 

If $n=2^r{p}_1^{s_1}{p}_2^{s_2}\dots p_k^{s_k}$, then the Laplacian spectrum of $G\left({\mathbb{Z}}_n\right)$ is as follows:

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

Directly following from Theorem 5, the subsequent results are obtained.

Corollary 3.

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

Corollary 4.

Let $2\ne p_1<p_2<\dots <p_k$. The Laplacian spectral radius of $G\left({\mathbb{Z}}_n\right)$ is as follows:   

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

Corollary 5.

Let $2\ne p_1<p_2<\dots <p_k$. The algebraic connectivity of $G\left({\mathbb{Z}}_n\right)$ is as follows:

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

5. Vertex Connectivity of $\mathit{G}\left({\mathbb{Z}}_{\mathit{n}}\right)$

In this section, we obtain the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ when $n=pq$ and $n=p^r{q}^s$, where $2\ne p<q$. To achieve this goal, we calculate the number of internally disjoint paths between any two nonadjacent vertices in $G\left({\mathbb{Z}}_{pq}\right)$, which allows us to be ready to explore $\kappa \left(G\left({\mathbb{Z}}_{pq}\right)\right)$ by using Menger’s theorem.

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

The subsequent result will be utilized in the upcoming analysis.

Lemma 5

([19]). Suppose $n=pq$. Then, the following statements hold:

1. 

Let $i\in \{1,2,\dots ,p-1\}$. The induced subgraph $G(i+⟨p⟩)$ of $G\left({\mathbb{Z}}_n\right)$ is isomorphic to $K_1\vee CP(q-1)$ ($CP(q-1)$ is the cocktail party graph, which is obtained from the complete graph $K_{2s}$, $2s=q-1$, by deleting the perfect matching, where a perfect matching in a graph G is a spanning subgraph W of G, where each vertex has a degree of 1). If $i=0$, then $G(i+⟨p⟩)$ is ${\overline{K}}_q$.

2. 

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

3. 

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

Now, we study the structure of $G\left({\mathbb{Z}}_{pq}\right)$, where $2\ne p<q$, analogous to Section 3. In this case, we choose the maximal ideal $⟨p⟩$. Let $S^{\prime }=\{0,1,\dots ,p-1\}$ be the set of distinct representatives of ${\mathbb{Z}}_{pq}/⟨p⟩$. For $i\in S^{\prime }$, we denote the following:

$${\widehat{B}}_i=i+⟨p⟩.$$

Note that the sets ${\widehat{B}}_0,{\widehat{B}}_1,\dots ,{\widehat{B}}_{p-1}$ form a partition of the vertex set of $G\left({\mathbb{Z}}_{pq}\right)$. Thus, we have the following:

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

This implies that any two vertices x and y that belong to the above union are adjacent if and only if $\gcd (x+y,pq)=1$. Let ${\widehat{B}}_0=⟨p⟩$. Then, $|{\widehat{B}}_i|=|{\widehat{B}}_0|=|⟨p⟩|=q$ for $i\in S^{\prime }$. Let $x\in {\mathbb{Z}}_{pq}$, and say $x\in {\widehat{B}}_i$. Denote by $x^{*}$ the unique element in ${\widehat{B}}_i$ which is a multiple of q.

Note that, Lemma 5 implies that the partition ${\cup }_{i=0}^{p-1}{\widehat{B}}_i$ of $V\left(G\left({\mathbb{Z}}_{pq}\right)\right)$ forms an almost equitable partition. This means that every vertex in ${\widehat{B}}_i$ has the same number of neighbors in ${\widehat{B}}_j$ for $i\ne j$ and $i,j\in S^{\prime }$. Also, ${\widehat{B}}_0$ is isomorphic to ${\overline{K}}_q$ and ${\widehat{B}}_i$ 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 ${\widehat{B}}_i$, where $i\in S^{\prime }$, so that $\left|V\right(G_{pq}^{\prime }\left)\right|=p$. Two distinct vertices ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$ are adjacent if and only if $\gcd (i+j,p)=1$, which is equivalent to each vertex in ${\widehat{B}}_i$ being adjacent to $(q-1)$ vertices in ${\widehat{B}}_j$.

Example 2.

The vertex set of $G\left({\mathbb{Z}}_{15}\right)$ is the union ${\widehat{B}}_0\cup {\widehat{B}}_1\cup {\widehat{B}}_2$, where ${\widehat{B}}_0=0+⟨3⟩$, ${\widehat{B}}_1=1+⟨3⟩$, and ${\widehat{B}}_2=2+⟨3⟩$. From Figure 3 below, we observe the following:

Figure 3. The graph $G\left({\mathbb{Z}}_{15}\right)$.

1. 

${\widehat{B}}_1$ is nonadjacent to ${\widehat{B}}_2$ in $G_{15}^{\prime }$ since $\gcd (1+2,3)\ne 1$.

2. 

Each vertex in $G({\widehat{B}}_1)$ and $G({\widehat{B}}_2)$ is adjacent to 4 vertices in $G({\widehat{B}}_0)$.

3. 

$G({\widehat{B}}_0)$ is isomorphic to ${\overline{K}}_5$ and $G({\widehat{B}}_1)$ and $G({\widehat{B}}_2)$ are isomorphic to $K_1\vee CP(4)$. The diamond-shaped vertices 0, 5, and 10 represent $K_1$ in $G({\widehat{B}}_0)$, $G({\widehat{B}}_2)$, and $G({\widehat{B}}_1)$, respectively. Note that these vertices are multiples of 5.

From now on, throughout the rest of this paper, when we write ${\widehat{B}}_i$ and ${\widehat{B}}_j$, it is assumed that $i\ne j$. This assumption ensures that the sets ${\widehat{B}}_i$ and ${\widehat{B}}_j$ are distinct. 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 ${\cup }_{i=0}^{p-1}{\widehat{B}}_i$.

Lemma 6.

Let $i\in S^{\prime }$. If $i\ne 0$, then there are $(p-2)$ neighbors of ${\widehat{B}}_i$ in $G_{pq}^{\prime }$. On the other hand, ${\widehat{B}}_0$ has $(p-1)$ neighbors in $G_{pq}^{\prime }$.

Proof. 

Suppose $i\ne 0$. For $j\ne i$, $\gcd (i+j,p)=1$ for all $j\in S^{\prime }$ except $j=p-i$. Consequently, ${\widehat{B}}_i$ is adjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$ when $j\ne i,p-i$. Then, there are $(p-2)$ neighbors of $G\left(B_i\right)$ in $G_{pq}^{\prime }$. If $i=0$, then $\gcd (0+j,p)=1$ for all $j\ne i$. Thus, ${\widehat{B}}_0$ is adjacent to all vertices in $G_{pq}^{\prime }$. Therefore, there are $(p-1)$ neighbors of ${\widehat{B}}_0$ in $G_{pq}^{\prime }$.   □

Lemma 7.

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

1. 

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

2. 

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

3. 

The number of common neighbors between ${\widehat{B}}_0$ and ${\widehat{B}}_i$ in $G_{pq}^{\prime }$ is $(p-3)$.

Proof. 

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

1.

Let ${\widehat{B}}_i$ and ${\widehat{B}}_j$ be nonadjacent in $G_{pq}^{\prime }$. By Lemma 6, $j=p-i$ and $i=p-j$. Hence, ${\widehat{B}}_i$ is adjacent to all vertices in $G_{pq}^{\prime }$ except ${\widehat{B}}_{p-i}$. Similarly, ${\widehat{B}}_j$ is adjacent to all vertices in $G_{pq}^{\prime }$ except ${\widehat{B}}_{p-j}$. So, ${\widehat{B}}_i$ and ${\widehat{B}}_j$ are adjacent to all vertices in $G_{pq}^{\prime }$ except ${\widehat{B}}_j$ and ${\widehat{B}}_i$. Therefore, there are $(p-2)$ common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

2.

Let ${\widehat{B}}_i$ and ${\widehat{B}}_j$ be adjacent in $G_{pq}^{\prime }$, where $i,j\ne 0$. According to Lemma 6, ${\widehat{B}}_i$ and ${\widehat{B}}_j$ are nonadjacent to ${\widehat{B}}_{p-i}$ and ${\widehat{B}}_{p-j}$ in $G_{pq}^{\prime }$, respectively. So, ${\widehat{B}}_i$ and ${\widehat{B}}_j$ are adjacent to all vertices in $G_{pq}^{\prime }$ except ${\widehat{B}}_{p-i}$ and ${\widehat{B}}_{p-j}$, respectively. Then, the set of common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$ is

$$V\left(G_{pq}^{\prime }\right)-\{{\widehat{B}}_i,{\widehat{B}}_j,{\widehat{B}}_{p-i},{\widehat{B}}_{p-j}\}.$$

Thus, there are $(p-4)$ common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

3.

By Lemma 6, ${\widehat{B}}_0$ is adjacent to all vertices in $G_{pq}^{\prime }$. Also, ${\widehat{B}}_i$ is adjacent to all vertices in $G_{pq}^{\prime }$ except ${\widehat{B}}_{p-i}$. So, the set of common neighbors between ${\widehat{B}}_0$ and ${\widehat{B}}_i$ in $G_{pq}^{\prime }$ is

$$V\left(G_{pq}^{\prime }\right)-\{{\widehat{B}}_0,{\widehat{B}}_i,{\widehat{B}}_{p-i}\}.$$

Then, there are $(p-3)$ common neighbors between ${\widehat{B}}_0$ and ${\widehat{B}}_i$ in $G_{pq}^{\prime }$.

   □

The next result finds the number of common neighbors between nonadjacent vertices $x,y\in {\widehat{B}}_i$ through ${\widehat{B}}_z$, where ${\widehat{B}}_z$ is a neighbor of ${\widehat{B}}_i$ in $G_{pq}^{\prime }$.

Lemma 8.

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

Proof. 

By Part 2 of Lemma 5, both x and y are adjacent to $(q-1)$ vertices in ${\widehat{B}}_z$. Suppose that x and y have the same neighbors in ${\widehat{B}}_z$. Then, x and y are adjacent to all vertices in ${\widehat{B}}_z$ except $z^{\prime }$. This implies that $z^{\prime }$ is adjacent to $(q-2)$ vertices in ${\widehat{B}}_i$, a contradiction with Part 2 of Lemma 5. Then, x and y are adjacent to all vertices in ${\widehat{B}}_z$ except $a_z$ and $b_z$, respectively. So, the number of common neighbors between x and y in ${\widehat{B}}_z$ is $(q-2)$.   □

The following result determines the number of common neighbors between nonadjacent vertices $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$ through ${\widehat{B}}_z$, where ${\widehat{B}}_z$ is a common neighbor between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

Lemma 9.

Let $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$ be nonadjacent and ${\widehat{B}}_z$ be a common neighbor between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

1. 

If x and y have the same neighbors in ${\widehat{B}}_z$, then the number of common neighbors in ${\widehat{B}}_z$ between x and y is $(q-1)$.

2. 

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

Proof. 

Let $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$ be nonadjacent and ${\widehat{B}}_z$ be a common neighbor between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

1.

The proof is direct from Part 2 of Lemma 5.

2.

Suppose x and y do not have the same neighbors in ${\widehat{B}}_z$. By Part 2 of Lemma 5, both x and y have $(q-1)$ neighbors in ${\widehat{B}}_z$. That is, x and y are adjacent to all vertices in ${\widehat{B}}_z$ except $a_z$ and $b_z$, respectively. So, the number of the common neighbors in ${\widehat{B}}_z$ between x and y is $(q-2)$.

   □

The next result characterizes the nonadjacent vertices of $G\left({\mathbb{Z}}_{pq}\right)$ for which the relation in Part 1 of Lemma 9 holds when ${\widehat{B}}_i$ is adjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

Proposition 1.

Let $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$ be nonadjacent, ${\widehat{B}}_i$ be adjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$, and ${\widehat{B}}_z$ be a common neighbor between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. Then, x and y have the same neighbors in ${\widehat{B}}_z$ if and only if $x=x^{*}$ and $y=y^{*}$.

Proof. 

$(\Rightarrow )$ Suppose that x and y have the same neighbors in ${\widehat{B}}_z$. Then, x and y are adjacent to all vertices in ${\widehat{B}}_z$ except $z^{\prime }$ by Part 2 of Lemma 5. 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 $a^{\prime },b^{\prime },c^{\prime }\in \{0,1,\dots ,q-1\}$. Since ${\widehat{B}}_i$ is adjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$ and x is nonadjacent to y, then $(i+j,p)=1$ and $\gcd ((i+j)+(a^{\prime }+b^{\prime })p,pq)\ne 1$, this implies that $\gcd ((i+j)+(a^{\prime }+b^{\prime })p,q)\ne 1$. Similarly, since ${\widehat{B}}_z$ is adjacent to ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$, then $\gcd ((i+z^{\prime })+(a^{\prime }+c^{\prime })p,q)\ne 1$ and $\gcd ((j+z^{\prime })+(b^{\prime }+c^{\prime })p,q)\ne 1$. So, we have the following:

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

So, $2z^{\prime }=(a_2+a_3-a_1)q$; this implies that 2 divides $(a_2+a_3-a_1)$ and, hence, $z^{\prime }$ is a multiple of q, in this case $z^{\prime }=z^{*}$. But $z^{*}$ is adjacent to all vertices in ${\widehat{B}}_i$ and ${\widehat{B}}_j$ except $x=x^{*}$ and $y=y^{*}$, respectively. This contradicts 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 ${\widehat{B}}_z$ by Part 2 of Lemma 5, that is, x is adjacent to all vertices of ${\widehat{B}}_z$ except $z^{*}$, where $z^{*}=cq$ and $c\in \{0,1,\dots ,q-1\}$. Similarly, y is adjacent to all vertices of ${\widehat{B}}_z$ except $z^{*}$. Then, the result is obtained.   □

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

The next two results calculate the number of internally disjoint paths (IDPs) between nonadjacent vertices x and y in ${\widehat{B}}_i$.

Lemma 10.

Let $x,y\in {\widehat{B}}_i$ be nonadjacent. Then, there are $(p-1)(q-2)$ IDPs of length 2 between x and y.

Proof. 

If $i\ne 0$, then ${\widehat{B}}_i$ is isomorphic to $K_1\vee CP(q-1)$ by Part 1 of Lemma 5. Since $x^{*}$ represents $K_1$, then $x^{*}$ is adjacent to all vertices in ${\widehat{B}}_i$. Since x is nonadjacent to y, then neither x nor y is $x^{*}$ in ${\widehat{B}}_i$. So, there are $(q-2)$ common neighbors between x and y and, hence, there are $(q-2)$ IDPs of length 2 between x and y in ${\widehat{B}}_i$. Let ${\widehat{B}}_z$ be a neighbor of ${\widehat{B}}_i$ in $G_{pq}^{\prime }$. By Lemma 8, there are $(q-2)$ common neighbors between x and y in ${\widehat{B}}_z$. Hence, there are $(q-2)$ IDPs of length 2 through ${\widehat{B}}_z$. By Lemma 6, there are $(p-2)$ neighbors of ${\widehat{B}}_i$ in $G_{pq}^{\prime }$ and, hence, there are $(p-2)(q-2)$ IDPs of length 2 between x and y through all neighbors of ${\widehat{B}}_i$ in $G_{pq}^{\prime }$. Thus, the total number of IDPs of length 2 between x and y is as follows:

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

If $i=0$, then ${\widehat{B}}_0$ is isomorphic to ${\overline{K}}_q$, by Part 1 of Lemma 5. So, there is no path between x and y in ${\widehat{B}}_0$. By Lemma 8, there are $(q-2)$ common neighbors between x and y in ${\widehat{B}}_z$, where ${\widehat{B}}_z$ is a neighbor of ${\widehat{B}}_0$ in $G_{pq}^{\prime }$. Hence, there are $(q-2)$ IDPs of length 2 through ${\widehat{B}}_z$. By Lemma 6, there are $(p-1)$ neighbors of ${\widehat{B}}_0$ in $G_{pq}^{\prime }$ and, hence, there are $(p-1)(q-2)$ IDPs of length 2 between x and y.   □

Lemma 11.

Let $x,y\in {\widehat{B}}_i$ be nonadjacent.

1. 

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

2. 

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

Proof. 

Let $x,y\in {\widehat{B}}_i$ be nonadjacent and ${\widehat{B}}_z$ be a neighbor of ${\widehat{B}}_i$ in $G_{pq}^{\prime }$. By proof of Lemma 8, x and y are adjacent to all vertices of ${\widehat{B}}_z$ except $a_z$ and $b_z$, respectively.

1.

Suppose $i\ne 0$. By Part 2 of Lemma 5, $a_z$ and $b_z$ in ${\widehat{B}}_z$ are adjacent to $(q-1)$ vertices of ${\widehat{B}}_{p-i}$. Since we investigate the IDPs between x and y through $a_z$ and $b_z$, we can choose a vertex w from ${\widehat{B}}_{p-i}$ that is adjacent to both $a_z$ and $b_z$. This path will be of length 4, as illustrated in Figure 4a. Similarly, for each neighbor ${\widehat{B}}_z$ of ${\widehat{B}}_i$ in $G_{pq}^{\prime }$ there is one internally disjoint path of length 4 between x and y. By Lemma 6, there are $(p-2)$ neighbors of ${\widehat{B}}_i$ in $G_{pq}^{\prime }$. Therefore, there are $(p-2)$ IDPs of length 4 between x and y through all neighbors of ${\widehat{B}}_i$.

Figure 4. $x-b_z-w-a_z-y$ when $x,y\in {\widehat{B}}_i$. (a) $x-b_z-w-a_z-y$ when $i\ne 0$. (b) $x-b_z-w-a_z-y$ when $i=0$.

2.

Suppose $i=0$. By Part 2 of Lemma 5, $a_z$ and $b_z$ in ${\widehat{B}}_z$ are adjacent to $(q-1)$ vertices of ${\widehat{B}}_0$. Following a similar approach to the proof in Part 1, we identify one internally disjoint path of length 4, illustrated in Figure 4b. By Lemma 6, the number of neighbors of ${\widehat{B}}_0$ in $G_{pq}^{\prime }$ is $(p-1)$. So, there are $(p-1)$ IDPs of length 4 between x and y through all neighbors of ${\widehat{B}}_0$.

   □

The next two results determine the number of IDPs between nonadjacent vertices $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$, where ${\widehat{B}}_i$ is nonadjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

Lemma 12.

Let $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$ be nonadjacent, ${\widehat{B}}_i$ be nonadjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$, and ${\widehat{B}}_z$ be a common neighbor between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

1. 

If x and y have the same neighbors in ${\widehat{B}}_z$, then there are $(p-2)(q-1)$ IDPs of length 2 between x and y.

2. 

If x and y do not have the same neighbors in ${\widehat{B}}_z$, then there are $(p-2)(q-2)$ IDPs of length 2 between x and y.

Proof. 

Let ${\widehat{B}}_i$ be nonadjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. By Part 1 of Lemma 7, there are $(p-2)$ common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

1.

If x and y have the same neighbors in ${\widehat{B}}_z$, then there are $(q-1)$ common neighbors between x and y in ${\widehat{B}}_z$ by Lemma 9. Thus, there are $(q-1)$ IDPs of length 2 between x and y through ${\widehat{B}}_z$. Hence, there are $(p-2)(q-1)$ IDPs of length 2 between x and y through all common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

2.

If x and y do not have the same neighbors in ${\widehat{B}}_z$, then there are $(q-2)$ common neighbors between x and y in ${\widehat{B}}_z$ by Lemma 9. So, there are $(q-2)$ IDPs of length 2 between x and y through ${\widehat{B}}_z$. Therefore, there are $(p-2)(q-2)$ IDPs of length 2 between x and y through all common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

   □

Lemma 13.

Let $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$ be nonadjacent, ${\widehat{B}}_i$ be nonadjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$, and ${\widehat{B}}_z$ be a common neighbor between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

1. 

If x and y have the same neighbors in ${\widehat{B}}_z$, then there are $(p-2)$ IDPs of length 4 between x and y.

2. 

If x and y do not have the same neighbors in ${\widehat{B}}_z$, then there are $2(p-2)$ IDPs of length 3 between x and y.

Proof. 

Let ${\widehat{B}}_i$ be nonadjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. By Part 1 of Lemma 7, there are $(p-2)$ common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

1.

Suppose x and y have the same neighbors in ${\widehat{B}}_z$. By Lemma 5, $x\in {\widehat{B}}_i$ has at least $(q-2)$ neighbors in ${\widehat{B}}_i$ and each of them is adjacent to $(q-1)$ vertices of ${\widehat{B}}_z$. Then, we can choose a neighbor, say $x^{\prime }$, of x in ${\widehat{B}}_i$ such that $x^{\prime }\sim z^{\prime }$, where $z^{\prime }\in {\widehat{B}}_z$ is nonadjacent to both x and y. Similarly, we can choose a neighbor $y^{\prime }$ of y in ${\widehat{B}}_j$ 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 5a. So, there are $(p-2)$ IDPs of length 4 between x and y through all common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

Figure 5. Paths between $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$, where ${\widehat{B}}_i\nsim {\widehat{B}}_j$ in $G_{pq}^{\prime }$. (a) $x-x^{\prime }-z^{\prime }-y^{\prime }-y$ when x and y have the same neighbors in ${\widehat{B}}_z$. (b) $x-x^{\prime }-a_z-y\mathrm{and}x-b_z-y^{\prime }-y$ when x and y do not have the same neighbors in ${\widehat{B}}_z$.

2.

Suppose x and y do not have the same neighbors in ${\widehat{B}}_z$. By the proof of Lemma 9, x and y are adjacent to all vertices in ${\widehat{B}}_z$ except $a_z$ and $b_z$, respectively. Approaching the proof similarly to Part 1, we can choose a neighbor $x^{\prime }$ of x in ${\widehat{B}}_i$ and a neighbor $y^{\prime }$ of y in ${\widehat{B}}_j$, where $x^{\prime }\sim a_z$ and $y^{\prime }\sim b_z$. Since $a_z\sim y$ and $b_z\sim x$, two IDPs of length 3 exist. These paths are described in Figure 5b. Hence, there are $2(p-2)$ IDPs of length 3 between x and y through all common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

   □

The following results find the number of IDPs between nonadjacent vertices $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$, where ${\widehat{B}}_i$ is adjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$.

Lemma 14.

Let $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$ be nonadjacent and ${\widehat{B}}_i$ be adjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. The number of IDPs of length 2 between x and y is either $(p-2)(q-1)$ or $(p-2)(q-2)$.

Proof. 

By Lemma 7, there are $(p-4)$ common neighbors between the adjacent vertices ${\widehat{B}}_i$ and ${\widehat{B}}_j$, where $i,j\ne 0$, in $G_{pq}^{\prime }$ and there are $(p-3)$ common neighbors between ${\widehat{B}}_0$ and ${\widehat{B}}_i$ in $G_{pq}^{\prime }$. Let ${\widehat{B}}_z$ be a common neighbor between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. Since x and y are nonadjacent, then x and y have the following possibilities:

1.

Assume that x and y have the same neighbors in ${\widehat{B}}_z$. By Proposition 1 and Lemma 9, $x=x^{*}$ and $y=y^{*}$ and there are $(q-1)$ common neighbors between x and y through ${\widehat{B}}_z$. So, there are $(q-1)$ IDPs of length 2 between x and y through ${\widehat{B}}_z$. If $i,j\ne 0$, then there are $(p-4)(q-1)$ IDPs of length 2 between x and y through all common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. Further, x (resp. y) is adjacent to all vertices in ${\widehat{B}}_i$ (resp. ${\widehat{B}}_j$). Also, x (resp. y) is adjacent to all vertices in ${\widehat{B}}_j$ (resp. ${\widehat{B}}_i$) except y (resp. x). So, the set of common neighbors between x and y in ${\widehat{B}}_i$ and ${\widehat{B}}_j$ is ${\widehat{B}}_i-\left\{x\right\}$ union ${\widehat{B}}_j-\left\{y\right\}$. Thus, there are $2(q-1)$ common neighbors between x and y in ${\widehat{B}}_i$ and ${\widehat{B}}_j$. Consequently, there are $2(q-1)$ IDPs of length 2 between x and y in ${\widehat{B}}_i$ and ${\widehat{B}}_j$. Therefore, the total number of IDPs of length 2 between x and y is as follows:   

$$(p-4)(q-1)+2(q-1)=(p-2)(q-1).$$

If $j=0$, then there are $(p-3)(q-1)$ IDPs of length 2 between x and y through all common neighbors between ${\widehat{B}}_0$ and ${\widehat{B}}_i$ in $G_{pq}^{\prime }$. Further, x is adjacent to all vertices in ${\widehat{B}}_i$ and y is nonadjacent to any vertex in ${\widehat{B}}_0$. Also, x (resp. y) is adjacent to all vertices in ${\widehat{B}}_0$ (resp. ${\widehat{B}}_i$) except y (resp. x). So, the set of common neighbors between x and y in ${\widehat{B}}_0$ and ${\widehat{B}}_i$ are ${\widehat{B}}_i-\left\{x\right\}$. Thus, there are $(q-1)$ common neighbors between x and y in ${\widehat{B}}_0$ and ${\widehat{B}}_i$. Then, there are $(q-1)$ IDPs of length 2 between x and y in ${\widehat{B}}_0$ and ${\widehat{B}}_i$. Thus, the total number of IDPs of length 2 between x and y is as follows:

$$(p-3)(q-1)+(q-1)=(p-2)(q-1).$$

2.

Assume that x and y do not have the same neighbors in ${\widehat{B}}_z$. By Proposition 1 and Lemma 9, $x\ne x^{*}$ and $y\ne y^{*}$ and there are $(q-2)$ common neighbors between x and y in ${\widehat{B}}_z$. Thus, there are $(q-2)$ IDPs of length 2 between x and y through ${\widehat{B}}_z$. If $i,j\ne 0$, then there are $(p-4)(q-2)$ IDPs of length 2 between x and y through all common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. Furthermore, x (resp. y) is adjacent to all vertices except only one vertex $x^{\prime }$ (resp. $y^{\prime }$) in ${\widehat{B}}_i$ (resp. ${\widehat{B}}_j$). Also, x (resp. y) is adjacent to all vertices in ${\widehat{B}}_j$ (resp. ${\widehat{B}}_i$) except y (resp. x). Thus, the set of common neighbors between x and y in ${\widehat{B}}_i$ and ${\widehat{B}}_j$ is ${\widehat{B}}_i-\{x,x^{\prime }\}$ union ${\widehat{B}}_j-\{y,y^{\prime }\}$. So, there are $2(q-2)$ common neighbors between x and y in ${\widehat{B}}_i$ and ${\widehat{B}}_j$. As a result, there are $2(q-2)$ IDPs of length 2 between x and y in ${\widehat{B}}_i$ and ${\widehat{B}}_j$. Therefore, the total number of IDPs of length 2 between x and y is as follows:

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

If $j=0$, then there are $(p-3)(q-2)$ IDPs of length 2 between x and y through all common neighbors between ${\widehat{B}}_0$ and ${\widehat{B}}_i$ in $G_{pq}^{\prime }$. Further, x is adjacent to all vertices except only one vertex $x^{\prime }$ in ${\widehat{B}}_i$, and y is nonadjacent to any vertex in ${\widehat{B}}_0$. Also, x (resp. y) is adjacent to all vertices in ${\widehat{B}}_0$ (resp. ${\widehat{B}}_i$) except y (resp. x). Thus, the set of common neighbors between x and y in ${\widehat{B}}_0$ and ${\widehat{B}}_i$ is ${\widehat{B}}_i-\{x,x^{\prime }\}$. So, there are $(q-2)$ common neighbors between x and y in ${\widehat{B}}_0$ and ${\widehat{B}}_i$. Then, there are $(q-2)$ IDPs of length 2 between x and y in ${\widehat{B}}_0$ and ${\widehat{B}}_i$. Thus, the total number of IDPs of length 2 between x and y is as follows:

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

   □

Lemma 15.

Let $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$ be nonadjacent, where ${\widehat{B}}_i$ is adjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. The number of IDPs of length 3 between x and y is either $(q-1)$, $(p+q-5)$, or $(p+q-4)$.

Proof. 

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

1.

Assume that x and y have the same neighbors in ${\widehat{B}}_z$. Indeed, $x=x^{*}$ and $y=y^{*}$ by Proposition 1. If $i,j\ne 0$, there are $(q-1)$ neighbors of x in ${\widehat{B}}_{p-j}$, denote these neighbors by $x_k^{\prime }$ such that $k=1,2,\dots ,q-1$, and each of them is adjacent to $(q-1)$ vertices of ${\widehat{B}}_{p-i}$ by Part 2 of Lemma 5. Similarly, there are $(q-1)$ neighbors of y in ${\widehat{B}}_{p-i}$, and each of them is adjacent to $(q-1)$ vertices of ${\widehat{B}}_{p-j}$. To obtain the IDPs of length 3 between x and y, we choose one of the neighbors of $x_k^{\prime }$, say $y_k^{\prime }$, in ${\widehat{B}}_{p-i}$ such that $y_k^{\prime }$ is a neighbor of y. Indeed, for each $x_k^{\prime }$ in ${\widehat{B}}_{p-j}$ there is one internally disjoint path between x and y through ${\widehat{B}}_{p-j}$ and ${\widehat{B}}_{p-i}$. Therefore, the total number of IDPs of length 3 between x and y through ${\widehat{B}}_{p-j}$ and ${\widehat{B}}_{p-i}$ together is equal to the number of neighbors of x in ${\widehat{B}}_{p-j}$, which is $(q-1)$. Now suppose $j=0$. There are $(q-1)$ neighbors of x in ${\widehat{B}}_0$ and each of them is adjacent to $(q-1)$ vertices of ${\widehat{B}}_{p-i}$. Since there are $(q-1)$ neighbors of y in ${\widehat{B}}_{p-i}$, so there are more than $(q-1)$ paths of length 3 between x and y through ${\widehat{B}}_0$ and ${\widehat{B}}_{p-i}$ together. By applying the same method in the case where $i,j\ne 0$, there are $(q-1)$ IDPs of length 3 between x and y.

2.

Assume that x and y do not have the same neighbors in ${\widehat{B}}_z$. So, $x\ne x^{*}$ and $y\ne y^{*}$ by Proposition 1. Suppose that ${\widehat{B}}_{z_k}$ is a common neighbor between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. By the proof of Lemma 9, x and y are adjacent to all vertices in ${\widehat{B}}_{z_k}$ except $a_{z_k}$ and $b_{z_k}$, respectively. Suppose $i,j\ne 0$. Since x has $(q-1)$ neighbors in ${\widehat{B}}_{p-j}$ and each of these neighbors is adjacent to $(q-1)$ vertices of ${\widehat{B}}_{z_k}$, then we can choose a neighbor $x_k^{\prime }$ of x in ${\widehat{B}}_{p-j}$ such that $x_k^{\prime }\sim a_{z_K}$. Similarly, we can choose a neighbor $y_k^{\prime }$ of y in ${\widehat{B}}_{p-i}$ such that $y_k^{\prime }\sim b_{z_K}$. Since $a_{z_k}\sim y$ and $b_{z_k}\sim x$, there exist two IDPs of length 3 between x and y, as illustrated in Figure 6a. By Part 2 of Lemma 7, there are $(p-4)$ common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. Then, there are $2(p-4)$ IDPs of length 3 between x and y through all common neighbors between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. After removing all $x_k^{\prime }$ and $y_k^{\prime }$ from ${\widehat{B}}_{p-j}$ and ${\widehat{B}}_{p-i}$, respectively, then the number of remaining neighbors of x and y in ${\widehat{B}}_{p-j}$ and ${\widehat{B}}_{p-i}$, respectively, is $(q-1)-(p-4)=q-p+3$. So, there are $(q-p+3)$ IDPs of length 3 between x and y that pass through the remaining neighbors of x and y in ${\widehat{B}}_{p-j}$ and ${\widehat{B}}_{p-i}$, respectively. So, the total number of IDPs of length 3 between x and y is

$$2(p-4)+(q-p+3)=p+q-5.$$

Figure 6. $x-x_k^{\prime }-a_{z_k}-y$ and $x-b_{z_k}-y_k^{\prime }-y$ when $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$, where ${\widehat{B}}_i\sim {\widehat{B}}_j$ in $G_{pq}^{\prime }$. (a) $x-x_k^{\prime }-a_{z_k}-y$ and $x-b_{z_k}-y_k^{\prime }-y$ when $i,j\ne 0$. (b) $x-x_k^{\prime }-a_{z_k}-y$ and $x-b_{z_k}-y_k^{\prime }-y$ when $j=0$.

Suppose $j=0$. Since x has $(q-1)$ neighbors in ${\widehat{B}}_0$ and each of these neighbors is adjacent to $(q-1)$ vertices of ${\widehat{B}}_{z_k}$, then we can choose a neighbor $x_k^{\prime }$ of x in ${\widehat{B}}_0$ such that $x_k^{\prime }\sim a_{z_k}$. Similarly, we can choose a neighbor $y_k^{\prime }$ of y in ${\widehat{B}}_{p-i}$ 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 6b. By Part 3 of Lemma 7, there are $(p-3)$ common neighbors between ${\widehat{B}}_0$ and ${\widehat{B}}_i$ in $G_{pq}^{\prime }$. Consequently, there are $2(p-3)$ IDPs of length 3 between x and y through all common neighbors between ${\widehat{B}}_0$ and ${\widehat{B}}_i$ in $G_{pq}^{\prime }$. After removing all $x_k^{\prime }$ and $y_k^{\prime }$ from ${\widehat{B}}_0$ and ${\widehat{B}}_{p-i}$, respectively, then the number of remaining neighbors of x and y in ${\widehat{B}}_0$ and ${\widehat{B}}_{p-i}$, respectively, is $(q-1)-(p-3)=q-p+2$. So, there are $(q-p+2)$ IDPs of length 3 between x and y that pass through the rest of the neighbors of x and y in ${\widehat{B}}_0$ and ${\widehat{B}}_{p-i}$, respectively, together. So, the total number of IDPs of length 3 between x and y is as follows:

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

   □

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

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

Theorem 6.

Suppose $2\ne p<q$ are distinct primes. The vertex connectivity of $G\left({\mathbb{Z}}_{pq}\right)$ is as follows:

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

Proof. 

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

Case 1: Suppose $x,y\in {\widehat{B}}_i$. By Lemma 10, there are $(p-1)(q-2)$ IDPs of length 2 between x and y. In addition, there are other IDPs depending on the following cases for i:

(a) Suppose $i\ne 0$. By Lemma 11, there are $(p-2)$ IDPs of length 4 between x and y. So,

$$\rho (x,y)=(p-1)(q-2)+(p-2)=pq-p-q.$$

(b) Suppose $i=0$. By Lemma 11, there are $(p-1)$ IDPs of length 4 between x and y. So,

$$\rho (x,y)=(p-1)(q-2)+(p-1)=(p-1)(q-1).$$

Case 2: Let $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$, where ${\widehat{B}}_i$ is nonadjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. Let ${\widehat{B}}_z$ be a common neighbor between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. The following cases arise for x and y:

(a) If y has the same neighbors as x in ${\widehat{B}}_z$, then there are $(p-2)(q-1)$ IDPs of length 2 between x and y by Lemma 12. According to Lemma 13, there are $(p-2)$ IDPs of length 4 between x and y. Hence,

$$\rho (x,y)=(p-2)(q-1)+(p-2)=(p-2)q.$$

(b) If x and y do not have the same neighbors in ${\widehat{B}}_z$, there are $(p-2)(q-2)$ IDPs of length 2 between x and y by Lemma 12. According to Lemma 13, there are $2(p-2)$ IDPs of length 3 between x and y. Hence,

$$\rho (x,y)=(p-2)(q-2)+2(p-2)=(p-2)q.$$

Case 3: Let $x\in {\widehat{B}}_i$ and $y\in {\widehat{B}}_j$, where ${\widehat{B}}_i$ is adjacent to ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. Let ${\widehat{B}}_z$ be a common neighbor between ${\widehat{B}}_i$ and ${\widehat{B}}_j$ in $G_{pq}^{\prime }$. There are the following cases for x and y:

(a) If y has the same neighbors as x in ${\widehat{B}}_z$, then there are $(p-2)(q-1)$ IDPs of length 2 between x and y by the proof of Lemma 14. According to the proof of Lemma 15, there are $(q-1)$ IDPs of length 3 between x and y. Hence,

$$\rho (x,y)=(p-2)(q-1)+(q-1)=(p-1)(q-1).$$

(b) If x and y do not have the same neighbors in ${\widehat{B}}_z$, then there are $(p-2)(q-2)$ IDPs of length 2 between x and y by the proof of Lemma 14. In addition, there are other IDPs, depending on the following cases for i and j:

(1) Suppose $i,j\ne 0$. According to the proof of Lemma 15, there are $(p+q-5)$ IDPs of length 3 between x and y. Therefore,

$$\rho (x,y)=(p-2)(q-2)+p+q-5=pq-p-q-1.$$

(2) Suppose $j=0$. According to the proof of Lemma 15, there are $(p+q-4)$ IDPs of length 3 between x and y. So,   

$$\rho (x,y)=(p-2)(q-2)+(p+q-4)=pq-p-q.$$

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

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

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Now, let us explore the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ where $n=p^r{q}^s$, with $2\ne p<q$ and at least one of r or s must be greater than 1.

Theorem 7.

Let $n=p^r{q}^s$, where $2\ne p<q$. Then, the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ is

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

Proof. 

By Lemma 3, the unit graph $G\left({\mathbb{Z}}_n\right)$ is

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

According to Lemma 2, $G_{p^r{q}^s}$ is isomorphic to $G\left({\mathbb{Z}}_{pq}\right)$. Hence, by Theorem 6, we obtain the following:

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

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

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

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6. Conclusions and Conjectures

In this paper, we investigated the structure of $G\left({\mathbb{Z}}_n\right)$ and its associated properties. We determined the Laplacian spectrum and vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ for various values of n, providing detailed proofs and utilizing the structure of $G\left({\mathbb{Z}}_n\right)$ for $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$. Specifically, we showed that $G\left({\mathbb{Z}}_n\right)$ can be expressed as a generalized join graph. We established that $G\left({\mathbb{Z}}_n\right)$ is the Laplacian integral, deduced its algebraic connectivity, and identified its Laplacian spectral radius. Using Menger’s theorem and the graph structure, we determined the vertex connectivity of $G\left({\mathbb{Z}}_{pq}\right)$ and $G\left({\mathbb{Z}}_{p^r{q}^s}\right)$. The results presented in this paper have been rigorously verified through Python 3.12 programming, as detailed in Appendix A.

Based on the results for the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ in Theorems 6 and 7 and on some actual experimental data (as with the Python verifications), we present the following conjectures:

• Conjecture I: Let $n=p_1{p}_2\dots p_k$, where $2\ne p_1<p_2<\dots <p_k$. Then,

  $$\kappa \left(G\left({\mathbb{Z}}_n\right)\right)=(p_1-2)\prod _{2\le i\le k}{p}_i.$$
• Conjecture II: Let $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$, where $2\ne p_1<p_2<\cdots <p_k$. Then,

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

These conjectures provide a foundation for future exploration of the vertex connectivity of $G\left({\mathbb{Z}}_n\right)$ for larger values of n and may lead to further generalizations in algebraic graph theory.