On the eigenvalues of zero-divisor graph associated to finite commutative ring

Abstract

Let Z(R) be the set of zero-divisors of a commutative ring R with non-zero identity and $Z^{*}(R)=Z(R)\setminus \left\{0\right\}$ be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by $\Gamma (R),$ is a simple graph whose vertex set is $Z^{*}(R)$ and two vertices $u,v\in Z^{*}(R)$ are adjacent if and only if $uv=vu=0.$ In this paper, we investigate the adjacency matrix and the spectrum of the zero-divisor graphs $\Gamma ({\mathbb{Z}}_n)$ for $n=p^M{q}^N,$ where $p<q$ are primes and M, N are positive integers. Moreover, we obtain the clique number, stability number and girth of $\Gamma ({\mathbb{Z}}_{p^M{q}^N}).$

Keywords:

• Adjacency matrix
• zero-divisor graph
• finite commutative ring
• Euler's totient function

AMS SUBJECT CLASSIFICATION:

• 05C50
• 05C12
• 15A18

1. Introduction

Let G be a finite simple connected graph with vertex set $V(G)=\{v_1,v_2,\dots ,v_n\}$ and edge set E(G). For $1\le i\ne j\le n,$ we write $v_i\sim v_j$ if v_i is adjacent to v_j in G. The cardinality of V(G) and E(G) are called the order and size of G, respectively. If $u\in V(G),$ then N(u) is the set of neighbors of u in G, that is, $N(u)=\{v\in V(G):uv\in E(G)\}.$ The cardinality of N(u) is said to be the degree of u and is denoted by d_v. A graph G is called p-regular when every vertex has the same degree equal to p. For any two distinct vertices u and v of G, d(u, v) denotes the length of a shortest (u, v)-path. Clearly, $d(u,u)=0$ and $d(u,v)=\infty ,$ if there is no path connecting u and v. The diameter of G is defined as $\mathit{diam}(G)=\mathit{max}\{d(u,v):u\text{and}v\text{are}\mathrm{}\text{vertices}\mathrm{}\text{of}G.$ A graph G is said to be complete if any two distinct vertices are adjacent. A complete graph on n vertices is denoted by K_n. The complement of K_n is a null graph and is denoted by ${\overline{K}}_n.$ A clique is a maximal induced complete subgraph. The maximum size of a clique of a graph G is called the clique number of G and is denoted by $\omega (G).$ For a graph G, a stable set is a set of vertices, no two of which are adjacent. A stable (or independent) set in a graph is maximum if the graph contains no larger stable set. The cardinality of a maximum stable set in a graph G is called the stability number and is denoted by $\alpha (G).$ The girth of G, denoted by gr(G), is the length of a shortest cycle in G. (If G contains no cycles, $gr(G)=\infty$). Other notations and terminology can be seen in [10].

The adjacency matrix $A(G)=(a_{ij})$ of G is a square matrix of order n, where $a_{ij}=1\text{or}0$ according as $v_i\sim v_j$ or not. The eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ of A(G) are the eigenvalues of G. We denote the spectra (multiset of eigenvalues) of a square matrix A(G) by $\sigma (G)=\{{\lambda }_1^{s_1},\dots ,{\lambda }_m^{s_m}\},$ where s_j is the multiplicity of the eigenvalue of λ_j, for $1\le j\le m.$

Let R be a commutative ring with multiplicative identity $1\ne 0.$ A non-zero element $x\in R$ is called a zero-divisor of R if there exists a non-zero element $y\in R$ such that $xy=0.$ The zero-divisor graphs of commutative rings were first introduced by Beck [3], in the definition he included the additive identity and was interested mainly in coloring the zero divisor graph of a commutative ring. Later Anderson and Livingston [2] modified the definition of zero-divisor graphs and excluded the additive identity of the ring in the zero-divisor set. For a commutative ring R with identity, let the set Z(R) denote the set of zero-divisors and let $Z^{*}(R)=Z(R)\setminus \left\{0\right\}.$ The zero-divisor graph of R, denoted by $\Gamma (R),$ is a simple graph whose vertex set is $Z^{*}(R)$ and two vertices $u,v\in Z^{*}(R)$ are adjacent if and only if $uv=vu=0.$ We denote the ring of integers modulo n by ${\mathbb{Z}}_n.$ The order of the zero-divisor graph $\Gamma ({\mathbb{Z}}_n)$ is $n-\varphi (n)-1,$ where $\varphi$ is Euler’s totient function. The adjacency spectral analysis was done in [5, 12]. More literature about zero-divisor graphs can be found in [1, 2, 11] and the references therein. Recently, Magi et al. [8, 9] investigated the adjacency spectrum of $\Gamma ({\mathbb{Z}}_n)$ for $n=p^k$ and $n=p^2{q}^2,$ where p and q are distinct primes.

In Section 2, we obtain the adjacency spectrum of $\Gamma ({\mathbb{Z}}_{p^M{q}^N}),$ where p and q are distinct primes and M and N are positive integers. In Section 3, we obtain the clique number, stability number and the girth of $\Gamma ({\mathbb{Z}}_{p^M{q}^N}).$

2. Adjacency matrix and the spectrum of $\Gamma ({\mathbb{Z}}_{p^M{q}^N})$

To facilitate our discussion, we begin with the following known definitions and results, which will be used frequently in the proof of our main results.

For any two graphs G₁ and G₂ with disjoint vertex sets, the join $G_1\vee G_2$ of G₁ and G₂ is the graph obtained from the union of G₁ and G₂ by adding new edges from each vertex of G₁ to every vertex of G₂. The following is a generalization of the definition of join graph (see, [4] and [6]).

Definition 2.1.

Let G be a graph on k vertices with $V(G)=\{1,2,\dots ,n\}$ and G_i, $1\le i\le n$ be n pairwise disjoint graphs of order n_i, respectively. The G-generalized join graph $G[G_1,G_2,\dots ,G_n]$ is a graph formed by replacing each vertex i of G by the graph G_i and joining each vertex of G_i to every vertex of G_j whenever i and j are adjacent in G.

The following theorem was proved by Cardoso et al. [4, Theorem 5], in which the adjacency spectrum of a generalized join graph $G[G_1,G_2,\dots ,G_n]$ is expressed in terms of the adjacency spectrum of the graphs G_i and the spectrum of n × n matrix $C_A(G).$

Theorem 2.2.

[4] Let G be a graph with $V(G)=\{1,2,\dots ,n\}$ and let G_i, $1\le i\le n$, be n pairwise disjoint r_i-regular graphs of order n_i, respectively. Then the adjacency spectrum of $G=G[G_1,G_2,\dots ,G_n]$ is given by

$${\sigma }_A(G)=\left(\underset{i=1}{\overset{n}{\cup }}({\sigma }_A(G_i)\backslash \{r_i\})\right)\cup \sigma (C_A(G)),$$

where (2.1) $$C_A(G)={(c_{ij})}_{n\times n}=\{\begin{array}{cc}{r}_i\hfill & i=j,\hfill \\ \sqrt{n_i{n}_j}\hfill & ij\in E(G),\hfill \\ 0\hfill & \mathit{\text{otherwise}}.\hfill \end{array}$$(2.1)

For a positive integer n with canonical decomposition $n=p_1^{n_1}{p}_2^{n_2}\dots p_r^{n_r},$ where $p_1,p_2,\dots ,p_r$ are distinct primes and $\tau (n)$ is the number of positive factors of n, it is easy to see that $$\tau (n)=(n_1+1)(n_2+1)\dots (n_r+1)$$

The Euler’s totient function $\varphi (n)$ denotes the number of positive integers less than or equal to n and relatively prime to n. The following result gives some properties of Euler’s totient function.

Theorem 2.3.

[7] Let $\varphi$ be the Euler’s totient function. Then the following hold.

• $\varphi$ is multiplicative, i.e.,$\varphi (st)=\varphi (s)\varphi (t),$ whenever s and t are relatively prime.

• If n is a positive integer, then$\sum _{d|n}\varphi (d)=n,$ where$d|n$ denotes d is divisor of$n.$

• If p is a prime. Then${\sum }_{i=1}^l\varphi (p^l)=p^l-1.$

An integer d is called a $\mathit{proper}\mathit{divisor}$ of n if $1<d<n$ and $d|n.$ Let ${\Upsilon }_n$ be the simple graph with vertex set $\{d_1,d_2,\dots ,d_k\}$ as proper divisors of n, in which two distinct vertices are adjacent if and only if n divides $d_i{d}_j.$ It is easy to see that ${\Upsilon }_n$ is a connected graph [5]. If $p_1^{n_1}{p}_2^{n_2}\dots p_r^{n_r}$ is the canonical decomposition of n, then the order of ${\Upsilon }_n$ is given by $$|V({\Upsilon }_n)|=(n_1+1)(n_2+1)\dots (n_r+1)-2.$$ For $1\le i\le k,$ let ${\mathcal{A}}_{d_i}=\{x\in {\mathbb{Z}}_n:(x,n)=d_i\},$ where $(x,n)$ denotes the greatest common divisor of x and n. We observe that the sets ${\mathcal{A}}_{d_1},{\mathcal{A}}_{d_2},\dots ,{\mathcal{A}}_{d_k}$ are pairwise disjoint and partitions the vertex set of $\Gamma ({\mathbb{Z}}_n)$ as $V(\Gamma ({\mathbb{Z}}_n))={\mathcal{A}}_{d_1}\cup {\mathcal{A}}_{d_2}\cup \dots \cup {\mathcal{A}}_{d_k}.$ From the definition of ${\mathcal{A}}_{d_i},$ a vertex of ${\mathcal{A}}_{d_i}$ is adjacent to the vertex of ${\mathcal{A}}_{d_j}$ in $\Gamma ({\mathbb{Z}}_n)$ if and only if $n|d_i{d}_j,$ for $i,j\in \{1,2,\dots ,t\}$ (see; [5]). The following result can be found in [12], which gives the cardinality of ${\mathcal{A}}_{d_i}.$

Lemma 2.4.

Let d_i be a divisor of n. Then $\left|{\mathcal{A}}_{d_i}\right|=\varphi \left(\frac{n}{d_i}\right)$, for $1\le i\le k.$

The next lemma [5] shows that the induced subgraphs $\Gamma ({\mathcal{A}}_{d_i})$ of $\Gamma ({\mathbb{Z}}_n)$ are either cliques or null graphs.

Lemma 2.5.

Let n be a positive integer and d_i be its proper divisor. Then the following hold.

1. For $i\in \{1,2,\dots ,k\},$ the induced subgraph $\Gamma ({\mathcal{A}}_{d_i})$ of $\Gamma ({\mathbb{Z}}_n)$ on the vertex set ${\mathcal{A}}_{d_i}$ is either the complete graph $K_{\varphi \left(\frac{n}{d_i}\right)}$ or its complement ${\overline{K}}_{\varphi \left(\frac{n}{d_i}\right)}.$ Also, $\Gamma ({\mathcal{A}}_{d_i})$ is $K_{\varphi \left(\frac{n}{d_i}\right)}$ if and only $n|d_i^2.$

2. For $i,j\in \{1,2,\dots ,k\}$ with $i\ne j,$ a vertex of ${\mathcal{A}}_{d_i}$ is adjacent to either all or none of the vertices in ${\mathcal{A}}_{d_j}$ of $\Gamma ({\mathbb{Z}}_n).$

The following lemma says that $\Gamma ({\mathbb{Z}}_n)$ is a G-join of certain complete graphs and null graphs.

Lemma 2.6.

[5] Let $\Gamma ({\mathcal{A}}_{d_i})$ be the induced subgraph of $\Gamma ({\mathbb{Z}}_n)$ on the vertex set ${\mathcal{A}}_{d_i}$ for $1\le i\le k.$ Then $\Gamma ({\mathbb{Z}}_n)={{\rm Y}}_n[\Gamma ({\mathcal{A}}_{d_1}),\Gamma ({\mathcal{A}}_{d_2}),\dots ,\Gamma ({\mathcal{A}}_{d_k})].$

Now, we find the adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n),$ for $n=p^M{q}^N,$ where p and q, $p<q$ are primes. This generalizes the results obtained in [8, 9]. We will prove the case when M and N are positive even integers with $M\le N,$ the other cases can be obtained similarly.

Theorem 2.7.

Let $\Gamma ({\mathbb{Z}}_n)$ be the zero-divisor graph of order n, where $n=p^M{q}^N$ and $M=2m_1\le 2n_1=N$. The adjacency spectrum of $\Gamma ({\mathbb{Z}}_n)$ consists of the eigenvalues $\{0,-1\}$ with multiplicities $p^{M-1}{q}^{N-1}(p+q-1)-p^{m_1}{q}^{n_1}-(3m_1{n}_1+m_1+n_1-1)$ and $p^{m_1}{q}^{n_1}-(m_1+n_1+m_1{n}_1+1)$ respectively. The remaining adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ are the eigenvalues of the matrix given in(2.1)(2.1) $$C_A(G)={(c_{ij})}_{n\times n}=\{\begin{array}{cc}{r}_i\hfill & i=j,\hfill \\ \sqrt{n_i{n}_j}\hfill & ij\in E(G),\hfill \\ 0\hfill & \mathit{\text{otherwise}}.\hfill \end{array}$$(2.1) .

Proof.

Let $n=p^M{q}^N,$ with $2<p<q$ as primes and $2\le M=2m_1\le 2n_1=N$ be positive even integers. Then the proper divisors of n are

$$\begin{array}{c}\{p,p^2,\dots ,p^{m_1},\dots ,p^M,q,q^2,\dots ,q^{n_1},\dots ,q^N,pq,p{q}^2,\dots ,p{q}^{n_1},\dots ,p{q}^N,\dots ,p^{m_1}q,p^{m_1}{q}^2,\dots ,\\ p^{m_1}{q}^{n_1-1},p^{m_1}{q}^{n_1},\dots ,p^{m_1}{q}^N,\dots ,p^{M}q,p^M{q}^2,\dots ,p^M{q}^{n_1-1},p^M{q}^{n_1},\dots ,p^M{q}^{N-1}\}\end{array}$$

and the size of ${\Upsilon }_n$ is $(M+1)(N+1)-2=MN+M+N-1.$ By the definition of ${\Upsilon }_n,$ we see that $$\begin{array}{c}p\sim p^{M-1}{q}^N,\\ p^2\sim p^{M-2}{q}^N,p^{M-1}{q}^N,\\ p^3\sim p^{M-3}{q}^N,p^{M-2}{q}^N,p^{M-1}{q}^N,\\ ⋮\\ p^{m_1}\sim p^{m_1}{q}^N,p^{m_1+1}{q}^N,\dots ,p^{M-1}{q}^N,\\ ⋮\\ p^M\sim q^N,p{q}^N,p^2{q}^N,\dots ,p^{m_1}{q}^N,\dots ,p^{M-1}{q}^N.\end{array}$$ That is, $$p^i\sim p^j{q}^N,i+j\ge M,\text{for}i=1,2,\dots ,M.$$ Now, following the similar procedure, we have $$\begin{array}{c}{q}^i\sim p^M{q}^j,i+j\ge N,\text{for}i=1,2,\dots ,N,\\ p{q}^i\sim p^t{q}^j,i+j\ge N,\text{for}i=1,2,\dots ,N\text{and}t\ge 2m_1-1,\\ ⋮\\ p^{m_1}{q}^i\sim p^t{q}^j,i+j\ge N,\text{for}i=1,2,\dots ,N\text{and}t\ge m_1,\\ ⋮\\ p^M{q}^i\sim p^t{q}^j,i+j\ge N,\text{for}i=1,2,\dots ,N-1\text{and}t\ge 0.\end{array}$$ By Lemma 2.4, for $i=1,2,\dots ,M$ and $j=1,2,\dots ,N,$ we see that $$\begin{array}{c}\left|A_{p^i}\right|=\varphi (p^{M-i}{q}^N),\left|A_{q^j}\right|=\varphi (p^M{q}^{N-j}),\left|A_{p{q}^j}\right|=\varphi (p^{M-1}{q}^{N-j}),\dots ,\left|A_{p^{m_1}{q}^j}\right|=\varphi (p^{m_1}{q}^{N-j}),\\ \dots ,\left|A_{p^{M-1}{q}^j}\right|=\varphi (p{q}^{N-j})\end{array}$$ and $$\left|A_{p^M{q}^t}\right|=\varphi (q^{N-t}),\text{for}t=1,2,\dots ,N-1.$$

Also, by Lemma 2.5, we have (2.2) $$G_i=\{\begin{array}{cc}\Gamma (A_{p^i})={\overline{K}}_{\varphi (p^{M-i}{q}^N)},\hfill & 1\le i\le M,\hfill \\ \Gamma (A_{q^j})={\overline{K}}_{\varphi (p^M{q}^{N-j})},\hfill & 1\le j\le N,\hfill \\ \Gamma (A_{p^i{q}^j})={\overline{K}}_{\varphi (p^{M-i}{q}^{N-j})},\hfill & 1\le i\le m_1-1\text{and}1\le j\le N\hfill \\ \hfill & \text{or}{m}_1\le i\le M\text{and}1\le j\le n_1-1,\hfill \\ \Gamma (A_{p^i{q}^j})=K_{\varphi (p^{M-i}{q}^{N-j})},\hfill & m_1\le i\le M\text{and}{n}_1\le j\le N\hfill \end{array}$$(2.2) By using Lemma 2.6, the joined union of the zero-divisor graph $\Gamma ({\mathbb{Z}}_n)$ is given by $$\begin{array}{c}\Gamma ({\mathbb{Z}}_n)={\Upsilon }_n[{\overline{K}}_{\varphi (p^{M-1}{q}^N)},\dots ,{\overline{K}}_{\varphi (p^{m_1}{q}^N)},\dots ,{\overline{K}}_{\varphi (q^N)},{\overline{K}}_{\varphi (p^M{q}^{N-1})},\dots ,{\overline{K}}_{\varphi (p^M{q}^{n_1})},\dots ,{\overline{K}}_{\varphi (p^M)},\\ {\overline{K}}_{\varphi (p^{M-1}{q}^{N-1})},\dots ,{\overline{K}}_{\varphi (p^{M-1}{q}^{n_1})},\dots ,{\overline{K}}_{\varphi (p^{M-1})},\dots ,{\overline{K}}_{\varphi (p^{m_1}{q}^{N-1})},\dots ,{\overline{K}}_{\varphi (p^{m_1}{q}^{n_1+1})},\\ K_{\varphi (p^{m_1}{q}^{n_1})},\dots ,K_{\varphi (p^{m_1})},\dots ,{\overline{K}}_{\varphi (q^{N-1})},\dots ,{\overline{K}}_{\varphi (q^{n_1-1})},K_{\varphi (q^{n_1})},\dots ,K_{\varphi (q)}].\end{array}$$ It is well known that the zero-divisor graphs are of diameter at most three, so that $p^i\sim q^i$ if and only if i = j = n, otherwise $p^i\sim p^k{q}^n,i+k\ge n$ and $q^j\sim p^n{q}^h,j+h\ge n$ and finally $p^k{q}^n\sim p^n{q}^h,k\ge 1,h\ge 1.$ This implies that $d(p^i,q^j)=3,\text{if}1\le i,j\le n-1$ in ${\Upsilon }_n.$ Similarly the distance between other vertices is at-most 2. Moreover, it is to be noted that the eigenvalues of K_n are $\{{(n-1)}^1,{(-1)}^{n-1}\}$ and ${\overline{K}}_n$ has 0 as eigenvalue with multiplicity n. Using Theorem 2.2, to the above joined graph, we observe that the adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ is 0 with multiplicity $$\begin{array}{c}\sum _{i=1}^M(\varphi (p^{M-i}{q}^N)-1)+\sum _{j=1}^N(\varphi (p^M{q}^{N-j})-1)+\sum _{i=1}^{m_1-1}\sum _{j=1}^N(\varphi (p^{M-i}{q}^{N-j})-1)\\ +\sum _{i=m_1}^M\sum _{j=1}^{n_1-1}(\varphi (p^{M-i}{q}^{N-j})-1)\\ =p^{M-1}\varphi (q^N)-M+\varphi (p^M)q^{N-1}-N+\sum _{i=1}^{m_1-1}\varphi (p^{M-i})\sum _{j=1}^N\varphi (q^{N-j})\\ -(m_1-1)N+\sum _{i=m_1}^M\varphi (p^{M-i})\sum _{j=1}^{n_1-1}\varphi (q^{N-j})-(m_1+1)(n_1-1)\\ =p^{M-1}{q}^{N-1}(q-1)-M+p^{M-1}{q}^{N-1}(p-1)-N+(p^{M-1}-p^{m_1})q^{N-1}\\ -(m_1-1)N+p^{m_1}(q^{N-1}-q^{n_1})-(m_1+1)(n_1-1)\\ =p^{M-1}{q}^{N-1}(p+q-1)-p^{m_1}{q}^{n_1}-(M+N+(m_1-1)N+(m_1+1)(n_1-1))\\ =p^{M-1}{q}^{N-1}(p+q-1)-p^{m_1}{q}^{n_1}-(3m_1{n}_1+m_1+n_1-1)\end{array}$$ and –1 with multiplicity $$\begin{array}{c}\sum _{j=n_1}^N\varphi (p^{m_1}{q}^{N-j})+\sum _{j=n_1}^N\varphi (p^{m_1-1}{q}^{N-j})+\dots +\sum _{j=n_1}^N\varphi (p{q}^{N-j})+\sum _{j=n_1}^{N-1}\varphi (q^{N-j})-(m_1+n_1+m_1{n}_1)\\ =\varphi (p^{m_1})q^{n_1}+\varphi (p^{m_1-1})q^{n_1}+\dots +\varphi (p)q^{n_1}+q^{n_1}-1-(m_1+n_1+m_1{n}_1)\\ =(p^{m_1}-1)q^{n_1}+q^{n_1}-1-(m_1+n_1+m_1{n}_1)\\ =p^{m_1}{q}^{n_1}-(m_1+n_1+m_1{n}_1+1)\end{array}$$ By using the adjacency relations (2.5)(2.5) $$G_i=\{\begin{array}{cc}\Gamma (A_{p^i})={\overline{K}}_{\varphi (p^{M-i}{q}^N)},\hfill & 1\le i\le M,\hfill \\ \Gamma (A_{q^j})={\overline{K}}_{\varphi (p^M{q}^{N-j})},\hfill & 1\le j\le N,\hfill \\ \Gamma (A_{p^i{q}^j})={\overline{K}}_{\varphi (p^{M-i}{q}^{N-j})},\hfill & 1\le i\le m_1-1\text{and}1\le j\le N\hfill \\ \hfill & \text{or}{m}_1\le i\le M\text{and}1\le j\le n_1,\hfill \\ \Gamma (A_{p^i{q}^j})=K_{\varphi (p^{M-i}{q}^{N-j})},\hfill & m_1\le i\le M\text{and}{n}_1+1\le j\le N\hfill \end{array}$$(2.5) and value of r_i’s, the remaining adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ are the eigenvalues of the matrix given in (2.1)(2.1) $$C_A(G)={(c_{ij})}_{n\times n}=\{\begin{array}{cc}{r}_i\hfill & i=j,\hfill \\ \sqrt{n_i{n}_j}\hfill & ij\in E(G),\hfill \\ 0\hfill & \mathit{\text{otherwise}}.\hfill \end{array}$$(2.1) . □

In particular, if q = 1 in Theorem 2.7, we get the adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_{p^{2m}}).$

Corollary 2.8.

If $n=p^{2m}$ for some positive integer $m\ge 2$, then the adjacency spectrum of $\Gamma ({\mathbb{Z}}_n)$ consists of the eigenvalue 0 with multiplicity $p^{2m-1}-p^m-(m-1)$, the eigenvalue –1 with multiplicity $p^m-(m+1)$ and the remaining adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ are the zeros of the characteristic polynomial of the matrix(2.3)(2.3) $$\left(\begin{array}{ccccccccc}0& 0& \cdots & 0& 0& 0& \cdots & 0& y_{1,2m-1}\\ 0& 0& \cdots & 0& 0& 0& \cdots & y_{2,2m-2}& y_{2,2m-1}\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& \cdots & 0& 0& y_{m-1,m+1}& \cdots & y_{m-1,2m-2}& y_{m-1,2m-1}\\ 0& 0& \cdots & 0& r_m& y_{m,m+1}& \cdots & y_{m,2m-2}& y_{m,2m-1}\\ 0& 0& \cdots & y_{m+1,m-1}& y_{m+1,m}& r_{m+1}& \cdots & y_{m+1,2m-2}& y_{m+1,2m-1}\\ ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& y_{2m-2,2}& \cdots & y_{2m-2,m-1}& y_{2m-2,m}& y_{2m-2,m+1}& \cdots & r_{2m-2}& y_{2m-2,2m-1}\\ y_{2m-1,1}& y_{2m-1,2}& \cdots & y_{2m-1,m-1}& y_{2m-1,m}& y_{2m-1,m+1}& \cdots & y_{2m-1,2m-2}& r_{2m-1}\end{array}\right),$$(2.3) .

Proof.

We know that the proper divisors of $n=2m$ are $\{p,p^2,\dots ,p^{2m-1}\}$ and so by definition of ${\Upsilon }_{p^{2m}},$ the vertex pⁱ is adjacent to the vertex p^j if and only if $j\ge 2m-i$ with $1\le i\le 2m-1$ and $i\ne j.$ Using Lemma 2.5 and noting the fact that n does not divide ${(p^i)}^2,$ for $i=1,2,\dots ,m-1$ and n divides ${(p^i)}^2,$ for $i=m,m+1,\dots ,2m-2,2m-1,$ we have $$G_i=\{\begin{array}{cc}{\overline{K}}_{\varphi (p^{2m-i})}\hfill & 1\le i\le m-1,\hfill \\ K_{\varphi (p^{2m-i})}\hfill & m\le i\le 2m-1.\hfill \end{array}$$ By using Lemma 2.6, the joined union of the zero-divisor graph $\Gamma ({\mathbb{Z}}_n)$ is given by $$\Gamma ({\mathbb{Z}}_n)={\Upsilon }_n[{\overline{K}}_{\varphi (p^{2m-1})},\dots ,{\overline{K}}_{\varphi (p^{m+1})},K_{\varphi (p^m)},\dots ,K_{\varphi (p)}].$$ Using Theorem 2.2, the adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ are 0 and –1 with multiplicities as $$\sum _{i=1}^{m-1}(\varphi (p^{2m-i})-1)=p^{2m-1}-p^m-(m-1)$$ and $$\sum _{i=m}^{2m-1}(\varphi (p^{2m-i})-1)=p^m-1-m$$ respectively. Therefore, we have $${\sigma }_A(G)=\{0^{p^m(p^{m-1}-1)},(-1^{p^m-(m+1)})\}\cup \sigma (C_A(G)),$$ and the eigenvalues of matrix (2.3)(2.3) $$\left(\begin{array}{ccccccccc}0& 0& \cdots & 0& 0& 0& \cdots & 0& y_{1,2m-1}\\ 0& 0& \cdots & 0& 0& 0& \cdots & y_{2,2m-2}& y_{2,2m-1}\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& \cdots & 0& 0& y_{m-1,m+1}& \cdots & y_{m-1,2m-2}& y_{m-1,2m-1}\\ 0& 0& \cdots & 0& r_m& y_{m,m+1}& \cdots & y_{m,2m-2}& y_{m,2m-1}\\ 0& 0& \cdots & y_{m+1,m-1}& y_{m+1,m}& r_{m+1}& \cdots & y_{m+1,2m-2}& y_{m+1,2m-1}\\ ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& y_{2m-2,2}& \cdots & y_{2m-2,m-1}& y_{2m-2,m}& y_{2m-2,m+1}& \cdots & r_{2m-2}& y_{2m-2,2m-1}\\ y_{2m-1,1}& y_{2m-1,2}& \cdots & y_{2m-1,m-1}& y_{2m-1,m}& y_{2m-1,m+1}& \cdots & y_{2m-1,2m-2}& r_{2m-1}\end{array}\right),$$(2.3) . (2.3) $$\left(\begin{array}{ccccccccc}0& 0& \cdots & 0& 0& 0& \cdots & 0& y_{1,2m-1}\\ 0& 0& \cdots & 0& 0& 0& \cdots & y_{2,2m-2}& y_{2,2m-1}\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& \cdots & 0& 0& y_{m-1,m+1}& \cdots & y_{m-1,2m-2}& y_{m-1,2m-1}\\ 0& 0& \cdots & 0& r_m& y_{m,m+1}& \cdots & y_{m,2m-2}& y_{m,2m-1}\\ 0& 0& \cdots & y_{m+1,m-1}& y_{m+1,m}& r_{m+1}& \cdots & y_{m+1,2m-2}& y_{m+1,2m-1}\\ ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& y_{2m-2,2}& \cdots & y_{2m-2,m-1}& y_{2m-2,m}& y_{2m-2,m+1}& \cdots & r_{2m-2}& y_{2m-2,2m-1}\\ y_{2m-1,1}& y_{2m-1,2}& \cdots & y_{2m-1,m-1}& y_{2m-1,m}& y_{2m-1,m+1}& \cdots & y_{2m-1,2m-2}& r_{2m-1}\end{array}\right),$$(2.3) where $y_{i,j}=y_{j,i}=\sqrt{n_i{n}_j}=\sqrt{\varphi (p^{2m-i})\varphi (p^{2m-j})},\text{for}1\le i,j\le 2m-1\text{and}{r}_i=\varphi (p^{2m-i})-1,$ for $i=m,m+1,\dots ,2m-1.$

If $m_1=2$ and q = 1 in Theorem 2.7, we have $\Gamma ({\mathbb{Z}}_{p^2})=K_{\varphi (p)}$ and its adjacency spectrum is given by the following observation.

Corollary 2.9.

If $n=p^4$, then the adjacency spectrum of $\Gamma ({\mathbb{Z}}_n)$ is $$\{0^{p^3-p^2-1},{(-1)}^{p^2-3}\}.$$

The other eigenvalues are the roots of the characteristic polynomial $$-{\lambda }^3+(p^3-p-2){\lambda }^2+(p^4-p^3+p^2-p-1)\lambda -(p^6-3p^5+2p^4+p^3-p^2).$$

If $m_1=m_2=0,$ then $n=pq.$ So, by Lemmas 2.5 and 2.6, we have (2.4) $$\Gamma ({\mathbb{Z}}_{pq})={\Upsilon }_{pq}[\Gamma (A_p),\Gamma (A_q)]=K_2[{\overline{K}}_{\varphi (p)},{\overline{K}}_{\varphi (q)}]={\overline{K}}_{\varphi (p)}▽{\overline{K}}_{\varphi (q)}=K_{\varphi (p),\varphi (q)}.$$(2.4) The next consequence of Theorem 2.7 gives the adjacency spectrum of the complete bipartite graph $\Gamma ({\mathbb{Z}}_{pq}).$

Corollary 2.10.

The adjacency spectrum of $\Gamma ({\mathbb{Z}}_{pq})$ is

$$\{0^{p+q-4},\pm \sqrt{(p-1)(q-1)}\}.$$

The following result can be obtained by arguments similar to those used in the proof of Theorem 2.7, and therefore the proof is omitted.

Theorem 2.11.

Let $\Gamma ({\mathbb{Z}}_n)$ be the zero-divisor graph of order n, where $n=p^M{q}^N$ and $M=2m_1+1\le 2n_1+1=N$. The adjacency spectrum of $\Gamma ({\mathbb{Z}}_n)$ consists of the eigenvalues 0,-1 with multiplicities $p^{M-1}{q}^{N-1}(p+q-1)-p^{m_1}{q}^{n_1}-(3m_1{n}_1+3m_1+3n_1+2)$ and $p^{m_1}{q}^{n_1}-(m_1+n_1+m_1{n}_1+1)$ respectively. The remaining adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ are the eigenvalues of the matrix given in(2.1)(2.1) $$C_A(G)={(c_{ij})}_{n\times n}=\{\begin{array}{cc}{r}_i\hfill & i=j,\hfill \\ \sqrt{n_i{n}_j}\hfill & ij\in E(G),\hfill \\ 0\hfill & \mathit{\text{otherwise}}.\hfill \end{array}$$(2.1) .

Now, consider the case when M is even and N is odd or M is odd and N is even. In the following result, we discuss the first case and the second case can be treated similarly.

Theorem 2.12.

Let $\Gamma ({\mathbb{Z}}_n)$ be the zero-divisor graph of order n, where $n=p^M{q}^N$ and $M=2m_1\le 2n_1+1=N$. The adjacency spectrum of $\Gamma ({\mathbb{Z}}_n)$ consists of the eigenvalues 0,-1 with multiplicities $p^{M-1}{q}^{N-1}(p+q-1)-p^{m_1}{q}^{n_1}-(3m_1{n}_1+3m_1+n_1)$ and $p^{m_1}{q}^{n_1}-(m_1+n_1+m_1{n}_1+1)$ respectively. The remaining adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ are the eigenvalues of the matrix given in (2.1)(2.1) $$C_A(G)={(c_{ij})}_{n\times n}=\{\begin{array}{cc}{r}_i\hfill & i=j,\hfill \\ \sqrt{n_i{n}_j}\hfill & ij\in E(G),\hfill \\ 0\hfill & \mathit{\text{otherwise}}.\hfill \end{array}$$(2.1) .

Proof.

Let $n=p^M{q}^N,$ with $2<p<q$ as primes and $2\le M=2m_1\le 2n_1+1=N$ be positive even integers. Then the proper divisors of n are $$\begin{array}{c}\{p,p^2,\dots ,p^{m_1},\dots ,p^M,q,q^2,\dots ,q^{n_1+1},\dots ,q^N,pq,p{q}^2,\dots ,p{q}^{n_1+1},\dots ,p{q}^N,\dots ,p^{m_1}q,p^{m_1}{q}^2,\dots ,\\ p^{m_1}{q}^{n_1},p^{m_1}{q}^{n_1+1}\dots ,p^{m_1}{q}^N,\dots ,p^{M}q,p^M{q}^2,\dots ,p^M{q}^{n_1},p^M{q}^{n_1+1},\dots ,p^M{q}^{N-1}\}\end{array}$$ and the size of ${\Upsilon }_n$ is $(M+1)(N+1)-2=MN+M+N-1.$ By the definition of ${\Upsilon }_n,$ we see that the adjacency relations are $$\begin{array}{c}{p}^i\sim p^j{q}^N,i+j\ge M,\text{for}i=1,2,\dots ,M,\\ q^i\sim p^M{q}^j,i+j\ge N,\text{for}i=1,2,\dots ,N,\\ p{q}^i\sim p^t{q}^j,i+j\ge N,\text{for}i=1,2,\dots ,N\text{and}t\ge 2m_1-1,\\ ⋮\\ p^{m_1}{q}^i\sim p^t{q}^j,i+j\ge N,\text{for}i=1,2,\dots ,N\text{and}t\ge m_1\\ ⋮\\ p^M{q}^i\sim p^t{q}^j,i+j\ge N,\text{for}i=1,2,\dots ,N-1\text{and}t\ge 0.\end{array}$$ By Lemma 2.4, for $i=1,2,\dots ,M$ and $j=1,2,\dots ,N,$ and $t=1,2,\dots ,M_2-1,$ we see that $$\begin{array}{c}\left|A_{p^i}\right|=\varphi (p^{M-i}{q}^N),\left|A_{q^j}\right|=\varphi (p^M{q}^{N-j}),\left|A_{p{q}^j}\right|=\varphi (p^{M-1}{q}^{N-j}),\dots ,\left|A_{p^{m_1}{q}^j}\right|=\varphi (p^{m_1}{q}^{N-j}),\\ \dots ,\left|A_{p^{M-1}{q}^j}\right|=\varphi (p{q}^{N-j}),\left|A_{p^M{q}^t}\right|=\varphi (q^{N-t}).\end{array}$$ Also, by Lemma 2.5, we have (2.5) $$G_i=\{\begin{array}{cc}\Gamma (A_{p^i})={\overline{K}}_{\varphi (p^{M-i}{q}^N)},\hfill & 1\le i\le M,\hfill \\ \Gamma (A_{q^j})={\overline{K}}_{\varphi (p^M{q}^{N-j})},\hfill & 1\le j\le N,\hfill \\ \Gamma (A_{p^i{q}^j})={\overline{K}}_{\varphi (p^{M-i}{q}^{N-j})},\hfill & 1\le i\le m_1-1\text{and}1\le j\le N\hfill \\ \hfill & \text{or}{m}_1\le i\le M\text{and}1\le j\le n_1,\hfill \\ \Gamma (A_{p^i{q}^j})=K_{\varphi (p^{M-i}{q}^{N-j})},\hfill & m_1\le i\le M\text{and}{n}_1+1\le j\le N\hfill \end{array}$$(2.5) Thus, by Lemma 2.6, the joined union of $\Gamma ({\mathbb{Z}}_n)$ is $$\begin{array}{c}\Gamma ({\mathbb{Z}}_n)={\Upsilon }_n[{\overline{K}}_{\varphi (p^{M-1}{q}^N)},\dots ,{\overline{K}}_{\varphi (p^{m_1}{q}^N)},\dots ,{\overline{K}}_{\varphi (q^N)},{\overline{K}}_{\varphi (p^M{q}^{N-1})},\dots ,{\overline{K}}_{\varphi (p^M{q}^{n_1})},\dots ,{\overline{K}}_{\varphi (p^M)},\\ {\overline{K}}_{\varphi (p^{M-1}{q}^{N-1})},\dots ,{\overline{K}}_{\varphi (p^{M-1}{q}^{n_1})},\dots ,{\overline{K}}_{\varphi (p^{M-1})},\dots ,{\overline{K}}_{\varphi (p^{m_1}{q}^{N-1})},\dots ,{\overline{K}}_{\varphi (p^{m_1}{q}^{n_1})},\\ K_{\varphi (p^{m_1}{q}^{n_1-1})},\dots ,K_{\varphi (p^{m_1})},\dots ,K_{(\varphi q^{N-1})},\dots ,K_{\varphi (q^{n_1})},\dots ,K_{\varphi (q)}].\end{array}$$ Using Theorem 2.2, to the above joined graph, we find that the adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ is 0 with multiplicity $$\begin{array}{c}\sum _{i=1}^M(\varphi (p^{M-i}{q}^N)-1)+\sum _{j=1}^N(\varphi (p^M{q}^{N-j})-1)+\sum _{i=1}^{m_1-1}\sum _{j=1}^N(\varphi (p^{M-i}{q}^{N-j})-1)\\ +\sum _{i=m_1}^M\sum _{j=1}^{n_1}(\varphi (p^{M-i}{q}^{N-j})-1)\end{array}$$ $$\begin{array}{c}=p^{M-1}\varphi (q^N)-M+\varphi (p^M)q^{N-1}-N+\sum _{i=1}^{m_1-1}\varphi (p^{M-i})\sum _{j=1}^N\varphi (q^{N-j})-(m_1-1)N\\ +\sum _{i=m_1}^M\varphi (p^{M-i})\sum _{j=1}^{n_1}\varphi (q^{N-j})-(m_1+1)n_1\\ =p^{M-1}{q}^{N-1}(q-1)-M+p^{M-1}{q}^{N-1}(p-1)-N+(p^{M-1}-p^{m_1})q^{N-1}-(m_1-1)N\\ +p^{m_1}(q^{N-1}-q^{n_1})-(m_1+1)n_1\\ =p^{M-1}{q}^{N-1}(p+q-1)-p^{m_1}{q}^{n_1}-(M+N+(m_1-1)N+(m_1+1)n_1)\\ =p^{M-1}{q}^{N-1}(p+q-1)-p^{m_1}{q}^{n_1}-(3m_1{n}_1+3m_1+n_1)\end{array}$$ and –1 with multiplicity $$\begin{array}{c}\sum _{j=n_1+1}^N\varphi (p^{m_1}{q}^{N-j})+\sum _{j=n_1+1}^N\varphi (p^{m_1-1}{q}^{N-j})+\dots +\sum _{j=n_1+1}^N\varphi (p{q}^{N-j})+\sum _{j=n_1+1}^{N-1}\varphi (q^{N-j})\\ -(m_1+n_1+m_1{n}_1)\\ =\varphi (p^{m_1})q^{n_1}+\varphi (p^{m_1-1})q^{n_1}+\dots +\varphi (p)q^{n_1}+q^{n_1}-1-(m_1+n_1+m_1{n}_1)\\ =(p^{m_1}-1)q^{n_1}+q^{n_1}-1-(m_1+n_1+m_1{n}_1)\\ =p^{m_1}{q}^{n_1}-(m_1+n_1+m_1{n}_1+1)\end{array}$$ By using the adjacency relations (2.5)(2.5) $$G_i=\{\begin{array}{cc}\Gamma (A_{p^i})={\overline{K}}_{\varphi (p^{M-i}{q}^N)},\hfill & 1\le i\le M,\hfill \\ \Gamma (A_{q^j})={\overline{K}}_{\varphi (p^M{q}^{N-j})},\hfill & 1\le j\le N,\hfill \\ \Gamma (A_{p^i{q}^j})={\overline{K}}_{\varphi (p^{M-i}{q}^{N-j})},\hfill & 1\le i\le m_1-1\text{and}1\le j\le N\hfill \\ \hfill & \text{or}{m}_1\le i\le M\text{and}1\le j\le n_1,\hfill \\ \Gamma (A_{p^i{q}^j})=K_{\varphi (p^{M-i}{q}^{N-j})},\hfill & m_1\le i\le M\text{and}{n}_1+1\le j\le N\hfill \end{array}$$(2.5) and value of r_i’s, the remaining adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ are the eigenvalues of the matrix given in (2.1)(2.1) $$C_A(G)={(c_{ij})}_{n\times n}=\{\begin{array}{cc}{r}_i\hfill & i=j,\hfill \\ \sqrt{n_i{n}_j}\hfill & ij\in E(G),\hfill \\ 0\hfill & \mathit{\text{otherwise}}.\hfill \end{array}$$(2.1) .□

In particular, if q = 1 in Theorem 2.11, we have the following observation.

Corollary 2.13.

If $n=p^{2m+1}$ for some positive integer $m\ge 2$, then the adjacency spectrum of $\Gamma ({\mathbb{Z}}_n)$ consists of the eigenvalue $\{0,-1\}$ with multiplicities $p^{2m}-p^m-m$ and $p^m-(m+1)$, the remaining adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ are the zeros of the characteristic polynomial of the matrix given in(2.6)(2.6) $$\left(\begin{array}{ccccccccc}0& 0& \cdots & 0& 0& 0& \cdots & 0& b_{1,2m}\\ 0& 0& \cdots & 0& 0& 0& \cdots & b_{2,2m-1}& b_{2,2m}\\ ⋮& ⋮& \ddots & \cdots & ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& \cdots & 0& 0& b_{m,m+1}& \cdots & b_{m,2m-1}& b_{m,2m}\\ 0& 0& \cdots & 0& a_{m+1}& b_{m+1,m+1}& \cdots & b_{m+1,2m-1}& b_{m,2m}\\ 0& 0& \cdots & b_{m+2,m}& b_{m+2,m+1}& a_{m+2}& \cdots & b_{m+2,2m-1}& b_{m+2,2m}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& b_{2m-1,2}& \cdots & b_{2m-1,m}& b_{2m-1,m+1}& b_{2m-1,m+2}& \cdots & a_{2m-1}& b_{2m-1,2m}\\ b_{2m,1}& b_{2m,2}& \cdots & b_{2m,m}& b_{2m,m+1}& b_{2m,m+2}& \cdots & b_{2m,2m-1}& a_{2m}\end{array}\right),$$(2.6) .

Proof.

The proper divisors of $n=2m+1$ are $\{p,p^2,\dots ,p^{2m-1},p^{2m}\}$ and so by definition of ${\Upsilon }_{p^{2m+1}},$ the vertex pⁱ is adjacent to the vertex p^j if and only if $j\ge 2m+1-i$ with $1\le i\le 2m$ and $i\ne j.$ Using Lemma 2.5 and noting the fact that n does not divide ${(p^i)}^2,$ for $i=1,2,\dots ,m-1,m$ and n divides ${(p^i)}^2,$ for $i=m+1,\dots ,2m-1,2m,$ we have

$$G_i=\{\begin{array}{cc}{\overline{K}}_{\varphi (p^{2m+1-i})}\hfill & 1\le i\le m,\hfill \\ K_{\varphi (p^{2m+1-i})}\hfill & m+1\le i\le 2m.\hfill \end{array}$$

By using Lemma 2.6, the joined union of the zero-divisor graph $\Gamma ({\mathbb{Z}}_n)$ is given by $$\Gamma ({\mathbb{Z}}_n)={\Upsilon }_n[{\overline{K}}_{\varphi (p^{2m})},\dots ,{\overline{K}}_{\varphi (p^{m+1})},K_{\varphi (p^m)},\dots ,K_{\varphi (p)}].$$ Using Theorem 2.2, the adjacency eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ are 0 and –1 with respective multiplicity as $$\sum _{i=1}^m(\varphi (p^{2m+1-i})-1)=p^{2m}-p^m-m$$ and $$\sum _{i=m+1}^{2m}(\varphi (p^{2m+1-i})-1)=p^m-1-m$$

Therefore, we have $${\sigma }_A(G)=\{0^{p^{2m}-p^m-m},-1^{p^m-(m+1)}\}\cup \sigma (C_A(G)),$$ and the eigen values of (2.6)(2.6) $$\left(\begin{array}{ccccccccc}0& 0& \cdots & 0& 0& 0& \cdots & 0& b_{1,2m}\\ 0& 0& \cdots & 0& 0& 0& \cdots & b_{2,2m-1}& b_{2,2m}\\ ⋮& ⋮& \ddots & \cdots & ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& \cdots & 0& 0& b_{m,m+1}& \cdots & b_{m,2m-1}& b_{m,2m}\\ 0& 0& \cdots & 0& a_{m+1}& b_{m+1,m+1}& \cdots & b_{m+1,2m-1}& b_{m,2m}\\ 0& 0& \cdots & b_{m+2,m}& b_{m+2,m+1}& a_{m+2}& \cdots & b_{m+2,2m-1}& b_{m+2,2m}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& b_{2m-1,2}& \cdots & b_{2m-1,m}& b_{2m-1,m+1}& b_{2m-1,m+2}& \cdots & a_{2m-1}& b_{2m-1,2m}\\ b_{2m,1}& b_{2m,2}& \cdots & b_{2m,m}& b_{2m,m+1}& b_{2m,m+2}& \cdots & b_{2m,2m-1}& a_{2m}\end{array}\right),$$(2.6) , (2.6) $$\left(\begin{array}{ccccccccc}0& 0& \cdots & 0& 0& 0& \cdots & 0& b_{1,2m}\\ 0& 0& \cdots & 0& 0& 0& \cdots & b_{2,2m-1}& b_{2,2m}\\ ⋮& ⋮& \ddots & \cdots & ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& \cdots & 0& 0& b_{m,m+1}& \cdots & b_{m,2m-1}& b_{m,2m}\\ 0& 0& \cdots & 0& a_{m+1}& b_{m+1,m+1}& \cdots & b_{m+1,2m-1}& b_{m,2m}\\ 0& 0& \cdots & b_{m+2,m}& b_{m+2,m+1}& a_{m+2}& \cdots & b_{m+2,2m-1}& b_{m+2,2m}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& b_{2m-1,2}& \cdots & b_{2m-1,m}& b_{2m-1,m+1}& b_{2m-1,m+2}& \cdots & a_{2m-1}& b_{2m-1,2m}\\ b_{2m,1}& b_{2m,2}& \cdots & b_{2m,m}& b_{2m,m+1}& b_{2m,m+2}& \cdots & b_{2m,2m-1}& a_{2m}\end{array}\right),$$(2.6) where, $b_{i,j}=b_{j,i}=\sqrt{n_i{n}_j},\text{for}1\le i,j\le 2m\text{and}{r}_i=\varphi (p^{2m+1-i})-1,$ for $i=m+1,m+2\dots ,2m.$ □

For $n=p^3,$ zero-divisor graph is $$\Gamma ({\mathbb{Z}}_{p^3})={\Upsilon }_{p^3}[\Gamma (A_p),\Gamma (A_{p^2})]=K_2[{\overline{K}}_{\varphi (p^2)},{\overline{K}}_{\varphi (p)}]={\overline{K}}_{p(p-1)}▽K_{p-1},$$ which is the complete split graph with independence number $p^2-p$ and clique number p – 1. The next observation of Theorem 2.12 gives the adjacency spectra of $\Gamma ({\mathbb{Z}}_{p^3}).$

Corollary 2.14.

If $n=p^3$, then the adjacency spectrum of $\Gamma ({\mathbb{Z}}_n)$ is $$\{0^{p(p-1)},{(-1)}^{p-2}\}\cup \sigma \left(\left(\begin{array}{cc}0& \sqrt{(p-1)(q-1)}\\ \sqrt{(p-1)(q-1)}& 0\end{array}\right)\right)$$

Additional information

Funding

The research of S. Pirzada is supported by the SERB-DST research project number MTR/2017/000084. Also the research of Bilal Ahmad Wani is supported by Dr. D.S. Kothari Post-Doctoral Fellowship Scheme Award Letter No. F.4-2/2006 (BSR)/MA/18-19/0037.