On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Abstract

Given a commutative ring R with identity $1\ne 0$, let the set $Z\left(R\right)$ denote the set of zero-divisors and let $Z^{*}\left(R\right)=Z\left(R\right)\backslash \left\{0\right\}$ be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by $\mathrm{\Gamma }\left(R\right)$, is a simple graph whose vertex set is $Z^{*}\left(R\right)$ and each pair of vertices in $Z^{*}\left(R\right)$ are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ for $n=p^{N_1}{q}^{N_2}$, where $p<q$ are primes and $N_1,N_2$ are positive integers.

1. Introduction

All graphs considered in the present article are connected, undirected, simple and finite. A graph is denoted by $G=G\left(V\right(G),E(G\left)\right)$, where $V\left(G\right)$ is the vertex set and $E\left(G\right)$ is the edge set of G. The order and the size of G are the cardinalities of $V\left(G\right)$ and $E\left(G\right)$, respectively. The neighborhood of a vertex $v,$ denoted by $N\left(v\right)$, is the set of vertices of G adjacent to v. The degree of v, denoted by $d_v$, is the cardinality of $N\left(v\right)$. A graph G is called r-regular if degree of every vertex is r. The adjacency matrix $A\left(G\right)=\left(a_{ij}\right)$ of G is a square matrix of order n, whose $(i,j)$-entry is 1, if $v_i$ and $v_j$ are adjacent and is 0, otherwise. Let $D\left(G\right)=Diag(d_1,d_2,\cdots ,d_n)$ be the diagonal matrix, where $d_i$ are the degrees of the vertices of G. The matrix $L\left(G\right)=D\left(G\right)-A\left(G\right)$ is the Laplacian matrix and its eigenvalues with multiplicities is known as the Laplacian spectrum of G. This matrix is real symmetric and positive semi-definite matrix, so the eigenvalues can be ordered as ${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_n$. Also, we note that each row (column) sum is zero, so 0 is the Laplacian eigenvalue of G. Furthermore, it is well known that the Laplacian eigenvalue ${\mu }_{n-1}$ is positive if and only if G is connected and is known as the algebraic connectivity of $G.$ More about the matrix $L\left(G\right)$ can be seen in [1,2].

Let R be a commutative ring with non-zero identity. An element $x\in R$, $x\ne 0$, is known as the zero-divisor of R if we can find $y\in R$, $y\ne 0$, such that $xy=0.$ Beck [3] introduced the concept of the zero-divisor graphs of commutative rings and included 0 in the definition. He was mainly interested in colorings of these rings. Later Anderson and Livingston [4] modified the definition of the zero-divisor graphs by excluding 0 of the ring in the zero-divisor set and defined the edges between two non-zero zero-divisors if and only if their product is zero. The adjacency, the Laplacian, the signless Laplacian, distance Laplacian and the signless Laplacian spectral analysis can be seen in [5,6,7,8,9,10,11]. More literature about zero-divisor graphs can be seen in [4,12,13,14] and the references therein.

In G, $x\sim y$ denotes that the vertices x and y are adjacent and $xy$ denotes an edge. We use the standard notation, for $K_n$ and $K_{a,b}$ respectively denote the complete graph and the complete bipartite graph. Other undefined notations and terminology can be seen in [1,15].

The remaining part of the paper is organized as follows. In Section 2, we present some preliminaries and investigate the structure of $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{N_1}{q}^{N_2}}\right)$ and discuss some of its graph invariants. In Section 3, we obtain the Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{N_1}{q}^{N_2}}\right)$, for $n=p^{N_1}{q}^{N_2}$, where p and q are primes. We deduce several consequences from these results, which include the determination of the eigenvalues of the graphs $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{2m}}\right)$, $\mathrm{\Gamma }\left({\mathbb{Z}}_{2^m}\left[i\right]\right)$ (zero-divisor graph of Gaussian integers modulo $2^m$), $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{2m+1}}\right)$, $\mathrm{\Gamma }\left({\mathbb{Z}}_{pq}\right)$ and $\mathrm{\Gamma }\left({\mathbb{Z}}_{pqr}\right)$. At the end of the article, we give the conclusion and discussion for possible further work.

2. Structure of the Zero-Divisor Graph $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{N_1}{q}^{N_2}}\right)$

We begin with the following definition.

Definition 1.

Let G be a graph of order n with vertex set $\{1,2,\cdots ,k\}$ and $G_i$ be disjoint graphs of order $n_i$, $1\le i\le k.$ The graph $G[G_1,G_2,\cdots ,G_n]$ is formed by taking the graphs $G_1,G_2,\cdots ,G_n$ and joining each vertex of $G_i$ to every vertex of $G_j$ whenever i and j are adjacent in $G.$

This graph operation is known by different names in the literature, such as G-join, generalized composition, generalized join, joined union, and here we follow the latter name.

Let n be a positive integer and let $\tau \left(n\right)$ denote the number of positive factors of n. Please note that $d|n$ denotes d divides n. The Euler’s totient function or Euler’s phi function, denoted by $\varphi \left(n\right)$, is the number of positive integers less or equal to n and relatively prime to n. We say that n is in canonical decomposition if $n=p_1^{n_1}{p}_2^{n_2}\cdots p_l^{n_l}$, where $l,n_1,n_2,\cdots ,n_l$ are positive integers and $p_1,p_2,\cdots ,p_l$ are distinct primes.

The following observations will be used in the sequel.

Lemma 1

([16]). If n is in canonical decomposition $p_1^{n_1}{p}_2^{n_2}\cdots p_r^{n_r}$, then

$$\tau \left(n\right)=(n_1+1)(n_2+1)\cdots (n_r+1)$$

Theorem 1

([16]). The Euler’s totient function ϕ satisfies the following.

(i) 

ϕ is multiplicative, i.e., $\varphi \left(pq\right)=\varphi \left(p\right)\varphi \left(q\right),$ whenever p and q are relatively prime.

(ii) 

${\displaystyle \sum _{d|n}}\varphi \left(d\right)=n.$

(iii) 

For prime p, ${\displaystyle \sum _{i=1}^l}\varphi \left(p^l\right)=p^l-1.$

For positive integer n, ${\mathbb{Z}}_n$ represents the set of congruence classes $\{\overline{0},\overline{1},\cdots ,\overline{n-1}\}$ of integer modulo $n.$ The ring of Gaussian integers modulon, denoted by ${\mathbb{Z}}_n\left[i\right]$, is represented by ${\mathbb{Z}}_n\left[i\right]=\{\overline{a}+i\overline{b}:\overline{a},\overline{b}\in {\mathbb{Z}}_n\}$.

An integer d dividing n is a proper divisor of n if and only if $1<d<n.$ Let ${\mathrm{{\rm Y}}}_n$ be the simple graph with vertex set as the proper divisor set $\{d_1,d_2,\cdots ,d_t\}$ of n, where two vertices are adjacent provided $d_i{d}_j$ is a multiple of n. Evidently, this graph is a connected graph [5]. If $p_1^{n_1}{p}_2^{n_2}\cdots p_r^{n_r}$ is the canonical decomposition of n, by Lemma 1, it follows that the order of ${\mathrm{{\rm Y}}}_n$ is given by

$$|V\left({\mathrm{{\rm Y}}}_n\right)|=(n_1+1)(n_2+1)\cdots (n_r+1)-2.$$

For $1\le i\le t$, let $A_{d_i}=\{r\in {\mathbb{Z}}_n:(r,n)=d_i\},$ where $(r,n)$ is the greatest common divisor of r and $n.$ We observe that $A_{d_i}\cap A_{d_j}=\varphi$, when $i\ne j$, so, the sets $A_{d_1},A_{d_2},\cdots ,A_{d_t}$ are pairwise disjoint and partitions the vertex set of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ as $V\left(\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)\right)=A_{d_1}\cup A_{d_2}\cup \cdots \cup A_{d_t}.$ From the definition of $A_{d_i}$, a vertex of $A_{d_i}$ is adjacent [5] to the vertex of $A_{d_j}$ in $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ provided that $n|d_i{d}_j$, for $i,j\in \{1,2,\cdots ,t\}$. The cardinality of $A_{d_i}$ is given as follows.

Lemma 2

([11]). For a divisor d of n, the cardinality of the set $A_d$ is equal to $|A_d|=\varphi \left(\frac{n}{d_i}\right)$.

We note that that the induced subgraphs $\mathrm{\Gamma }\left(A_{d_i}\right)$ of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ are either cliques or null graphs, as can be seen below [5].

Lemma 3.

For the positive integer n and its proper $d_i$, the following hold.

(i) 

If $i\in \{1,2,\cdots ,t\},$ then the subgraph $\mathrm{\Gamma }\left(A_{d_i}\right)$ of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ on $A_{d_i}$ 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, $\mathrm{\Gamma }\left(A_{d_i}\right)$ is $K_{\varphi \left(\frac{n}{d_i}\right)}$ provided $d_i^2$ is a multiple of $n.$

(ii) 

For distinct $i,j$ in $\{1,2,\cdots ,t\}$, a vertex of $A_{d_i}$ is adjacent to all $A_{d_j}$ or none of the vertices in $A_{d_j}$.

(iii) 

For distinct $i,j$ in $\{1,2,\cdots ,t\}$, a vertex of $A_{d_i}$ is adjacent to a vertex of $A_{d_j}$ in $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ provided $d_i{d}_j$ is a multiple of n.

The graph formed in part (iii) of Lemma 3 is known as $\mathcal{G}\left(A\left(d_i\right)\right)$ graph. Clearly, $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ can be expressed as a joined union of complete graphs and empty graphs.

Lemma 4.

[5] For the induced subgraph $\mathrm{\Gamma }\left(A_{d_i}\right)$ of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ on vertices $A_{d_i}$ for $1\le i\le t$, the zero-divisor graph $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)={\mathrm{{\rm Y}}}_n[\mathrm{\Gamma }\left(A_{d_1}\right),\mathrm{\Gamma }\left(A_{d_2}\right),\cdots ,\mathrm{\Gamma }\left(A_{d_t}\right)].$

For a commutative ring R with non-zero identity $1\ne 0$, and $a\in R$, the annihilator of a, denoted by $ann\left(a\right)$, is the set of those elements of R that annihilates a, and we write $ann\left(a\right)=\{b\in R:ab=0\}$. Define a relation on R by $a\sim b$ whenever $ann\left(a\right)=ann\left(b\right)$. Obviously, $ann\left(a\right)=ann\left(a\right)$ and if $ann\left(a\right)=ann\left(b\right)$ then $ann\left(b\right)=ann\left(a\right)$ implying that ∼ is symmetric relation. Also, if $ann\left(a\right)=ann\left(b\right)$ and $ann\left(b\right)=ann\left(c\right)$, then ∼ is transitive and is an equivalence relation on R which partitions R into equivalence classes. Furthermore, $\left[a\right]$ represents the class of $a\in R$, that is, $\left[a\right]=\{b\in R:ann(a)=ann(b\left)\right\}$.

The compressed zero-divisor graph of a commutative ring R, denoted by ${\mathrm{\Gamma }}_E\left(R\right)$, is the undirected, simple graph with the vertex set $Z\left(R_E\right)-\left\{\left[0\right]\right\}$ = $R_E-\{\left[0\right],\left[1\right]\}$ and is defined by $R_E=\{\left[a\right]:a\in R\}$, where $\left[a\right]=\{b\in R:ann(a)=ann(b\left)\right\}$ and the two vertices $\left[a\right]$ and $\left[b\right]$ are adjacent provided $\left[a\right]\left[b\right]=\left[0\right]=\left[ab\right]$. This graph was first defined in [17] and their properties for ${\mathbb{Z}}_{p^n}$ were investigated in [13].

For example, consider ${\mathbb{Z}}_{12}$ with non-zero zero-divisor set $\left\{2,3,4,6,8,9,10\right\}.$ The annihilators of this set are

$$\begin{array}{cc}\hfill ann\left(2\right)=\left\{6\right\},& ann\left(3\right)=\{4,8\},ann\left(4\right)=\{3,6,9\},ann\left(6\right)=\{2,4,6,8,10\},\hfill \\ \hfill & ann\left(8\right)=\{3,6,9\},ann\left(9\right)=\{4,8\},ann\left(10\right)=\left\{6\right\}.\hfill \end{array}$$

The compressed zero-divisor graph ${\mathbb{Z}}_{12}$ with the vertex set $\left\{\left[2\right],\left[3\right],\left[4\right],\left[6\right]\right\}$ and the proper divisor graph of ${\mathbb{Z}}_{12}$ with the vertex set $\left\{2,3,4,6\right\}$ are shown in Figure 1.

In case of $R={\mathbb{Z}}_n$, we observe that the vertex sets of $R_E,\mathcal{G}\left(A\left(d_i\right)\right)$ and ${\mathrm{{\rm Y}}}_n$ are in one-one correspondence.

Proposition 1.

If ${\mathbb{Z}}_n$ is the finite commutative ring, then ${\mathrm{\Gamma }}_E\left(R\right)\cong \mathcal{G}\left(A\left(d_i\right)\right)\cong {\mathrm{{\rm Y}}}_n.$

Now, we find the structure of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$, for $n=p^{N_1}{q}^{N_2}$, where p and q, $p<q$, are primes. This generalizes the results obtained in [13]. We prove the cases when $N_1$ and $N_2,N_1\le N_2,$ are positive even integers, $N_1$ and $N_2$ are positive odd integers and the other possible cases can be similarly proved.

Theorem 2.

Let $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ be the zero-divisor graph of order N, where $n=p^{N_1}{q}^{N_2}$ and $N_1=2m_1\le 2m_2=N_2$. Then

$$\begin{array}{cc}\hfill \mathrm{\Gamma }\left({\mathbb{Z}}_n\right)={\mathrm{{\rm Y}}}_n[& {\overline{K}}_{\varphi \left(p^{N_1-1}{q}^{N_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{m_1}{q}^{N_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(q^{N_2}\right)},{\overline{K}}_{\varphi \left(p^{N_1}{q}^{N_2-1}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{N_1}{q}^{m_2}\right)},\hfill \\ \hfill & \cdots ,{\overline{K}}_{\varphi \left(p^{N_1}\right)},{\overline{K}}_{\varphi \left(p^{N_1-1}{q}^{N_2-1}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{N_1-1}{q}^{m_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{N_1-1}\right)},\cdots ,\hfill \\ \hfill & {\overline{K}}_{\varphi \left(p^{m_1}{q}^{N_2-1}\right)},\cdots ,K_{\varphi \left(p^{m_1}{q}^{m_2-1}\right)},K_{\varphi \left(p^{m_1}{q}^{m_2}\right)},\cdots ,K_{\varphi \left(p^{m_1}\right)},\cdots ,{\overline{K}}_{\varphi \left(q^{N_2-1}\right)},\cdots ,\hfill \\ \hfill & {\overline{K}}_{\varphi \left(q^{m_2-1}\right)},K_{\varphi \left(q^{m_2}\right)},\cdots ,K_{\varphi \left(q\right)}].\hfill \end{array}$$

(1)

Proof. 

Let $n=p^{N_1}{q}^{N_2}$, where p and q, $2<p<q$, are primes and $N_1$ and $N_2$, $2\le N_1=2m_1\le 2m_2=N_2$, are positive even integers. The proper divisors of n are

$$\begin{array}{cc}\hfill \{p,p^2,\cdots ,& p^{m_1},\cdots ,p^{N_1},q,q^2,\cdots ,q^{m_2},\cdots ,q^{N_2},pq,p{q}^2,\cdots ,p{q}^{m_2},\cdots ,p{q}^{N_2},\cdots ,p^{m_1}q,p^{m_1}{q}^2,\cdots ,\hfill \\ \hfill & p^{m_1}{q}^{m_2-1},p^{m_1}{q}^{m_2},\cdots ,p^{m_1}{q}^{N_2},\cdots ,p^{N_1}q,p^{N_1}{q}^2,\cdots ,p^{N_1}{q}^{m_2-1},p^{N_1}{q}^{m_2},\cdots ,p^{N_1}{q}^{N_2-1}\}.\hfill \end{array}$$

By Lemma 1, order of ${\mathrm{{\rm Y}}}_n$ is $(N_1+1)(N_2+1)-2=N_1{N}_2+N_1+N_2-1$. From the definition of ${\mathrm{{\rm Y}}}_n$, we have

$$\begin{array}{cc}\hfill p& \sim p^{N_1-1}{q}^{N_2},p^2\sim p^{N_1-2}{q}^{N_2},p^{N_1-1}{q}^{N_2},\cdots ,p^{m_1}\sim p^{m_1}{q}^{N_2},p^{m_1+1}{q}^{N_2},\cdots ,p^{N_1-1}{q}^{N_2},\hfill \\ \hfill & \cdots p^{N_1}\sim q^{N_2},p{q}^{N_2},p^2{q}^{N_2},\cdots ,p^{m_1}{q}^{N_2},\cdots ,p^{N_1-1}{q}^{N_2},\hfill \end{array}$$

which in iteration form can be read as

$$p^i\sim p^j{q}^{N_2},i+j\ge N_1,\mathrm{for}i=1,2,\cdots ,N_1.$$

Arguing as above, other adjacency relations are

$$\begin{array}{cc}\hfill q^i& \sim p^{N_1}{q}^j,i+j\ge N_2,\mathrm{for}i=1,2,\cdots ,N_2,\hfill \\ \hfill p{q}^i& \sim p^k{q}^j,i+j\ge N_2,\mathrm{for}i=1,2,\cdots ,N_2\mathrm{and}k\ge 2m_1-1,\hfill \\ \hfill & ⋮\hfill \\ \hfill p^{m_1}{q}^i& \sim p^k{q}^j,i+j\ge N_2,\mathrm{for}i=1,2,\cdots ,N_2\mathrm{and}k\ge m_1,\hfill \\ \hfill & ⋮\hfill \\ \hfill p^{N_1}{q}^i& \sim p^k{q}^j,i+j\ge N_2,\mathrm{for}i=1,2,\cdots ,N_2-1\mathrm{and}k\ge 0.\hfill \end{array}$$

For $i=1,2,\cdots ,N_1,j=1,2,\cdots ,N_2$ and $k=1,2,\cdots ,N_2-1$, by Lemma 2, the cardinalities of $A_{d_i}$ are

$$\begin{array}{cc}\hfill |A_{p^i}|=\varphi \left(p^{N_1-i}{q}^{N_2}\right),\left|A_{q^j}\right|=\varphi \left(p^{N_1}{q}^{N_2-j}\right),& |A_{p{q}^j}|=\varphi \left(p^{N_1-1}{q}^{N_2-j}\right),\cdots ,\left|A_{p^{m_1}{q}^j}\right|=\varphi \left(p^{m_1}{q}^{N_2-j}\right),\hfill \\ \hfill & \cdots ,|A_{p^{N_1-1}{q}^j}|=\varphi \left(p{q}^{N_2-j}\right),\left|A_{p^{N_1}{q}^k}\right|=\varphi \left(q^{N_2-k}\right).\hfill \end{array}$$

Also, by Lemma 3, the induced graphs $\mathrm{\Gamma }\left(A_{d_{p^i}}\right)$ are

$$G_i=\left\{\begin{array}{cc}\mathrm{\Gamma }\left(A_{d_{p^i}}\right)={\overline{K}}_{\varphi \left(p^{N_1-i}{q}^{N_2}\right)},\hfill & 1\le i\le N_1,\hfill \\ \mathrm{\Gamma }\left(A_{d_{q^j}}\right)={\overline{K}}_{\varphi \left(p^{N_1}{q}^{N_2-j}\right)},\hfill & 1\le j\le N_2,\hfill \\ \mathrm{\Gamma }\left(A_{d_{p^i{q}^j}}\right)={\overline{K}}_{\varphi \left(p^{N_1-i}{q}^{N_2-j}\right)},\hfill & 1\le i\le m_1-1\mathrm{and}1\le j\le N_2\hfill \\ \hfill & \mathrm{or}{m}_1\le i\le N_1\mathrm{and}1\le j\le m_2-1,\hfill \\ \mathrm{\Gamma }\left(A_{d_{p^i{q}^j}}\right)=K_{\varphi \left(p^{N_1-i}{q}^{N_2-j}\right)},\hfill & m_1\le i\le N_1\mathrm{and}{m}_2\le j\le N_2,\hfill \end{array}\right.$$

(2)

where we avoid $\mathrm{\Gamma }\left(A_{d_{p^{N_1}{q}^{N_2}}}\right)$ corresponding to the proper divisor $p^{N_1}{q}^{N_2}.$ Lastly, by Lemma 4, the structure of the zero-divisor graph of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ is as in Equation (1). This completes the proof.    □

In Theorem 2, taking $N_2=0$, we have the following consequence.

Corollary 1.

If $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ is the zero-divisor graph of order $N,$ where $n=p^{2m}$, then

$$\begin{array}{cc}\hfill \mathrm{\Gamma }\left({\mathbb{Z}}_n\right)=& {\mathrm{{\rm Y}}}_n\left[{\overline{K}}_{\varphi \left(p^{2m-1}\right)},{\overline{K}}_{\varphi \left(p^{2m-2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{m+1}\right)},K_{\varphi \left(p^m\right)},K_{\varphi \left(p^{m-1}\right)},\cdots ,K_{\varphi \left(p^2\right)},K_{\varphi \left(p\right)}\right].\hfill \end{array}$$

(3)

Proof. 

The proper divisors of $n=p^{2m}$ are $\{p,p^2,\cdots ,p^{m-1},p^m,p^{m+1},\cdots ,p^{2m-1}\}$. In ${\mathrm{{\rm Y}}}_{p^{2m}}$, vertex $p^i$ is adjacent to vertex $p^j$ if and only if $i+j\ge 2m$ with $1\le i\le 2m-1$ and to avoid loops, we assume $i\ne j.$ Also, n does not divide ${\left(p^i\right)}^2$, for $i=1,2,\cdots ,m-1$, so $G_i={\overline{K}}_{\varphi \left(p^{2m-i}\right)}$ and n divides ${\left(p^i\right)}^2$, for $i=m,m+1,\cdots ,2m-2,2m-1$, and thus $G_i=K_{\varphi \left(p^{2m-i}\right)}$. Thus, Equation (3) follows.    □

Another consequence gives the diameter of $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{2m_1}{q}^{2m_2}}\right)$.

Corollary 2.

The diameter of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ is 3 for $n=p^{2m_1}{q}^{2m_2}$, and is 2 if $m_2=0.$

Proof. 

In the proof of Theorem 2, we observe 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 $q^j\sim p^n{q}^h,j+h\ge n$. Lastly, $p^k{q}^n\sim p^n{q}^h,k\ge 1,h\ge 1$. This implies that $d(p^i,q^j)=3,\mathrm{if}1\le i,j\le n-1$ in ${\mathrm{{\rm Y}}}_n$. Similarly, from Corollary 1, distance in ${\mathrm{{\rm Y}}}_{p^{2m}}$ is at most $2.$    □

The following consequence gives the clique number of $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{2m_1}{q}^{2m_2}}\right).$

Corollary 3.

The clique number of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ is

$$\omega \left(\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)\right)=\left\{\begin{array}{cc}{p}^{m_1}{q}^{m_2}-1\hfill & ifn=p^{2m_1}{q}^{2m_2}\hfill \\ p^{m_1}-1\hfill & ifn=p^{2m}.\hfill \end{array}\right.$$

Proof. 

By the definition of ${\mathrm{{\rm Y}}}_{p^{2m_1}{q}^{2m_2}}$, we can easily see that $p^i{q}^j,i\ge m_1,j\ge m_2$ are the vertices of the clique of ${\mathrm{{\rm Y}}}_{p^{2m_1}{q}^{2m_2}}$ and the number of such vertices is

$$\underset{m_1}{\underbrace{m_2+1+m_2+1+\cdots +m_2+1}}+m_2=m_1(m_2+1)+m_2.$$

By Lemma 3, $\mathrm{\Gamma }\left(A_{d_i}\right)$ is $K_{\varphi \left(\frac{n}{d_i}\right)}$ if and only if n divides $d_i^2$, so that the clique number of $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{N_1}{q}^{N_2}}\right)$ is

$$\begin{array}{cc}\hfill & |\mathrm{\Gamma }\left(A_{p^{m_1}{q}^{m_2}}\right)|+|\mathrm{\Gamma }\left(A_{p^{m_1}{q}^{m_2-1}}\right)|+\cdots +|\mathrm{\Gamma }\left(A_{p^{m_1}q}\right)|+|\mathrm{\Gamma }\left(A_{p^{m_1}}\right)|\hfill \\ \hfill & +|\mathrm{\Gamma }\left(A_{p^{m_1-1}{q}^{m_2}}\right)|+|\mathrm{\Gamma }\left(A_{p^{m_1-1}{q}^{m_2-1}}\right)|+\cdots +|\mathrm{\Gamma }\left(A_{p^{m_1-1}q}\right)|+|\mathrm{\Gamma }\left(A_{p^{m_1-1}}\right)|\hfill \\ \hfill & ⋮\hfill \\ \hfill & +|\mathrm{\Gamma }\left(A_{p{q}^{m_2}}\right)|+|\mathrm{\Gamma }\left(A_{p{q}^{m_2-1}}\right)|+\cdots +|\mathrm{\Gamma }\left(A_{p{q}^2}\right)|+|\mathrm{\Gamma }\left(A_{pq}\right)|\hfill \\ \hfill & +|\mathrm{\Gamma }\left(A_{q^{m_2}}\right)|+|\mathrm{\Gamma }\left(A_{q^{m_2-1}}\right)|+\cdots +|\mathrm{\Gamma }\left(A_{q^2}\right)|+|\mathrm{\Gamma }\left(A_q\right)|\hfill \\ \hfill & =\varphi \left(p^{m_1}{q}^{m_2}\right)+\varphi \left(p^{m_1}{q}^{m_2-1}\right)+\cdots +\varphi \left(p^{m_1}q\right)+\varphi \left(p^{m_1}\right)\hfill \\ \hfill & +\varphi \left(p^{m_1-1}{q}^{m_2}\right)+\varphi \left(p^{m_1-1}{q}^{m_2-1}\right)+\cdots +\varphi \left(p^{m_1-1}q\right)+\varphi \left(p^{m_1-1}\right)\hfill \\ \hfill & ⋮\hfill \\ \hfill & +\varphi \left(p{q}^{m_2}\right)+\varphi \left(p{q}^{m_2-1}\right)+\cdots +\varphi \left(pq\right)+\varphi \left(p\right)\hfill \\ \hfill & +\varphi \left(q^{m_2}\right)+\varphi \left(q^{m_2-1}\right)+\cdots +\varphi \left(q^2\right)+\varphi \left(q\right)\hfill \\ \hfill & =\varphi \left(p^{m_1}\right)q^{m_2}+\varphi \left(p^{m_1-1}\right)q^{m_2}+\cdots +\varphi \left(p\right)q^{m_2}+q^{m_2}-1=p^{m_1}{q}^{m_2}-1.\hfill \end{array}$$

If $m_2=0$, then by definition of ${\mathrm{{\rm Y}}}_{p^{2m}}$, the vertices $p^i,i\ge m$, form the clique in it and its size is m. Thus, the sum cardinality of the cardinalities $\mathrm{\Gamma }\left(A_{p^i}\right),i\ge m,$ is the clique size of $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{2m}}\right)$. Using Lemma 1, we have

$$\begin{array}{cc}\hfill cl\left(\mathrm{\Gamma }\left(A_{p^i}\right)\right)=& |\mathrm{\Gamma }\left(A_{p^m}\right)|+|\mathrm{\Gamma }\left(A_{p^{m+1}}\right)|+\cdots +|\mathrm{\Gamma }\left(A_{p^{2m-1}}\right)|\hfill \\ \hfill =& \varphi \left(p^m\right)+\varphi \left(p^{m-1}\right)+\cdots +\varphi \left(p\right)=p^m-1.\hfill \end{array}$$

□

The following result gives the structure of $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{N_1}{q}^{N_2}}\right)$, where $N_1$ and $N_2$ are both odd.

Theorem 3.

Let $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ be the zero-divisor graph of order N where $n=p^{N_1}{q}^{N_2}$ and $N_1=2m_1+1\le 2m_2+1=N_2$. Then

$$\begin{array}{cc}\hfill \mathrm{\Gamma }\left({\mathbb{Z}}_n\right)={\mathrm{{\rm Y}}}_n[& {\overline{K}}_{\varphi \left(p^{2m_1}{q}^{N_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{m_1}{q}^{N_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(q^{N_2}\right)},{\overline{K}}_{\varphi \left(p^{N_1}{q}^{2m_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{N_1}{q}^{m_2}\right)},\hfill \\ \hfill & \cdots ,{\overline{K}}_{\varphi \left(p^{N_1}\right)},{\overline{K}}_{\varphi \left(p^{2m_1}{q}^{2m_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{2m_1}{q}^{m_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{2m_1}\right)},\cdots ,\hfill \\ \hfill & {\overline{K}}_{\varphi \left(p^{m_1}{q}^{2m_2}\right)},\cdots ,K_{\varphi \left(p^{m_1}{q}^{m_2}\right)},K_{\varphi \left(p^{m_1}{q}^{m_2-1}\right)},\cdots ,K_{\varphi \left(p^{m_1}\right)},\cdots ,{\overline{K}}_{\varphi \left(q^{2m_2}\right)},\cdots ,\hfill \\ \hfill & {\overline{K}}_{\varphi \left(q^{m_2}\right)},K_{\varphi \left(q^{m_2-1}\right)},\cdots ,K_{\varphi \left(q\right)}].\hfill \end{array}$$

(4)

Proof. 

Let $n=p^{N_1}{q}^{N_2}$, where p and q, $2<p<q$, are primes and $N_1$ and $N_2$, $2\le N_1=2m_1+1\le 2m_2+1=N_2$, are positive even integers. Then the proper divisors of n are

$$\begin{array}{cc}\hfill \{p,p^2,\cdots ,& p^{m_1+1},\cdots ,p^{2m_1+1},q,q^2,\cdots ,q^{m_2+1},\cdots ,q^{2m_2+},pq,p{q}^2,\cdots ,p{q}^{m_2+1},\cdots ,p{q}^{2m_2+1},\cdots ,\hfill \\ \hfill & p^{m_1+1}q,p^{m_1+1}{q}^2,\cdots ,p^{m_1+1}{q}^{m_2},p^{m_1+1}{q}^{m_2+1},\cdots ,p^{m_1+1}{q}^{2m_2+1},\cdots ,p^{2m_1+1}q,p^{2m_1+1}{q}^2,\cdots ,\hfill \\ \hfill & p^{2m_1+1}{q}^{m_2},p^{2m_1+1}{q}^{m_2+1},\cdots ,p^{2m_1+1}{q}^{2m_2}\}.\hfill \end{array}$$

Therefore, by the definition of ${\mathrm{{\rm Y}}}_n$, we have

$$\begin{array}{cc}\hfill p^i& \sim p^j{q}^{2m_2+1},i+j\ge 2m_1+1,\mathrm{for}i=1,2,\cdots ,2m_1+1,\hfill \\ \hfill q^i& \sim p^{2m_1}{q}^j,i+j\ge 2m_2+1,\mathrm{for}i=1,2,\cdots ,2m_2+1,\hfill \\ \hfill p{q}^i& \sim p^k{q}^j,i+j\ge 2m_2+1,\mathrm{for}i=1,2,\cdots ,2m_2+1\mathrm{and}k\ge 2m_1,\hfill \\ \hfill & ⋮\hfill \\ \hfill p^{m_1+1}{q}^i& \sim p^k{q}^j,i+j\ge 2m_2+1,\mathrm{for}i=1,2,\cdots ,2m_2+1\mathrm{and}k\ge m_1,\hfill \\ \hfill & ⋮\hfill \\ \hfill p^{2m_1+1}{q}^i& \sim p^k{q}^j,i+j\ge 2m_2+1,\mathrm{for}i=1,2,\cdots ,2m_2\mathrm{and}k\ge 0.\hfill \end{array}$$

By Lemma 2, for $i=1,2,\cdots ,2m_1+1,j=1,2,\cdots ,2m_2+1$ and $k=1,2,\cdots ,2m_2$, the cardinalities of $A_{d_i}$ are

$$\begin{array}{cc}\hfill |A_{p^i}|& =\varphi \left(p^{2m_1+1-i}{q}^{2m_2+1}\right),|A_{q^j}|=\varphi \left(p^{2m_1+1}{q}^{2m_2+1-j}\right),\left|A_{p{q}^j}\right|=\varphi \left(p^{2m_1}{q}^{2m_2+1-j}\right),\cdots ,\hfill \\ \hfill & |A_{p^{m_1+1}{q}^j}|=\varphi \left(p^{m_1}{q}^{2m_2+1-j}\right),\cdots ,|A_{p^{2m_1}{q}^j}|=\varphi \left(p{q}^{2m_2+1-j}\right),\left|A_{p^{2m_1+1}{q}^k}\right|=\varphi \left(q^{2m_2+1-k}\right).\hfill \end{array}$$

Also, by Lemma 3, the induced graphs $\mathrm{\Gamma }\left(A_{d_{p^i}}\right)$ are

$$G_i=\left\{\begin{array}{cc}\mathrm{\Gamma }\left(A_{d_{p^i}}\right)={\overline{K}}_{\varphi \left(p^{2m_1+1-i}{q}^{2m_2+1}\right)},\hfill & 1\le i\le 2m_1+1,\hfill \\ \mathrm{\Gamma }\left(A_{d_{q^j}}\right)={\overline{K}}_{\varphi \left(p^{2m_1+1}{q}^{2m_2+1-j}\right)},\hfill & 1\le j\le 2m_2+1,\hfill \\ \mathrm{\Gamma }\left(A_{d_{p^i{q}^j}}\right)={\overline{K}}_{\varphi \left(p^{2m_1+1-i}{q}^{2m_2+1-j}\right)},\hfill & 1\le i\le m_1\mathrm{and}1\le j\le 2m_2+1\hfill \\ \hfill & \mathrm{or}1\le i\le 2m_1+1\mathrm{and}1\le j\le m_2,\hfill \\ \mathrm{\Gamma }\left(A_{d_{p^i{q}^j}}\right)=K_{\varphi \left(p^{2m_1+1-i}{q}^{2m_2+1-j}\right)},\hfill & m_1+1\le i\le 2m_1+1\mathrm{and}{m}_2+1\le j\le N_2,\hfill \end{array}\right.$$

(5)

where we avoid $\mathrm{\Gamma }\left(A_{d_{p^{2m_1+1}{q}^{2m_2}+1}}\right)$ corresponding to proper divisor $p^{2m_1+1}{q}^{2m_2+1}.$ Therefore, by Lemma 4, the structure of zero-divisor graph of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ is

$$\begin{array}{cc}\hfill \mathrm{\Gamma }\left({\mathbb{Z}}_n\right)={\mathrm{{\rm Y}}}_n[& {\overline{K}}_{\varphi \left(p^{2m_1}{q}^{N_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{m_1}{q}^{N_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(q^{N_2}\right)},{\overline{K}}_{\varphi \left(p^{N_1}{q}^{2m_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{N_1}{q}^{m_2}\right)},\hfill \\ \hfill & \cdots ,{\overline{K}}_{\varphi \left(p^{N_1}\right)},{\overline{K}}_{\varphi \left(p^{2m_1}{q}^{2m_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{2m_1}{q}^{m_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{2m_1}\right)},\cdots ,\hfill \\ \hfill & {\overline{K}}_{\varphi \left(p^{m_1}{q}^{2m_2}\right)},\cdots ,K_{\varphi \left(p^{m_1}{q}^{m_2}\right)},K_{\varphi \left(p^{m_1}{q}^{m_2-1}\right)},\cdots ,K_{\varphi \left(p^{m_1}\right)},\cdots ,{\overline{K}}_{\varphi \left(q^{2m_2}\right)},\cdots ,\hfill \\ \hfill & {\overline{K}}_{\varphi \left(q^{m_2}\right)},K_{\varphi \left(q^{m_2-1}\right)},\cdots ,K_{\varphi \left(q\right)}].\hfill \end{array}$$

□

If $N_2=0$, in Theorem 3, we have the following consequence.

Corollary 4.

Let $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ be the zero-divisor graph of order $N,$ where $n=p^{2m+1}$. Then

$$\begin{array}{cc}\hfill \mathrm{\Gamma }\left({\mathbb{Z}}_n\right)=& {\mathrm{{\rm Y}}}_n\left[{\overline{K}}_{\varphi \left(p^{2m}\right)},{\overline{K}}_{\varphi \left(p^{2m-1}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{m+1}\right)},K_{\varphi \left(p^m\right)},K_{\varphi \left(p^{m-1}\right)},\cdots ,K_{\varphi \left(p^2\right)},K_{\varphi \left(p\right)}\right].\hfill \end{array}$$

Proof. 

The proper divisors of $n=p^{2m+1}$ are $\{p,p^2,\cdots ,p^m,p^{m+1},p^{m+2},\cdots ,p^{2m}\}$. In the graph ${\mathrm{{\rm Y}}}_{p^{2m+1}}$, the vertex $p^i$ is adjacent to the vertex $p^j$ if and only if $i+j\ge 2m+1$ with $1\le i\le 2m$ and to avoid loops we assume $i\ne j.$ Also, n does not divide ${\left(p^i\right)}^2$, for $i=1,2,\cdots ,m$, this implies that $G_i={\overline{K}}_{\varphi \left(p^{2m+1-i}\right)}$ and n divides ${\left(p^i\right)}^2$, for $i=m+1,m+2,\cdots ,2m-1,2m$, so that $G_i=K_{\varphi \left(p^{2m+1-i}\right)}$. Now result follows.    □

Other graph invariants of $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{N_1}{q}^{N_2}}\right)$, like automorphism group, chromatic number, domination number, independence number, matching number can similarly be investigated.

3. Laplacian Eigenvalues of the Zero-Divisor Graph $\mathbf{\Gamma }\left({\mathbb{Z}}_{\mathbf{n}}\right)$

Consider an $n\times n$ matrix

$$M=\left(\begin{array}{cccc}{\mathcal{A}}_{1,1}& {\mathcal{A}}_{1,2}& \cdots & {\mathcal{A}}_{1,l}\\ {\mathcal{A}}_{2,1}& {\mathcal{A}}_{2,2}& \cdots & {\mathcal{A}}_{2,l}\\ ⋮& ⋮& \cdots & ⋮\\ {\mathcal{A}}_{l,1}& {\mathcal{A}}_{l,2}& \cdots & {\mathcal{A}}_{l,l}\end{array}\right),$$

whose rows and columns are partitioned according to a partition $P=\{P_1,P_2,\cdots ,P_l\}$ of the set $X=\{1,2,\cdots ,n\}.$ The quotient matrix Q is a matrix of order l whose $(i,j)$th entry is the average row sums of the blocks ${\mathcal{A}}_{i,j}$ of M. If each block ${\mathcal{A}}_{i,j}$ has constant row (column) sum, then the partition P is called equitable and the matrix Q is known as equitable quotient matrix. In general, the spectrum of Q is interlaced by the spectrum of M, equality holds in case of the equitable partition [1].

The following lemma gives a different method of finding determinant (det) of a matrix.

Lemma 5

([18]). Let $A_1,A_2,A_3$ and $A_4$ be respectively $p\times p,p\times q,q\times p$ and $q\times q$ matrices with $A_1$ and $A_4$ invertible. Then

$$\begin{array}{cc}\hfill det\left(\begin{array}{cc}{A}_1& A_2\\ A_3& A_4\end{array}\right)& =det\left(A_1\right)det(A_4-A_3{A}_1^{-1}{A}_2)\hfill \\ \hfill & =det\left(A_4\right)det(A_1-A_2{A}_4^{-1}{A}_3),\hfill \end{array}$$

where $A_4-A_3{A}_1^{-1}{A}_2$ and $A_1-A_2{A}_4^{-1}{A}_3$ are known as the Schur complement of $A_1$ and $A_4$, respectively.

The following result gives the Laplacian spectrum of $G[G_1,\dots ,G_n]$ in terms of the Laplacian spectrum of $G_i$’s and the eigenvalues of the quotient matrix.

Theorem 4.

Let G be a graph of order n and let $G_i$ be regular graphs of order $n_i$ with Laplacian eigenvalues ${\mu }_{i1}\ge {\mu }_{i2}\ge \dots \ge {\mu }_{i{n}_i},$ where $i=1,2,\dots ,n$. Then the Laplacian eigenvalues of $G[G_1,\dots ,G_n]$ are the eigenvalues ${\alpha }_i+{\mu }_{ik}\left(G_i\right)$ for $i=1,\dots ,n$ and $k=2,3,\dots ,n_i$, where ${\alpha }_i={\displaystyle \sum _{v_j\in N_G\left(v_i\right)}}{n}_i$ is the sum of the cardinalities of the graphs $G_j,j\ne i$, which corresponds to the neighbors of vertex $v_i\in G$ and n eigenvalues of the following matrix

$$\mathcal{Q}=\left(\begin{array}{cccc}{\alpha }_1& -{\psi }_{12}& \dots & -{\psi }_{1n}\\ -{\psi }_{21}& {\alpha }_2& \dots & -{\psi }_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ -{\psi }_{n1}& -{\psi }_{n2}& \dots & {\alpha }_n\end{array}\right),$$

(6)

where for $i\ne j$, ${\psi }_{ij}=n_j$, if $v_i\sim v_j$, while as ${\psi }_{ij}=0$, if $v_i\nsim v_j$.

An equivalent statement of Theorem 4 can be seen in [19], so we omit the proof here.

Usually it is difficult to obtain the Laplacian eigenvalues of graphs in general. So, researchers attempt to get the Laplacian eigenvalues of particular class of graphs. It is important to mention that the structure of the zero-graphs associated with ${\mathbb{Z}}_n$ for $n=p^{N_1}{q}^{N_2}$ has not been obtained earlier. Therefore, it becomes essential to write graphs in some known structure and obtain their Laplacian spectrum.

Now, we will find the Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$, for $n=p^{N_1}{q}^{N_2}$, where p and q, $p<q$, are primes. This generalizes the results obtained in [5] and that too by using different technique. We prove the case when $N_1$ and $N_2$, $N_1\le N_2$, are positive even integers and the odd case can be similarly proved.

Theorem 5.

Let $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ be the zero-divisor graph of order $N,$ where $n=p^{N_1}{q}^{N_2}$ and $N_1=2m_1\le 2m_2=N_2$. The Laplacian spectrum of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ consists of the eigenvalues

$$\begin{array}{cc}\hfill \left\{\right(p^i-1)& {}^{[\varphi \left(p^{N_1-i}{q}^{N_2}\right)-1]},{(q^j-1)}^{[\varphi \left(p^{N_1}{q}^{N_2-j}\right)-1]},{(p{q}^j-1)}^{[\varphi \left(p^{N_1-1}{q}^{N_2-j}\right)-1]},\cdots ,{(p^{m_1}{q}^k-1)}^{[\varphi \left(p^{m_1}{q}^{N_2-k}\right)-1]}\hfill \\ \hfill & {(p^{m_1}{q}^l-1)}^{[\varphi \left(p^{m_1}{q}^{N_2-l}\right)-1]},\cdots ,{(p^{2m_1}{q}^k-1)}^{[\varphi \left(q^{N_2-k}\right)-1]},{(p^{2m_1}{q}^t-1)}^{[\varphi \left(q^{N_2-t}\right)-1]},\}\hfill \end{array}$$

where $i=1,2,\cdots ,m_1,\cdots ,N_1,j=1,2,\cdots ,N_2,k=1,2,\cdots ,m_2-1,l=m_2,\cdots ,2m_2$ and $t=m_2,\cdots ,2m_2-1.$ The remaining Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ are the eigenvalues of the matrix given in (6).

Proof. 

By using Theorems 1 and 4, the value of ${\alpha }_i$’s are

$$\begin{array}{cc}\hfill & {\alpha }_1=\varphi \left(p\right)=p-1,{\alpha }_2=\varphi \left(p\right)+\varphi \left(p^2\right)=p^2-1,\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\alpha }_{m_1}=\varphi \left(p^{m_1}\right)+\varphi \left(p^{m_1-1}\right)+\cdots +\varphi \left(p\right)=p^{m_1}-1,\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\alpha }_{N_1}=\varphi \left(p^{N_1}\right)+\varphi \left(p^{N_1-1}\right)+\cdots +\varphi \left(p\right)=p^{N_1}-1,\hfill \end{array}$$

that is,

$$\begin{array}{c}\hfill {\alpha }_i=p^i-1,\mathrm{for}i=1,2,\cdots ,N_1.\end{array}$$

For $i\ge m_2$ and $j\ge m_2$, we note that $\mathrm{\Gamma }\left(A_{p^i{q}^j}\right)$ as vertex of ${\mathrm{{\rm Y}}}_n$ are adjacent to itself, so we add and subtract cardinalities of such type of $\mathrm{\Gamma }\left(A_{p^i{q}^j}\right)$’s so that ${\alpha }_i$’s are easy to calculate. Now, as above other ${\alpha }_i$’s are given by

$$\begin{array}{cc}\hfill {\alpha }_i=& q^j-1,\mathrm{for}i=N_1+1,\cdots ,N_1+N_2,\mathrm{and}j=1,2,\cdots ,m_2,\cdots ,N_1,\hfill \\ \hfill {\alpha }_i=& p{q}^j-1\mathrm{for}i=N_1+N_2+1,\cdots ,N_1+2N_2\mathrm{and}j=1,2,\cdots ,m_2,\cdots ,N_1,\hfill \\ \hfill & ⋮\hfill \\ \hfill {\alpha }_i=& p^{m_1}{q}^j-1,\mathrm{for}i=N_1+m_1{N}_2+1,\cdots ,N_1+m_1{N}_2+m_2-1\mathrm{and}j=1,2,\cdots ,m_2-1,\hfill \\ \hfill {\alpha }_i=& p^{m_1}{q}^j-1-\varphi \left(p^{m_1}{q}^j\right),\mathrm{for}i=N_1+m_1{N}_2,\cdots ,N_1+(m_1+1)N_2\mathrm{and}j=m_2,\cdots ,N_2,\hfill \\ \hfill & ⋮\hfill \\ \hfill {\alpha }_i=& p^{N_1}{q}^j-1,\mathrm{for}i=N_1+N_1{N}_2+1,\cdots ,N_1+N_1{N}_2+m_2-1\mathrm{and}j=1,2,\cdots ,m_2-1,\hfill \\ \hfill {\alpha }_i=& p^{N_1}{q}^j-1-\varphi \left(q^{N_2-j}\right),\mathrm{for}i=N_1+N_1{N}_2+m_2,\cdots ,N_1+N_1{N}_2+N_2-1\hfill \\ \hfill & \mathrm{and}j=m_2,\cdots ,N_2-1.\hfill \end{array}$$

By using Theorem 4, Equation (2) and the fact that Laplacian spectrum of ${\overline{K}}_{\omega }$ is $\left\{0^{\left[\omega \right]}\right\}$, we have

$$\begin{array}{c}\hfill {\alpha }_i+{\lambda }_{ik}\left(G_i\right)={\alpha }_i+{\lambda }_{ik}\left({\overline{K}}_{\varphi \left(p^{N_1-i}{q}^{N_2}\right)}\right)={\alpha }_i=p^i-1,\mathrm{for}i=1,2,\cdots ,N_1.\end{array}$$

Thus, for $i=1,2,\cdots ,N_1$, we see that $p^i-1$ is the Laplacian eigenvalue of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ with multiplicity $\varphi \left(p^{N_1-i}{q}^{N_2}\right)-1.$

Now, following similar steps, it is easy to see that

$${(q^j-1)}^{[\varphi \left(p^{N_1}{q}^{N_2-j}\right)-1]},{(p{q}^j-1)}^{[\varphi \left(p^{N_1-1}{q}^{N_2-j}\right)-1]},\cdots ,{(p^{m_1}{q}^k-1)}^{[\varphi \left(p^{m_1}{q}^{N_2-k}\right)-1]}$$

are also the Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right).$ Again, by Equation (2), $G_i=K_{\varphi \left(p^{N_1-i}{q}^{N_2-j}\right)}$, when $i\ge m_1$ and $j\ge m_2$ and Laplacian spectrum of $K_{\omega }$ is $\{0,{\omega }^{\omega -1}\}$, so

$${\alpha }_i+{\lambda }_{ik}\left(G_i\right)=p^{m_1}{q}^l-1-\varphi \left(p^{m_1}{q}^l\right)+\varphi \left(p^{m_1}{q}^l\right)=p^{m_1}{q}^l-1$$

is the Laplacian eigenvalue of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ with multiplicity $\varphi \left(p^{m_1}{q}^l\right)-1$, where $l=m_2,\cdots ,N_2.$ Similarly, for $k=1,2,\cdots ,m_2-1$ and $t=m_2,\cdots ,N_2-1$, we see that $p^{N_1}{q}^k-1$ and $p^{N_1}{q}^t-1$ are also the Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ with multiplicities $\varphi \left(q^{N_2-t}\right)-1$ and $\varphi \left(q^{N_2-k}\right)-1$, respectively. The other Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ are the zeros of the characteristic polynomial of the quotient matrix (6).  □

If we put $m_2=0$ in Theorem 5, it reduces to the following result [5] with a different technique.

Corollary 5.

If $n=p^{2m}$ for some positive integer $m\ge 2$, then the Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ are

$$\begin{array}{cc}\hfill \{0,{(p-1)}^{\left[\varphi \left(p^{2m-1}\right)\right]},(p^2-1& {)}^{\left[\varphi \left(p^{2m-2}\right)1\right]},\cdots ,{(p^{m-1}-1)}^{\left[\varphi \left(p^{m+1}\right)\right]},{(p^m-1)}^{[\varphi \left(p^m\right)-1]},\hfill \\ \hfill & {(p^{m+1}-1)}^{\left[\varphi \left(p^{m-1}\right)\right]},\cdots ,{(p^{2m-2}-1)}^{\left[\varphi \left(p^2\right)\right]},{(p^{2m-1}-1)}^{\left[\varphi \right(p\left)\right]}\}.\hfill \end{array}$$

Proof. 

Using Corollary 1, $p\sim p^{2m-1}$ implies that ${\alpha }_1=\varphi \left(p\right)$. However, in general, we see that ${\alpha }_i=p^i-1$, for $i=1,2,\cdots ,m-1$, where from Theorem 1, we have used ${\displaystyle \sum _{i=1}^l}{p}^i=p^l-1$. As $p^m\sim p^m$, so we add and subtract cardinality of $\mathrm{\Gamma }\left(A_{p^m}\right)$ and thus ${\alpha }_m$ is given as

$$\begin{array}{cc}\hfill {\alpha }_m& =\varphi \left(p^{m-1}\right)+\varphi \left(p^{m-2}\right)+\cdots +\varphi \left(p^2\right)+\varphi \left(p\right)\hfill \\ \hfill & =\varphi \left(p^m\right)+\varphi \left(p^{m-1}\right)+\varphi \left(p^{m-2}\right)+\cdots +\varphi \left(p^2\right)+\varphi \left(p\right)-\varphi \left(p^m\right)\hfill \\ \hfill & =p^m-1-\varphi \left(p^m\right).\hfill \end{array}$$

Likewise, for $i=m+1,\cdots ,2m-2,2m-1$, it is clear that

$${\alpha }_i=\sum _{j=1}^i\varphi \left(p^j\right)-\varphi \left(p^{2m-i}\right)=p^i-1-\varphi \left(p^{2m-i}\right).$$

For $i=1,2,\cdots ,m-1$, clearly ${\alpha }_i=p^i-1$ are the Laplacian eigenvalue of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ with multiplicity $\varphi \left(p^{2m-i}\right)-1$. Also, from Theorem 4 and for $i=m,m+1,\cdots ,2m-1$, we see that

$${\alpha }_i+{\mu }_{ik}\left(G_i\right)=p^i-1-\varphi \left(p^{2m-i}\right)+{\mu }_{ik}\left(K_{\varphi \left(p^{2m-i}\right)}\right)=p^i-1$$

are also Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ with multiplicities $\varphi \left(p^{2m-i}\right)-1$. The remaining Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ are given by the following quotient matrix

$$\mathcal{Q}=\left(\begin{array}{cc}{A}_{m-1\times m-1}& B_{m-1\times m}\\ C_{m\times m-1}& D_{m\times m}\end{array}\right)$$

where $A=diag(p-1,p^2-1,\cdots ,p^{m-1}-1)$,

$$\begin{array}{cc}\hfill & B=\left(\begin{array}{ccccc}0& 0& \cdots & 0& -\varphi \left(p\right)\\ 0& 0& \cdots & -\varphi \left(p^2\right)& -\varphi \left(p\right)\\ ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& -\varphi \left(p^{m-1}\right)& \cdots & -\varphi \left(p^2\right)& -\varphi \left(p\right)\end{array}\right),C=\left(\begin{array}{cccc}0& 0& \cdots & 0\\ 0& 0& \cdots & -\varphi \left(p^{m+1}\right)\\ ⋮& ⋮& \cdots & ⋮\\ 0& -\varphi \left(p^{2m-2}\right)& \cdots & -\varphi \left(p^{m+1}\right)\\ -\varphi \left(p^{2m-1}\right)& -\varphi \left(p^{2m-2}\right)& \cdots & -\varphi \left(p^{m+1}\right)\end{array}\right)\hfill \\ \hfill & \mathrm{and}D=\left(\begin{array}{ccccc}{p}^m-1-\varphi \left(p^m\right)& -\varphi \left(p^{m-1}\right)& \cdots & -\varphi \left(p^2\right)& -\varphi \left(p\right)\\ -\varphi \left(p^m\right)& p^{m-1}-1-\varphi \left(p^{m-1}\right)& \cdots & -\varphi \left(p^2\right)& -\varphi \left(p\right)\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -\varphi \left(p^m\right)& -\varphi \left(p^{m-1}\right)& \cdots & p^{2m-2}-1-\varphi \left(p^2\right)& -\varphi \left(p\right)\\ -\varphi \left(p^m\right)& -\varphi \left(p^{m-1}\right)& \cdots & -\varphi \left(p^2\right)& p^{2m-1}-1-\varphi \left(p\right)\end{array}\right).\hfill \end{array}$$

Applying Lemma 5, we have

$$det(xI-\mathcal{Q})=det(xI-A)det((xI-D)-C{(xI-A)}^{-1}B).$$

(7)

By evaluating Equation (7), we can verify that

$$\begin{array}{c}\hfill \left\{0,p-1,p^2-1,\cdots ,p^{m-1}-1,p^{m+1}-1,\cdots ,p^{2m-2}-1,p^{2m-1}-1\right\}\end{array}$$

are the remaining Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$. We note that all the Laplacian eigenvalues of quotient matrix $\mathcal{Q}$ are repeated with the eigenvalues obtained by ${\alpha }_i+{\mu }_{ik}\left(G_i\right)$ except $p^m-1.$    □

As $\mathrm{\Gamma }\left({\mathbb{Z}}_{2^m}\left[i\right]\right)\cong \mathrm{\Gamma }\left({\mathbb{Z}}_{2^{2m}}\right)$, so for $p=2$ in Corollary 5, we get the following.

Corollary 6.

The Laplacian eigenvalues of the zero-divisor graph $\mathrm{\Gamma }\left({\mathbb{Z}}_{2^m}\left[i\right]\right)$ of Gaussian integers modulo $2^m$ is

$$\begin{array}{cc}\hfill \{0,1^{\left[\varphi \left(2^{2m-1}\right)\right]},& 2^{\left[\varphi \left(2^{2m-2}\right)1\right]},\cdots ,{(2^{m-1}-1)}^{\left[\varphi \left(2^{m+1}\right)\right]},{(2^m-1)}^{[\varphi \left(2^m\right)-1]},\hfill \\ \hfill & {(2^{m+1}-1)}^{\left[\varphi \left(2^{m-1}\right)\right]},\cdots ,{(2^{2m-2}-1)}^2,(2^{2m-1}-1)\}.\hfill \end{array}$$

If $m_1=1$ and $m_2=0$ in Theorem 5, we have $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^2}\right)=K_{\varphi \left(p\right)}$ and its Laplacian spectrum is given by the following observation.

Corollary 7.

If $n=p^2$, then the Laplacian spectrum of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ is

$$\{0,{(p-1)}^{[p-2]}\}.$$

The following result gives the Laplacian spectrum of $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^{N_1}{q}^{N_2}}\right)$, when both $N_1$ and $N_2$ are odd. Its proof is similar to that of Theorem 5.

Theorem 6.

Let $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ be the zero-divisor graph of order $N,$ where $n=p^{N_1}{q}^{N_2}$ and $N_1=2m_1+1\le 2m_2+1=N_2$. The Laplacian spectrum of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ consists of the eigenvalues

$$\begin{array}{cc}\hfill \left\{\right(p^i-1)& {}^{[\varphi \left(p^{N_1-i}{q}^{N_2}\right)-1]},{(q^j-1)}^{[\varphi \left(p^{N_1}{q}^{N_2-j}\right)-1]},{(p{q}^j-1)}^{[\varphi \left(p^{N_1-1}{q}^{N_2-j}\right)-1]},\cdots ,{(p^{m_1+1}{q}^k-1)}^{[\varphi \left(p^{m_1}{q}^{N_2-j}\right)-1]}\hfill \\ \hfill & ,\cdots ,{(p^{2m_1+1}{q}^k-1)}^{[\varphi \left(q^{N_2-k}\right)-1]}\},\hfill \end{array}$$

where $i=1,2,\cdots ,m_1,\cdots ,N_1,j=1,2,\cdots ,N_2$ and $k=1,2,\cdots ,2m_2$. The remaining Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ are the eigenvalues of the matrix given in (6).

In particular, if $q=1$ in Theorem 6, we have the following result of [5].

Corollary 8.

If $n=p^{2m+1}$ for some positive integer $m\ge 2$, then the Laplacian spectrum of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ is

$$\begin{array}{cc}\hfill \{0,{(p-1)}^{\left[\varphi \left(p^{2m}\right)\right]},(p^2-1& {)}^{\left[\varphi \left(p^{2m-1}\right)1\right]},\cdots ,{(p^{m-1}-1)}^{\left[\varphi \left(p^{m+2}\right)\right]},{(p^m-1)}^{[\varphi \left(p^{m+1}\right)-1]},\hfill \\ \hfill & {(p^{m+1}-1)}^{\left[\varphi \left(p^m\right)\right]},\cdots ,{(p^{2m-1}-1)}^{\left[\varphi \left(p^2\right)\right]},{(p^{2m}-1)}^{\left[\varphi \right(p\left)\right]}\}.\hfill \end{array}$$

If $m_1=m_2=0$, then $n=pq$. Therefore, by Lemmas 3 and 4, we have

$$\mathrm{\Gamma }\left({\mathbb{Z}}_{pq}\right)={\mathrm{{\rm Y}}}_{pq}[\mathrm{\Gamma }\left(A_p\right),\mathrm{\Gamma }\left(A_q\right)]=K_2[{\overline{K}}_{\varphi \left(p\right)},{\overline{K}}_{\varphi \left(q\right)}]={\overline{K}}_{\varphi \left(p\right)}▿{\overline{K}}_{\varphi \left(q\right)}=K_{\varphi \left(p\right),\varphi \left(q\right)}.$$

(8)

The next consequence of Theorem 6 gives the Laplacian spectrum of the complete bipartite graph $\mathrm{\Gamma }\left({\mathbb{Z}}_{pq}\right).$

Corollary 9.

The Laplacian spectrum of $\mathrm{\Gamma }\left({\mathbb{Z}}_{pq}\right)$ is

$$\left\{0,{(q-1)}^{[p-2]},{(p-1)}^{[q-2]},p+q-2\right\}.$$

For $m=1$ and $q=1$ in Theorem 6, we have the following observation for $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^3}\right)$.

Corollary 10.

If $n=p^3$, then the Laplacian spectrum of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ is

$$\left\{0,{(p-1)}^{[p^2-p-1]},{(p^2-1)}^{[p-2]}\right\}.$$

Proof. 

As the proper divisors of n are p and $p^2$, so ${\mathrm{{\rm Y}}}_n$ is $K_2:$$p\sim p^2$. By Lemma 4, we have

$$\mathrm{\Gamma }\left({\mathbb{Z}}_{p^3}\right)={\mathrm{{\rm Y}}}_{p^3}[\mathrm{\Gamma }\left(A_p\right),\mathrm{\Gamma }\left(A_{p^2}\right)]=K_2[{\overline{K}}_{\varphi \left(p^2\right)},{\overline{K}}_{\varphi \left(p\right)}]={\overline{K}}_{p(p-1)}▿K_{p-1}.$$

That is, $\mathrm{\Gamma }\left({\mathbb{Z}}_{p^3}\right)$ is a complete split graph of order $p^2-1$, with independence number $p(p-1)$. Therefore, by Theorem 4, we have $({\alpha }_1,{\alpha }_2)=(p-1,p^2-p)$, and

$$\mathcal{Q}\left(K_2\right)=\left(\begin{array}{cc}p-1& -\varphi \left(p\right)\\ -\varphi \left(p^2\right)& p^2-p\end{array}\right)$$

(9)

.

As $G_1={\overline{K}}_{p(p-1)}$, so the Laplacian spectrum of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ consists of the eigenvalue ${\alpha }_1=p-1$ with multiplicity $p(p-1)-1$, the eigenvalue ${\alpha }_2+{\mu }_{2k}\left(K_{p-1}\right)=p^2-p+p-1=p^2-1$ with multiplicity $p-2$ and the other two Laplacian eigenvalues are the eigenvalues of matrix (9).    □

Now, consider the case when one of $N_i$’s is even and other is odd, say $N_1$ is even and $N_2$ is odd or $N_1$ is odd and $N_2$ is even. In the following result, first case is given and the second case can be treated similarly.

Theorem 7.

Let $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ be the zero-divisor graph of order N, where $n=p^{N_1}{q}^{N_2}$ and $m_1<m_2$ so that $N_1=2m_1<2m_2+1=N_2$. The Laplacian spectrum of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ consists of the eigenvalues

$$\begin{array}{cc}\hfill \left\{\right(p^i-1)& {}^{[\varphi \left(p^{N_1-i}{q}^{N_2}\right)-1]},{(q^j-1)}^{[\varphi \left(p^{N_1}{q}^{N_2-j}\right)-1]},{(p{q}^j-1)}^{[\varphi \left(p^{N_1-1}{q}^{N_2-j}\right)-1]},\cdots ,{(p^{m_1}{q}^j-1)}^{[\varphi \left(p^{m_1}{q}^{N_2-j}\right)-1]}\hfill \\ \hfill & ,\cdots ,{(p^{N_1}{q}^k-1)}^{[\varphi \left(q^{N_2-k}\right)-1]}\},\hfill \end{array}$$

where $i=1,2,\cdots ,m_1,\cdots ,N_1,j=1,2,\cdots ,N_2$ and $k=1,2,\cdots ,2m_2$. The remaining Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ are the eigenvalues of the matrix given in (6).

Proof. 

For $n=p^{N_1}{q}^{N_2}$, with p and q, being primes and $2\le N_1=2m_1<2m_2+1=N_2$. The proper divisor set of n is

$$\begin{array}{cc}\hfill \{p,p^2,\cdots ,& p^{m_1},\cdots ,p^{N_1},q,q^2,\cdots ,q^{m_2+1},\cdots ,q^{N_2},pq,p{q}^2,\cdots ,p{q}^{m_2+1},\cdots ,p{q}^{N_2},\cdots ,p^{m_1}q,p^{m_1}{q}^2,\hfill \\ \hfill \cdots ,& p^{m_1}{q}^{m_2},p^{m_1}{q}^{m_2+1},\cdots ,p^{m_1}{q}^{N_2},\cdots ,p^{N_1}q,p^{N_1}{q}^2,\cdots ,p^{N_1}{q}^{m_2},p^{N_1}{q}^{m_2+1},\cdots ,p^{N_1}{q}^{N_2-1}\}.\hfill \end{array}$$

Now by the definition of ${\mathrm{{\rm Y}}}_n$, the adjacency relations are

$$\begin{array}{cc}\hfill p^i& \sim p^j{q}^{N_2},i+j\ge N_1,\mathrm{for}i=1,2,\cdots ,N_1\hfill \\ \hfill q^i& \sim p^{N_1}{q}^j,i+j\ge N_2,\mathrm{for}i=1,2,\cdots ,N_2,\hfill \\ \hfill p{q}^i& \sim p^k{q}^j,i+j\ge N_2,\mathrm{for}i=1,2,\cdots ,N_2\mathrm{and}k\ge 2m_1-1,\hfill \\ \hfill & ⋮\hfill \\ \hfill p^{m_1}{q}^i& \sim p^k{q}^j,i+j\ge N_2,\mathrm{for}i=1,2,\cdots ,N_2\mathrm{and}k\ge m_1\hfill \\ \hfill & ⋮\hfill \\ \hfill p^{M_1}{q}^i& \sim p^k{q}^j,i+j\ge N_2,\mathrm{for}i=1,2,\cdots ,N_2-1\mathrm{and}k\ge 0.\hfill \end{array}$$

Also, the cardinalities of $A_{d_i}$’s are

$$\begin{array}{cc}\hfill |A_{p^i}|=\varphi \left(p^{N_1-i}{q}^{N_2}\right),\left|A_{q^j}\right|& =\varphi \left(p^{N_1}{q}^{N_2-j}\right),|A_{p{q}^j}|=\varphi \left(p^{N_1-1}{q}^{N_2-j}\right),\cdots ,\left|A_{p^{m_1}{q}^j}\right|=\varphi \left(p^{m_1}{q}^{N_2-j}\right),\hfill \\ \hfill & \cdots ,|A_{p^{N_1-1}{q}^j}|=\varphi \left(p{q}^{N_2-j}\right),\left|A_{p^{N_1}{q}^k}\right|=\varphi \left(q^{N_2-k}\right),\hfill \end{array}$$

where $i=1,2,\cdots ,N_1,j=1,2,\cdots ,N_2$ and $k=1,2,\cdots ,N_2-1$. Further by using Lemma 3, we have

$$G_i=\left\{\begin{array}{cc}\mathrm{\Gamma }\left(A_{d_{p^i}}\right)={\overline{K}}_{\varphi \left(p^{N_1-i}{q}^{N_2}\right)},\hfill & 1\le i\le N_1,\hfill \\ \mathrm{\Gamma }\left(A_{d_{q^j}}\right)={\overline{K}}_{\varphi \left(p^{N_1}{q}^{N_2-j}\right)},\hfill & 1\le j\le N_2,\hfill \\ \mathrm{\Gamma }\left(A_{d_{p^i{q}^j}}\right)={\overline{K}}_{\varphi \left(p^{N_1-i}{q}^{N_2-j}\right)},\hfill & 1\le i\le m_1-1\mathrm{and}1\le j\le N_2\hfill \\ \hfill & \mathrm{or}1\le i\le N_1\mathrm{and}1\le j\le m_2,\hfill \\ \mathrm{\Gamma }\left(A_{d_{p^i{q}^j}}\right)=K_{\varphi \left(p^{N_1-i}{q}^{N_2-j}\right)},\hfill & m_1\le i\le N_1\mathrm{and}{m}_2\le j\le N_2.\hfill \end{array}\right.$$

(10)

Thus, by Lemma 4, the joined union of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ is

$$\begin{array}{cc}\hfill \mathrm{\Gamma }\left({\mathbb{Z}}_n\right)=& {\mathrm{{\rm Y}}}_n[{\overline{K}}_{\varphi \left(p^{N_1-1}{q}^{N_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{m_1}{q}^{N_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(q^{N_2}\right)},{\overline{K}}_{\varphi \left(p^{N_1}{q}^{N_2-1}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{N_1}{q}^{m_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{N_1}\right)},\hfill \\ \hfill & {\overline{K}}_{\varphi \left(p^{N_1-1}{q}^{N_2-1}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{N_1-1}{q}^{m_2}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{N_1-1}\right)},\cdots ,{\overline{K}}_{\varphi \left(p^{m_1}{q}^{N_2-1}\right)},\cdots ,K_{\varphi \left(p^{m_1}{q}^{m_2-11}\right)},\hfill \\ \hfill & K_{\varphi \left(p^{m_1}{q}^{m_2}\right)},\cdots ,K_{\varphi \left(p^{m_1}\right)},\cdots ,K_{\varphi \left(q^{N_2-1}\right)},\cdots ,K_{\varphi \left(q^{m_2-1}\right)},K_{\varphi \left(q^{m_2}\right)},\cdots ,K_{\varphi \left(q\right)}].\hfill \end{array}$$

By using Theorems 1 and 4, the value of ${\alpha }_i$’s are

$$\begin{array}{cc}\hfill {\alpha }_i=& p^i-1,\mathrm{for}i=1,2,\cdots ,N_1\hfill \\ \hfill {\alpha }_i=& q^j-1,\mathrm{for}i=N_1+1,\cdots ,N_1+N_2\mathrm{and}j=1,2,\cdots ,m_2+1,\cdots ,N_2,\hfill \\ \hfill {\alpha }_i=& p{q}^j-1\mathrm{for}i=N_1+N_2+1,\cdots ,N_1+2N_2\mathrm{and}j=1,2,\cdots ,m_2+1,\cdots ,N_2,\hfill \\ \hfill & ⋮\hfill \\ \hfill {\alpha }_i=& p^{m_1}{q}^j-1,\mathrm{for}i=N_1+m_1{N}_2+1,\cdots ,N_1+m_1{N}_2+m_2-1\mathrm{and}j=1,2,\cdots ,m_2,\hfill \\ \hfill {\alpha }_i=& p^{m_1}{q}^j-1-\varphi \left(p^{m_1}{q}^j\right),\mathrm{for}i=N_1+m_1{N}_2,\cdots ,N_1+(m_1+1)N_2\mathrm{and}j=m_2+1,\cdots ,N_2,\hfill \\ \hfill & ⋮\hfill \\ \hfill {\alpha }_i=& p^{N_1}{q}^j-1,\mathrm{for}i=N_1+N_1{N}_2+1,\cdots ,N_1+N_1{N}_2+m_2\mathrm{and}j=1,2,\cdots ,m_2,\hfill \\ \hfill {\alpha }_i=& p^{N_1}{q}^j-1-\varphi \left(q^{N_2-j}\right),\mathrm{for}i=N_1+N_1{N}_2+m_2+1,\cdots ,N_1+N_1{N}_2+N_2-1\hfill \\ \hfill & \mathrm{and}j=m_2+1,\cdots ,2m_2.\hfill \end{array}$$

Again, applying Theorem 4 and using Equation (10), we see that

$$\begin{array}{cc}\hfill {\alpha }_1+{\mu }_{1k}\left(G_1\right)& ={\alpha }_1+0=p-1\hfill \end{array}$$

is the Laplacian eigenvalue of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ with multiplicity $\varphi \left(p^{2m_1-1}{q}^{2m_2+1}\right)-1$. Similarly, the other Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ are as in the statement.    □

Next, we find the Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ when n is the product of three primes.

Theorem 8.

The Laplacian spectrum of $\mathrm{\Gamma }\left({\mathbb{Z}}_{pqr}\right)$ consists of the eigenvalues

$$\left\{{(p-1)}^{\left[\varphi \right(qr)-1]},{(q-1)}^{\left[\varphi \right(p1)-1]},{(r-1)}^{\left[\varphi \right(pq)-1]},{(pq-1)}^{\left[\varphi \right(r)-1]},{(pr-1)}^{\left[\varphi \right(q)-1]},{(qr-1)}^{\left[\varphi \right(p)-1]}\right\}.$$

Proof. 

Let $n=pqr$. Then $p,q,r,pq,pr$ and $qr$ are the proper divisors of n and ${\mathrm{{\rm Y}}}_n$ is the graph $G_6:$ $q\sim pr\sim pq\sim r$, $pr\sim qr\sim p$ and $pq\sim qr$, i.e., ${\mathrm{{\rm Y}}}_n$ is a unicyclic graph with pendent vertices at each vertex of cycle as shown in Figure 2. Ordering the vertices by increasing divisor sequence and applying Lemma 4, we have

$$\mathrm{\Gamma }\left({\mathbb{Z}}_{30}\right)={\mathrm{{\rm Y}}}_{30}[{\overline{K}}_8,{\overline{K}}_4,{\overline{K}}_{24},{\overline{K}}_4,{\overline{K}}_2,{\overline{K}}_1].$$

By Theorem 4, value of ${\alpha }_i$’s are

$$\begin{array}{cc}\hfill {\alpha }_1=\varphi \left(p\right)=p-1,{\alpha }_2& =\varphi \left(q\right)=q-1,{\alpha }_3=\varphi \left(r\right)=r-1,{\alpha }_4=\varphi \left(pq\right)+\varphi \left(p\right)+\varphi \left(q\right)=pq-1,\hfill \\ \hfill {\alpha }_5& =\varphi \left(pr\right)+\varphi \left(p\right)+\varphi \left(r\right)=pr-1,{\alpha }_6=\varphi \left(qr\right)+\varphi \left(q\right)+\varphi \left(r\right)=qr-1.\hfill \end{array}$$

Since each of $G_i$ is a null graph, so the Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_{pqr}\right)$ are $p-1$ with multiplicity $\varphi \left(qr\right)-1$, $q-1$ with multiplicity $\varphi \left(pr\right)-1$, $r-1$ with multiplicity $\varphi \left(pq\right)-1$, $pq-1$ with multiplicity $\varphi \left(r\right)-1$, $pr-1$ with multiplicity $\varphi \left(q\right)-1$, and $qr-1$ with multiplicity $\varphi \left(p\right)-1$. The remaining Laplacian eigenvalues of $\mathrm{\Gamma }\left({\mathbb{Z}}_{pqr}\right)$ are the eigenvalues of the following matrix

$$\left(\begin{array}{cccccc}\varphi \left(p\right)& 0& 0& 0& 0& -\varphi \left(p\right)\\ 0& \varphi \left(q\right)& 0& 0& -\varphi \left(q\right)& 0\\ 0& 0& \varphi \left(r\right)& -\varphi \left(q\right)& 0& 0\\ 0& 0& -\varphi \left(pq\right)& pq-1& -\varphi \left(q\right)& -\varphi \left(p\right)\\ 0& -\varphi \left(pr\right)& 0& -\varphi \left(r\right)& pr-1& -\varphi \left(p\right)\\ -\varphi \left(qr\right)& 0& 0& -\varphi \left(r\right)& -\varphi \left(q\right)& qr-1\end{array}\right).$$

□

Theorem 8 can be generalized for arbitrary product of distinct primes. Although it is hard to find the Laplacian spectra of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right)$ with canonical decomposition of n, it is interesting and can explore various properties of ${\mathbb{Z}}_n$ and the structure of its associated zero-divisor graph. The spectral study of zero-divisor graphs of rings may open research work as in the case of Cayley graphs.

4. Conclusions and Comments

Let ${\mathbb{M}}_n\left(\mathbb{C}\right)$ be the set of all square matrices of order n with complex entries. The trace norm of a matrix $M\in {\mathbb{M}}_n\left(\mathbb{C}\right)$ is defined as ${\parallel M\parallel }_{*}={\displaystyle \sum _{i=1}^n}{\sigma }_i\left(M\right)$, where ${\sigma }_1\left(M\right)\ge {\sigma }_2\left(M\right)\ge \cdots \ge {\sigma }_n\left(M\right)$ are the singular values of M (that is the square roots of the eigenvalues of $M{M}^{*}$, where $M^{*}$ is the complex conjugate of M). In case of symmetric matrices, the singular values coincide with the absolute values of the eigenvalues, i.e., if ${\sigma }_i\left(M\right)$ are the singular values and ${\lambda }_i\left(M\right)$, $i=1,2,\cdots ,n,$ are the eigenvalues of M, then ${\sigma }_i\left(M\right)=\left|{\lambda }_i\left(M\right)\right|$. Thus, the sum of the absolute values of eigenvalues of the matrix $L\left(\mathrm{\Gamma }\left(Z_n\right)\right)-\frac{2m}{n}{I}_n$ is the trace norm of $L\left(\mathrm{\Gamma }\left(Z_n\right)\right)-\frac{2m}{n}{I}_n$, where $I_n$ is the identity matrix of order n. It is an interesting problem in Matrix theory, to determine among a given class of matrices the matrix (or the matrices) which attain the maximum value and the minimum value for the trace norm. The trace norm of matrices associated with the graphs and digraphs are extensively studied. For some recent papers in this direction see [20,21] and the references therein.

In spectral graph theory, the trace norm is studied under the name graph energy. Gutman and Zhou [22] defined the Laplacian energy of G as

$$\begin{array}{c}\hfill LE\left(G\right)=\sum _{i=1}^n\left|{\mu }_i-\frac{2m}{n}\right|.\end{array}$$

Using the fact that ${\displaystyle \sum _{1=i}^{n-1}}{\mu }_i=2m$, from [23], we have

$$LE\left(G\right)=2\left(\sum _{i=1}^{\sigma }{\mu }_i-\sigma \overline{d}\right)=2\underset{1\le k\le n}{max}\left(\sum _{1=i}^k{\mu }_i-k\overline{d}\right),$$

(11)

where $\sigma$ is the number of Laplacian eigenvalues greater than or equal to the average degree $\overline{d}$. We note that ${\displaystyle \sum _{1=i}^k}{\mu }_i$ is actually the Ky Fan k-norm, which for positive semi-definite matrices is the sum of k largest eigenvalues. The parameter $\sigma$ is an active component of the present research and some work mostly on trees can be found in the literature [24]. In fact, it is shown in [25] that the Laplacian energy has remarkable chemical applications beyond the molecular orbital theory of conjugated molecules. For some recent works on Laplacian energy and related results, we refer to [2,26,27,28] and the references therein.

In case of $n=p^2,n=p^3$ and $n=pq$, the trace norm of $\mathcal{L}=L\left(\mathrm{\Gamma }\left(Z_n\right)\right)-\frac{2m}{n}{I}_n$ are

$$2(p-2),{\displaystyle \frac{2(2p+q^2-pq-2)}{p+q-2}},\mathrm{and}{\displaystyle \frac{2(2p^3-4p^2-p+3)}{p^3-1}}.$$

Similarly, Laplacian energy of $\mathrm{\Gamma }\left({\mathbb{Z}}_n\right))$ can be discussed for other values of n and various upper bounds and lower bounds can be obtained.

As zero-divisor graph of ${\mathbb{Z}}_n$ has been written in terms of the joined union, where components are either cliques or their complements, but, in general the zero-divisor graphs of ring R cannot be expressed as the joined union of graphs. So, their spectral analysis becomes difficult. No general method is yet available in discussing the spectra of zero-divisor graphs of rings such as ${\mathbb{Z}}_n\left[i\right],{\mathbb{Z}}_p\times {\mathbb{Z}}_q,(p\ne q),{\mathbb{Z}}_p\left[i\right]\times {\mathbb{Z}}_q\left[i\right],(p\ne q)$ and many other zero-divisor graphs associated with commutative as well as non-commutative rings. Also relating spectral properties with the graph invariants such as connectivity, chromatic number, matching number and other parameters are very interesting problems.