Laplacian eigenvalues of the zero divisor graph of the ring $\mathbb{Z}_{n}$

TL;DR

This paper investigates the Laplacian eigenvalues of the zero divisor graph of the ring _n, proving Laplacian integrality for certain cases and characterizing spectral properties and connectivity for various n.

Contribution

It establishes Laplacian eigenvalue properties of zero divisor graphs of _n, including integrality and spectral radius, and characterizes when algebraic and vertex connectivity coincide.

Findings

_{p^t} has Laplacian integral zero divisor graph for prime p and t2.

Laplacian spectral radius and algebraic connectivity relate to eigenvalues of a vertex weighted Laplacian.

Conditions identified when algebraic and vertex connectivity of _n graphs are equal.

Abstract

We study the Laplacian eigenvalues of the zero divisor graph $\Gamma\left(\mathbb{Z}_{n}\right)$ of the ring $\mathbb{Z}_{n}$ and prove that $\Gamma\left(\mathbb{Z}_{p^t}\right)$ is Laplacian integral for every prime $p$ and positive integer $t\geq 2$. We also prove that the Laplacian spectral radius and the algebraic connectivity of $\Gamma\left(\mathbb{Z}_{n}\right)$ for most of the values of $n$ are, respectively, the largest and the second smallest eigenvalues of the vertex weighted Laplacian matrix of a graph which is defined on the set of proper divisors of $n$. The values of $n$ for which algebraic connectivity and vertex connectivity of $\Gamma\left(\mathbb{Z}_{n}\right)$ coincide are also characterized.

Equations

$$\sigma (B)=\left\{\begin{array}{cccc}{\mu }_1& {\mu }_2& \cdots & {\mu }_t\\ m_1& m_2& \cdots & m_t\end{array}\right\}.$$

$$A_{d_i}=\{x\in {\mathbb{Z}}_n:(x,n)=d_i\}.$$

$$V(\Gamma ({\mathbb{Z}}_n))=A_{d_1}\cup A_{d_2}\cup \dots \cup A_{d_k}.$$

$$n|m_x{m}_y{d}_i{d}_j\iff \frac{n}{d_i}|m_x{m}_y{d}_j\iff \frac{n}{d_i}|m_y{d}_j\iff n|m_y{d}_i{d}_j\iff \frac{n}{d_j}|m_y{d}_i\iff \frac{n}{d_j}|d_i\iff n|d_i{d}_j.$$

$$|V({{\rm Y}}_n)|=\underset{i=1}{\prod ^r}(n_i+1)-2.$$

$$\Gamma ({\mathbb{Z}}_{18})={{\rm Y}}_{18}[\Gamma (A_2),\Gamma (A_3),\Gamma (A_6),\Gamma (A_9)],$$

$${\sigma }_L\left(G\left[H_1,H_2,\dots ,H_k\right]\right)=\left(\underset{j=1}{\bigcup ^k}\left(M_j+\left({\sigma }_L\left(H_j\right)\setminus \{0\}\right)\right)\right)\bigcup \sigma (C),$$

$$M_j=\{\begin{array}{ll}\sum _{v_i\sim v_j}{m}_i& \text{if NG(vj)=∅}\\ 0& \text{otherwise,}\end{array}$$

$$C=\left(\begin{array}{cccc}{M}_1& -s_{1,2}& \cdots & -s_{1,k}\\ -s_{1,2}& M_2& \cdots & -s_{2,k}\\ \cdots & \cdots & \cdots & \cdots \\ -s_{1,k}& -s_{2,k}& \cdots & M_k\end{array}\right),$$

$$s_{i,j}=\{\begin{array}{ll}\sqrt{m_i{m}_j}& \text{if vi∼vj in G}\\ 0& \text{otherwise.}\end{array}$$

$${\text{l}}_{i,j}=\{\begin{array}{ll}-m_j& \text{if i=j and vi∼vj}\\ \sum _{v_i\sim v_r}{m}_r& \text{if i=j}\\ 0& \text{otherwise.}\end{array}$$

$$M_{d_j}=\sum _{d_i\in N_{{{\rm Y}}_n}(d_j)}\varphi \left(\frac{n}{d_i}\right)$$

$$\text{L}({{\rm Y}}_n)=\left(\begin{array}{cccc}{M}_{d_1}& -t_{1,2}& \cdots & -t_{1,k}\\ -t_{2,1}& M_{d_2}& \cdots & -t_{2,k}\\ \cdots & \cdots & \cdots & \cdots \\ -t_{k,1}& -t_{k,2}& \cdots & M_{d_k}\end{array}\right),$$

$$t_{i,j}=\{\begin{array}{ll}\varphi \left(\frac{n}{d_j}\right)& \text{if di∼dj in Υn}\\ 0& \text{otherwise}\end{array}$$

$${\sigma }_L\left(\Gamma \left({\mathbb{Z}}_n\right)\right)=\underset{j=1}{\bigcup ^k}\left(M_{d_j}+\left({\sigma }_L\left(\Gamma \left(A_{d_j}\right)\right)\setminus \{0\}\right)\right)\bigcup \sigma \left(\text{L}\left({{\rm Y}}_n\right)\right),$$

$${\sigma }_L(K_m)=\left\{\begin{array}{cc}0& m\\ 1& m-1\end{array}\right\}\text{ and }{\sigma }_L({\overline{K}}_m)=\left\{\begin{array}{c}0\\ m\end{array}\right\}.$$

$${\sigma }_L(\Gamma ({\mathbb{Z}}_{pq}))$$

$$\text{L}({{\rm Y}}_{pq})=\left(\begin{array}{cc}\varphi (p)& -\varphi (p)\\ -\varphi (q)& \varphi (q)\end{array}\right)=\left(\begin{array}{cc}p-1& -(p-1)\\ -(q-1)& q-1\end{array}\right).$$

$${\sigma }_L(\Gamma ({\mathbb{Z}}_{pq}))=\left\{\begin{array}{ccccc}p-1& q-1& 0& p+q-2& \\ q-2& p-2& 1& 1& \end{array}\right\}.$$

$$\Gamma \left({\mathbb{Z}}_{p^{2}q}\right)={{\rm Y}}_{p^{2}q}\left[\Gamma \left(A_p\right),\Gamma \left(A_{pq}\right),\Gamma \left(A_{p^2}\right),\Gamma \left(A_q\right)\right].$$

$$M_p$$

$$M_{pq}$$

$$M_{p^2}$$

$$M_q$$

$${\sigma }_L\left(\Gamma \left({\mathbb{Z}}_{p^{2}q}\right)\right)$$

$$\text{L}\left({{\rm Y}}_{p^{2}q}\right)=\left(\begin{array}{cccc}\varphi (p)& -\varphi (p)& 0& 0\\ -\varphi (pq)& \varphi (pq)+\varphi (q)& -\varphi (q)& 0\\ 0& -\varphi (p)& \varphi (p)+\varphi (p^2)& -\varphi (p^2)\\ 0& 0& -\varphi (q)& \varphi (q)\end{array}\right).$$

$$x[x^3-(p^2+pq+q-3)x^2+((pq-1)(p^2+q-2)+(p-1)(q-1))x-(p-1)(q-1)(p^2+pq-p-1)].$$

$$\Theta (G_1\vee G_2,x)=\frac{x(x-n_1-n_2)}{(x-n_1)(x-n_2)}\Theta (G_1,x-n_2)\Theta (G_2,x-n_1).$$

$$\left\{\begin{array}{c}0\\ 1\end{array}\right\}\text{mbox}or\left\{\begin{array}{cc}p-1& 0\\ p-2& 1\end{array}\right\}$$

$$\left\{\begin{array}{ccccccccc}{p}^{2m-1}-1& p^{2m-2}-1& \cdots & p^{m+1}-1& p^m-1& p^{m-1}-1& \cdots & p-1& 0\\ \varphi (p)& \varphi (p^2)& \cdots & \varphi (p^{m-1})& \varphi (p^m)-1& \varphi (p^{m+1})& \cdots & \varphi (p^{2m-1})& 1\end{array}\right\}.$$

$$\left\{\begin{array}{ccccccccc}{p}^{2m}-1& p^{2m-1}-1& \cdots & p^{m+1}-1& p^m-1& p^{m-1}-1& \cdots & p-1& 0\\ \varphi (p)& \varphi (p^2)& \cdots & \varphi (p^m)& \varphi (p^{m+1})-1& \varphi (p^{m+2})& \cdots & \varphi (p^{2m})& 1\end{array}\right\}.$$

$$H_1$$

Full text

Key words: Zero divisor graph, Algebraic connectivity, Laplacian spectral radius, Vertex connectivity

AMS subject classification. 05C25, 05C50, 05C75

1 Introduction

Let $G$ be a finite simple graph with vertex set $V(G)=\{v_{1},v_{2},\ldots,v_{n}\}$. For $1\leq i\neq j\leq n$, we write $v_{i}\sim v_{j}$ if $v_{i}$ is adjacent to $v_{j}$ in $G$. The adjacency matrix of $G$ is the $n\times n$ matrix $A(G)=(a_{ij})$, where $a_{ij}=1$ or [math] according as $v_{i}\sim v_{j}$ in $G$ or not. The Laplacian matrix $L(G)$ of $G$ is defined by $L(G):=D(G)-A(G)$, where $D(G)$ is the diagonal matrix of vertex degrees of $G$. The eigenvalues of $L(G)$ are called the Laplacian eigenvalues of $G$. Since $L(G)$ is a real, symmetric and positive semidefinite matrix, all its eigenvalues are real and nonnegative. Since the sum of the entries in each row of $L(G)$ is zero, the smallest eigenvalue of $L(G)$ is [math] with corresponding eigenvector $\textbf{1}=[1,1,\ldots,1]^{T}$. The second smallest eigenvalue of $L(G)$, denoted by $\mu(G)$, is called the algebraic connectivity of $G$. Applying the Perron-Frobenius theorem to the matrix $(n-1)I-L(G)$, it follows that $\mu(G)$ is positive if and only if $G$ is connected. The largest eigenvalue of $L(G)$, denoted by $\lambda(G)$, is called the Laplacian spectral radius of $G$. Fiedler proved that $\lambda(G)=n-\mu(\overline{G})$ [7, 3.7($1^{\circ}$)], where $\overline{G}$ denotes the complement graph of $G$. Characteristic polynomial of $L(G)$ is called the Laplacian characteristic polynomial of $G$.

The graph $G$ is called Laplacian integral if all the Laplacian eigenvalues of $G$ are integers. The vertex connectivity of $G$, denoted by $\kappa(G)$, is the minimum number of vertices which need to be removed from $V(G)$ so that the induced subgraph of $G$ on the remaining vertices is disconnected or has only one vertex. Fiedler proved that $\mu(G)\leq\kappa(G)$ for a noncomplete graph $G$ [7, 4.1]. For a complete graph $K_{m}$ on $m$ vertices, $\mu(G)=m=\kappa(G)+1$.

The spectrum of a square matrix $B$, denoted by $\sigma(B)$, is the multiset of all the eigenvalues of $B$. If $\mu_{1},\mu_{2},\ldots,\mu_{t}$ are the distinct eigenvalues of $B$ with respective multiplicities $m_{1},m_{2},\ldots,m_{t}$, then we shall denote the spectrum of $B$ by

[TABLE]

For a graph $G$, the spectrum of $L(G)$ is called the Laplacian spectrum of $G$, which is denoted by $\sigma_{L}(G)$. The Laplaian spectrum of graphs have been widely studied in the literature, see [10] and the references therein.

Let $R$ be a commutative ring with multiplicative identity $1\neq 0$. A nonzero element $x\in R$ is called a zero divisor of $R$ if there exist a nonzero element $y\in R$ such that $xy=0$. The notion of zero divisor graph of a commutative ring was first introduced by I. Beck in [3] and it was later modified by Anderson and Livingston in [2] as the following. The zero divisor graph $\Gamma(R)$ of $R$ is the simple graph with vertex set consisting of the zero divisors of $R$, in which two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. Note that $\Gamma(R)$ is the empty graph (that is, no vertex) if $R$ is an integral domain.

For a positive integer $n$, let $\mathbb{Z}_{n}$ denote the ring of integers modulo $n$. Different aspects of the zero divisor graph $\Gamma\left(\mathbb{Z}_{n}\right)$ of $\mathbb{Z}_{n}$ are studied in [1, 2, 8, 12]. In this paper, we study the Laplacian eigenvalues of the zero divisor graph $\Gamma\left(\mathbb{Z}_{n}\right)$. In Section 2, we study the structure of $\Gamma\left(\mathbb{Z}_{n}\right)$ and prove that $\Gamma\left(\mathbb{Z}_{n}\right)$ is a generalized join of certain complete graphs and null graphs222By a null graph we mean a graph with no edges.. In Section 3, we discuss the Laplacian spectrum of $\Gamma\left(\mathbb{Z}_{n}\right)$. In Section 4, we prove that the graph $\Gamma\left(\mathbb{Z}_{p^{t}}\right)$ is Laplacian integral for every prime $p$ and positive integer $t\geq 2$. Finally, in Section 5, we study the algebraic connectivity and Laplacian spectral radius of $\Gamma\left(\mathbb{Z}_{n}\right)$. We characterize the values of $n$ for which algebraic connectivity and vertex connectivity of $\Gamma\left(\mathbb{Z}_{n}\right)$ coincide. We also prove that the Laplacian spectral radius and the algebraic connectivity of $\Gamma\left(\mathbb{Z}_{n}\right)$ for most of the values of $n$ are, respectively, the largest and the second smallest eigenvalues of the vertex weighted Laplacian matrix of a graph which is defined on the set of proper divisors of $n$.

2 $\Gamma(\mathbb{Z}_{n})$ as a generalized join graph

2.1 Generalized join graphs

For two graphs $G_{1}$ and $G_{2}$ with disjoint vertex sets, recall that the join $G_{1}\vee G_{2}$ of $G_{1}$ and $G_{2}$ is the graph obtained from the union of $G_{1}$ and $G_{2}$ by adding new edges from each vertex of $G_{1}$ to every vertex of $G_{2}$. The following is a generalization of the definition of join graph (which is called generalized composition graph in [11]).

Definition 2.1**.**

Let $G$ be a graph on $k$ vertices with $V(G)=\{v_{1},v_{2},\ldots,v_{k}\}$ and let $H_{1},H_{2},\ldots,H_{k}$ be $k$ pairwise disjoint graphs. The $G$-generalized join graph $G[H_{1},H_{2},\ldots,H_{k}]$ of $H_{1},H_{2},\ldots,H_{k}$ is the graph formed by replacing each vertex $v_{i}$ of $G$ by the graph $H_{i}$ and then joining each vertex of $H_{i}$ to every vertex of $H_{j}$ whenever $v_{i}\sim v_{j}$ in $G$.

Note that if $G$ consists of two adjacent vertices only, then the $G$-generalized join graph $G[H_{1},H_{2}]$ coincides with the usual join $H_{1}\vee H_{2}$ of $H_{1}$ and $H_{2}$. The following lemma is useful for us.

Lemma 2.2**.**

Let $G$ be a graph with $V(G)=\{v_{1},v_{2},\ldots,v_{k}\}$ and let $H_{1},H_{2},\ldots,H_{k}$ be $k$ pairwise disjoint graphs. If $G$-generalized join graph $G[H_{1},H_{2},\ldots,H_{k}]$ is connected, then $G$ is connected. Conversely, if $k\geq 2$ and $G$ is connected, then $G[H_{1},H_{2},\ldots,H_{k}]$ is connected.

Proof.

Suppose that $k\geq 2$ and $G$ is connected. Let $x$ and $y$ be two distinct vertices of $G[H_{1},\ldots,H_{k}]$ with $x\in V(H_{i})$ and $y\in V(H_{j})$. First assume that $i\neq j$. Then $v_{i}\neq v_{j}$. Let $v_{i}\sim v_{i_{1}}\sim v_{i_{2}}\sim\cdots\sim v_{i_{l}}\sim v_{j}$ be a path between $v_{i}$ and $v_{j}$ in $G$. Take a vertex $a_{i_{r}}\in V(H_{i_{r}})$ for $1\leq r\leq l$. Then $x\sim a_{i_{1}}\sim a_{i_{2}}\sim\cdots\sim a_{i_{l}}\sim y$ is a path between $x$ and $y$ in $G[H_{1},H_{2},\ldots,H_{k}]$. Now assume that $i=j$. Then $x,y\in V(H_{i})$. If $H_{i}$ is connected, then there is nothing to prove. Otherwise, since $k\geq 2$, consider a neighbour $v_{l}$ of $v_{i}$ in $G$. Then, for $a\in V(H_{l})$, $x\sim a\sim y$ is a path in $G[H_{1},H_{2},\ldots,H_{k}]$. So $G[H_{1},H_{2},\ldots,H_{k}]$ is connected.

Conversely, assume that $G[H_{1},H_{2},\ldots,H_{k}]$ is connected. Let $v_{s}$ and $v_{t}$ be two distinct vertices of $G$ (so $s\neq t$). Take $x\in V(H_{s})$ and $y\in V(H_{t})$. Let $x=x_{1}\sim x_{2}\sim x_{3}\sim\cdots\sim x_{l-1}\sim x_{l}=y$ be a shortest path between $x$ and $y$ in $G[H_{1},H_{2},\ldots,H_{k}]$. Then observe that no two vertices of $x=x_{1},x_{2},\ldots,x_{l-1},x_{l}=y$ are in the same vertex set $V(H_{r})$ for any $r\in\{1,2,\ldots,k\}$. For $2\leq j\leq l-1$, assuming that $x_{j}\in V(H_{i_{j}})$ for some $i_{j}\in\{1,2,\ldots,k\}\setminus\{s,t\}$, we can see that $v_{s}\sim v_{i_{2}}\sim v_{i_{3}}\sim\cdots\sim v_{i_{l-1}}\sim v_{t}$ is a path between $v_{s}$ and $v_{t}$ in $G$. So $G$ is connected. ∎

2.2 Structure of $\Gamma(\mathbb{Z}_{n})$

For two integers $s,t$, the greatest common divisor of $s$ and $t$ is denoted by $(s,t)$. Throughout the paper, we denote the elements of the ring $\mathbb{Z}_{n}$ by $0,1,2,\cdots,n-1$. To avoid triviality of $\Gamma(\mathbb{Z}_{n})$ being an empty graph, we assume that $n\neq 1$ and that $\mathbb{Z}_{n}$ is not an integral domain. So $n\geq 4$ and $n$ is not a prime. A nonzero element $x$ of $\mathbb{Z}_{n}$ is called a unit if $xy=1$ for some element $y\in\mathbb{Z}_{n}$. Any nonzero element $a$ of $\mathbb{Z}_{n}$ is either a unit or a zero divisor according as $(a,n)=1$ or not. The number of vertices in $\Gamma(\mathbb{Z}_{n})$ is $n-\phi(n)-1$, where $\phi$ is the Euler’s totient function.

An integer $d$ is called a proper divisor of $n$ if $1<d<n$ and $d\mid n$. Let $d_{1},d_{2},\cdots,d_{k}$ be the distinct proper divisors of $n$. For $1\leq i\leq k$, we define the following sets:

[TABLE]

The sets $A_{d_{1}},A_{d_{2}},\ldots,A_{d_{k}}$ are pairwise disjoint and we can partition the vertex set of $\Gamma(\mathbb{Z}_{n})$ as

[TABLE]

The following result is proved in [12, Proposition 2.1].

Lemma 2.3**.**

[12]** $|A_{d_{i}}|=\phi\left(\frac{n}{d_{i}}\right)$ for $1\leq i\leq k$.

Note that any element $x$ of $A_{d_{i}}$ can be written as $x=m_{x}d_{i}$ for some integer $m_{x}$ with $0<m_{x}<\frac{n}{d_{i}}$ and $(m_{x},\frac{n}{d_{i}})=1$. The following lemma describes adjacency of vertices in $\Gamma(\mathbb{Z}_{n})$.

Lemma 2.4**.**

For $i,j\in\{1,2,\ldots,k\}$, a vertex of $A_{d_{i}}$ is adjacent to a vertex of $A_{d_{j}}$ in $\Gamma(\mathbb{Z}_{n})$ if and only if $n$ divides $d_{i}d_{j}$.

Proof.

Let $x\in A_{d_{i}}$ and $y\in A_{d_{j}}$. Then $x=m_{x}d_{i}$ and $y=m_{y}d_{j}$ for some integers $m_{x},m_{y}$ with $0<m_{x}<\frac{n}{d_{i}}$, $0<m_{y}<\frac{n}{d_{j}}$ and $(m_{x},\frac{n}{d_{i}})=1=(m_{y},\frac{n}{d_{j}})$. The vertices $x$ and $y$ are adjacent in $\Gamma(\mathbb{Z}_{n})$ if and only if $n$ divides $xy$, that is, if and only if $n$ divides $m_{x}m_{y}d_{i}d_{j}$. Since $(m_{x},\frac{n}{d_{i}})=1=(m_{y},\frac{n}{d_{j}})$, we have

[TABLE]

This completes the proof. ∎

As a consequence of Lemmas 2.3 and 2.4, we have the following.

Corollary 2.5**.**

The following hold:

1. (i)

For $i\in\{1,2,\ldots,k\}$, the induced subgraph $\Gamma(A_{d_{i}})$ of $\Gamma(\mathbb{Z}_{n})$ on the vertex set $A_{d_{i}}$ is either the complete graph $K_{\phi\left(\frac{n}{d_{i}}\right)}$ or its complement graph $\overline{K}_{\phi\left(\frac{n}{d_{i}}\right)}$. Indeed, $\Gamma(A_{d_{i}})$ is $K_{\phi\left(\frac{n}{d_{i}}\right)}$ if and only if $n$ divides $d_{i}^{2}$. 2. (ii)

For $i,j\in\{1,2,\ldots,k\}$ with $i\neq j$, a vertex of $A_{d_{i}}$ is adjacent to either all or none of the vertices of $A_{d_{j}}$ in $\Gamma(\mathbb{Z}_{n})$.

The above corollary implies that the partition $A_{d_{1}}\cup A_{d_{2}}\cup\cdots\cup A_{d_{k}}$ of the vertex set $V(\Gamma(\mathbb{Z}_{n}))$ of $\Gamma(\mathbb{Z}_{n})$ is an equitable partition [6, p.83], that is, every vertex in $A_{d_{i}}$ has the same number of neighbors in $A_{d_{j}}$ for all $i,j\in\{1,2,\ldots,k\}$.

Denote by $\Upsilon_{n}$ the simple graph with vertices the proper divisors $d_{1},d_{2},\ldots,d_{k}$ of $n$, in which two distinct vertices $d_{i}$ and $d_{j}$ are adjacent if and only if $n$ divides $d_{i}d_{j}$. If $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{r}^{n_{r}}$ is the prime power factorization of $n$, where $r,n_{1},n_{2},\ldots,n_{r}$ are positive integers and $p_{1},p_{2},\ldots,p_{r}$ are distinct prime numbers, then the number of vertices of $\Upsilon_{n}$ is given by:

[TABLE]

The graph $\Upsilon_{n}$ shall play an important role in the rest of the paper.

Lemma 2.6**.**

$\Upsilon_{n}$* is a connected graph.*

Proof.

Consider two vertices $d_{i}$ and $d_{j}$ of $\Upsilon_{n}$ with $i\neq j$ and let $(d_{i},d_{j})=l$. If $l\neq 1$, then $\frac{n}{l}$ is a vertex of $\Upsilon_{n}$ and $d_{i}\sim\frac{n}{l}\sim d_{j}$ in $\Upsilon_{n}$. If $l=1$, then $d_{i}d_{j}$ divides $n$ and so $d_{i}\sim\frac{n}{d_{i}}\sim\frac{n}{d_{j}}\sim d_{j}$ in $\Upsilon_{n}$. So $\Upsilon_{n}$ is connected. ∎

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

Lemma 2.7**.**

Let $\Gamma(A_{d_{i}})$ be the induced subgraph of $\Gamma(\mathbb{Z}_{n})$ on the vertex set $A_{d_{i}}$ for $1\leq i\leq k$. Then $\Gamma(\mathbb{Z}_{n})=\Upsilon_{n}[\Gamma(A_{d_{1}}),\Gamma(A_{d_{2}}),\cdots,\Gamma(A_{d_{k}})].$

Proof.

Replace the vertex $d_{i}$ of $\Upsilon_{n}$ by $\Gamma(A_{d_{i}})$ for $1\leq i\leq k$. Then the result can be seen using Lemma 2.4. ∎

Corollary 2.8**.**

$\Gamma(\mathbb{Z}_{n})$* is connected.*

Proof.

If $n$ has at least two proper divisors, then $|V(\Upsilon_{n})|\geq 2$ and so the corollary follows from Lemmas 2.2, 2.6 and 2.7. If $n$ has exactly one proper divisor, then $n=p^{2}$ for some prime $p$. In this case, $\Gamma\left(\mathbb{Z}_{p^{2}}\right)=\Gamma(A_{p})$ has $p-1$ vertices and $\Gamma(A_{p})$ is a complete graph by Corollary 2.5(i). ∎

We note that the above corollary also follows from a more general result by Anderson and Livingston in [2, Theorem 2.3] which says that the zero divisor graph of any commutative ring with multiplicative identity is connected.

Corollary 2.9**.**

$\Gamma(\mathbb{Z}_{n})$* is a complete graph if and only if $n=p^{2}$ for some prime $p$.*

Proof.

If $n=p^{2}$ for some prime $p$, then $\Gamma(\mathbb{Z}_{n})=\Gamma(A_{p})$ is the complete graph $K_{\phi(p)}$ by Corollary 2.5(i). Conversely, assume that $\Gamma(\mathbb{Z}_{n})$ is a complete graph. If $p$ is a prime divisor of $n$, then $\Gamma(A_{p})$ must be a complete graph. So $n\mid p^{2}$ by Corollary 2.5(i) and it follows that $n=p^{2}$. ∎

Example 2.10**.**

The zero divisor graph $\Gamma(\mathbb{Z}_{18})$ of $\mathbb{Z}_{18}$ is shown in Figure 1. Here $V(\Upsilon_{18})=\{2,3,6,9\}$ and $\Upsilon_{18}$ is the path $P_{4}:2\sim 9\sim 6\sim 3$. By Lemma 2.7, we have

[TABLE]

where $\Gamma(A_{2})=\overline{K}_{6}$, $\Gamma(A_{3})=\overline{K}_{2}$, $\Gamma(A_{6})=K_{2}$ and $\Gamma(A_{9})$ is an isolated vertex. In Figure 1, the dotted lines between two circles mean that each vertex in one circle is adjacent to every vertex in the other circle.

3 Laplacian Spectrum

For a vertex $v$ of a graph $G$, $N_{G}(v)$ denotes the neighbourhood of $v$ in $G$, that is, the set of vertices of $G$ which are adjacent to $v$ in $G$.

3.1 Laplacian spectrum of generalized join graphs

The following theorem was proved in [4, Theorem 8] by Cardoso et al., in which the Laplacian spectrum of a generalized join graph $G[H_{1},H_{2},\ldots,H_{k}]$ is expressed in terms of the Laplacian spectrum of the graphs $H_{i}$ and the spectrum of another $k\times k$ matrix.

Theorem 3.1**.**

[4]** Let $G$ be a graph on $k$ vertices with $V(G)=\{v_{1},v_{2},\ldots,v_{k}\}$ and let $H_{1},H_{2},\ldots,H_{k}$ be $k$ pairwise disjoint graphs on $m_{1},m_{2},\ldots,m_{k}$ vertices, respectively. Then the Laplacian spectrum of $G[H_{1},H_{2},\ldots,H_{k}]$ is given by

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

In (1), $M_{j}+(\sigma_{L}(H_{j})\setminus\{0\})$ means that $M_{j}$ is added to each element of $\sigma_{L}(H_{j})\setminus\{0\}$.

Consider $G$ as a vertex weighted graph by assigning the weight $m_{i}=|V(H_{i})|$ to the vertex $v_{i}$ of $G$ for $1\leq i\leq k$. Let $\textbf{L}(G)=(\textbf{l}_{i,j})$ be the $k\times k$ matrix, where

[TABLE]

The matrix $\textbf{L}(G)$ is called a vertex weighted Laplacian matrix of $G$, which is a zero row sum matrix but not symmetric in general.

Note that the matrix $C$ in Theorem 3.1 is precisely the matrix $\mathcal{L}(G)$ defined in [5, p. 317], which is symmetric but need not be a zero row sum matrix. Further, if $W$ is the $k\times k$ diagonal matrix with diagonal entries $m_{1},m_{2},\ldots,m_{k}$, then $\textbf{L}(G)=W^{-\frac{1}{2}}CW^{\frac{1}{2}}$ and so $\textbf{L}(G)$ and $C$ are similar. We thus have the following.

Proposition 3.2**.**

$\sigma(C)=\sigma(\textbf{L}(G))$.

3.2 Laplacian Spectrum of $\Gamma(\mathbb{Z}_{n})$

Let $d_{1},d_{2},\ldots,d_{k}$ be the proper divisors of $n$. For $1\leq i\leq k$, we assign the weight $\phi\left(\frac{n}{d_{i}}\right)=|A_{d_{i}}|$ to the vertex $d_{i}$ of the graph $\Upsilon_{n}$. Define

[TABLE]

for $1\leq j\leq k$. The $k\times k$ vertex weighted Laplacian matrix $\textbf{L}(\Upsilon_{n})$ of $\Upsilon_{n}$ defined in Section 3.1 is given by

[TABLE]

where

[TABLE]

for $1\leq i\neq j\leq k$. The following theorem describes the Laplacian spectrum of the zero-divisor graph of $\mathbb{Z}_{n}$.

Theorem 3.3**.**

If $d_{1},d_{2},\cdots,d_{k}$ are the proper divisors of $n$, then the Laplacian spectrum of $\Gamma(\mathbb{Z}_{n})$ is given by

[TABLE]

where $M_{d_{j}}+\left(\sigma_{L}\left(\Gamma\left(A_{d_{j}}\right)\right)\right)$ means that $M_{d_{j}}$ is added to each element of $\sigma_{L}\left(\Gamma\left(A_{d_{j}}\right)\right)\setminus\{0\}$.

Proof.

By Lemma 2.7, we have $\Gamma(\mathbb{Z}_{n})=\Upsilon_{n}[\Gamma(A_{d_{1}}),\Gamma(A_{d_{2}}),\ldots,\Gamma(A_{d_{k}})]$. Then the result follows from Theorem 3.1 and Proposition 3.2. ∎

The Laplacian spectrum of the complete graph $K_{m}$ on $m$ vertices and its complement graph $\overline{K}_{m}$ are known. Indeed,

[TABLE]

By Corollary 2.5(i), $\Gamma(A_{d_{i}})$ is either $K_{\phi\left(\frac{n}{d_{i}}\right)}$ or $\overline{K}_{\phi\left(\frac{n}{d_{i}}\right)}$ for $1\leq i\leq k$. Also, $M_{d_{j}}>0$ as $\Upsilon_{n}$ is connected by Lemma 2.6. Thus, by Theorem 3.3, out of the $n-\phi(n)-1$ number of Laplacian eigenvalues of $\Gamma(\mathbb{Z}_{n})$, $n-\phi(n)-1-k$ of them are known to be nonzero integer values. The remaining $k$ Laplacian eigenvalues of $\Gamma(\mathbb{Z}_{n})$ will come from the spectrum of $\textbf{L}(\Upsilon_{n})$.

Example 3.4**.**

*We discuss the Laplacian spectrum of $\Gamma(\mathbb{Z}_{n})$ for $n\in\{pq,p^{2}q\}$, where $p$ and $q$ are distinct primes.

(i) Let $n=pq$, where $p<q$ are distinct primes. The proper divisors of $n$ are $p$ and $q$. So $\Upsilon_{pq}:p\sim q$ is $K_{2}$ and by Lemma 2.7, $\Gamma(\mathbb{Z}_{pq})=\Upsilon_{pq}[\Gamma(A_{p}),\Gamma(A_{q})]$. By Corollary 2.5(i), $\Gamma(A_{p})=\overline{K}_{\phi(q)}$ and $\Gamma(A_{q})=\overline{K}_{\phi(p)}$. We have $M_{p}=\phi\left(\frac{n}{q}\right)=\phi(p)$ and $M_{q}=\phi\left(\frac{n}{p}\right)=\phi(q)$. So, by Theorem 3.3, the Laplacian spectrum of $\Gamma(\mathbb{Z}_{pq})$ is given by

[TABLE]

We have

[TABLE]

which has eigenvalues $p+q-2$ and [math]. Thus the Laplacian spectrum of $\Gamma(\mathbb{Z}_{pq})$ is

[TABLE]

*Note that $\Gamma(\mathbb{Z}_{pq})=\Upsilon_{pq}[\Gamma(A_{p}),\Gamma(A_{q})]=\overline{K}_{\phi(p)}\vee\overline{K}_{\phi(q)}=K_{\phi(p),\phi(q)}$. Using the result known for the Laplacian eigenvalues of complete bipartite graphs, the Laplacian spectrum of $\Gamma(\mathbb{Z}_{pq})$ can also be obtained as above.

(ii) Let $n=p^{2}q$, where $p$ and $q$ are distinct primes. The proper divisors of $n$ are $p$, $q$, $pq$ and $p^{2}$. So $\Upsilon_{p^{2}q}$ is the path $P_{4}:p\sim pq\sim p^{2}\sim q$. By Lemma 2.7,

[TABLE]

By Corollary 2.5(i), $\Gamma\left(A_{p}\right)=\overline{K}_{\phi(pq)}$, $\Gamma\left(A_{pq}\right)=K_{\phi(p)}$, $\Gamma\left(A_{p^{2}}\right)=\overline{K}_{\phi(q)}$ and $\Gamma\left(A_{q}\right)=\overline{K}_{\phi\left(p^{2}\right)}$. We have

[TABLE]

So, by Theorem 3.3, the Laplacian spectrum of $\Gamma(\mathbb{Z}_{p^{2}q})$ is given by

[TABLE]

We have

[TABLE]

The characteristic polynomial $Q(x)$ of $\textbf{L}\left(\Upsilon_{p^{2}q}\right)$ is

[TABLE]

If $K(x)=\frac{Q(x)}{x}$, then the algebraic connectivity and the Lapacian spectral radius of $\Gamma\left(\mathbb{Z}_{p^{2}q}\right)$ are the smallest and the largest roots of $K(x)=0,$ respectively. This follows from Theorem 5.8 in the last section.

4 Laplacian Integrality of $\Gamma\left(\mathbb{Z}_{p^{m}}\right)$

Recall that a graph $G$ is called Laplacian integral if all the Laplacian eigenvalues of $G$ are integers. The following proposition is an immediate consequence of the observation made in the paragraph after Theorem 3.3.

Proposition 4.1**.**

The zero-divisor graph $\Gamma(\mathbb{Z}_{n})$ is Laplacian integral if and only if all the eigenvalues of $\textbf{L}(\Upsilon_{n})$ are integers.

By Example 3.4(i), the graph $\Gamma(\mathbb{Z}_{pq})$ is Laplacian integral for distinct primes $p$ and $q$. In this section, we shall prove that $\Gamma(\mathbb{Z}_{p^{t}})$ is Lapacian integral for every prime $p$ and $t\geq 2$. One approach is to show that all the eigenvalues of $\textbf{L}\left(\Upsilon_{p^{t}}\right)$ are integers and then to use Proposition 4.1. However, if $t$ is large, then it is more difficult to find the eigenvalues of $\textbf{L}\left(\Upsilon_{p^{t}}\right)$. We shall adopt a different approach to find the Laplacian characteristic polynomial of $\Gamma\left(\mathbb{Z}_{p^{t}}\right)$. For this, we first express $\Gamma\left(\mathbb{Z}_{p^{t}}\right)$ as the union and join of certain complete graphs and null graphs and then use Theorem 4.2 below to find the Laplacian eigenvalues of $\Gamma\left(\mathbb{Z}_{p^{t}}\right)$.

For a graph $G$, we denote the characteristic polynomial of $L(G)$ by $\Theta(G,x)$. The following theorem gives the Laplacian characteristic polynomial of the join of two graphs, see [10, Corollary 3.7].

Theorem 4.2**.**

[10]** Let $G_{1}$ and $G_{2}$ be two vertex disjoint graphs on $n_{1}$ and $n_{2}$ vertices, respectively. Then the Laplacian characteristic polynomial of $G_{1}\vee G_{2}$ is given by

[TABLE]

Theorem 4.3**.**

Let $n=p^{t}$ where $p$ is a prime and $t\geq 2$ is a positive integer. Then the following hold.

1. (i)

If $t=2$, then the Laplacian spectrum of $\Gamma(\mathbb{Z}_{n})$ is given by

[TABLE]

according as $p=2$ or $p\geq 3$. 2. (ii)

If $t=2m$ for some integer $m\geq 2$, then the Laplacian spectrum of $\Gamma(\mathbb{Z}_{n})$ is given by

[TABLE] 3. (iii)

If $t=2m+1$ for some integer $m\geq 1$, then the Laplacian spectrum of $\Gamma(\mathbb{Z}_{n})$ is given by

[TABLE]

Proof.

(i) We have $\Gamma\left(\mathbb{Z}_{p^{2}}\right)=\Gamma\left(A_{p}\right)$ is the complete graph $K_{p-1}$ by Corollary 2.5(i) and so the results follows depending on $p=2$ or not.

(ii) Here $n=p^{2m}$ with $m\geq 2$ and the proper divisors of $n$ are $p,p^{2},\ldots,p^{2m-1}$. We shall express the graph $\Upsilon_{p^{2m}}$ as the join and union of certain graphs. Observe that the vertex $p^{i}$, $1\leq i\leq 2m-1$, of $\Upsilon_{p^{2m}}$ is adjacent to the vertex $p^{j}$ for every $j\geq 2m-i$ with $j\neq i$. Define the following graphs $H_{1},H_{2},\ldots,H_{m}$ recursively, where $\{x\}$ denotes the graph with one vertex $x$:

[TABLE]

It can be seen that $H_{m}$ is precisely the graph $\Upsilon_{p^{2m}}$. Now define the graphs $G_{1},G_{2},\ldots,G_{m}$ recursively as given below:

[TABLE]

We have $\Gamma\left(\mathbb{Z}_{p^{2m}}\right)=\Upsilon_{p^{2m}}\left[\Gamma\left(A_{p}\right),\Gamma\left(A_{p^{2}}\right),\ldots,\Gamma\left(A_{p^{2m-1}}\right)\right]$. Since $\Gamma\left(A_{p^{j}}\right)=K_{\phi\left(p^{2m-j}\right)}$ for $m\leq j\leq 2m-1$ and $\Gamma\left(A_{p^{j}}\right)=\overline{K}_{\phi\left(p^{2m-j}\right)}$ for $1\leq j\leq m-1$, it follows that $\Gamma(\mathbb{Z}_{p^{2m}})$ is precisely the graph $G_{m}$. The Laplacian characteristic polynomial of $G_{1}$ is

[TABLE]

and that of $\overline{K}_{\phi\left(p^{m+1}\right)}\cup G_{1}$ is

[TABLE]

Using Theorem 4.2, the Laplacian characteristic polynomial of $G_{2}$ is

[TABLE]

Now the Laplacian characteristic polynomial of $\overline{K}_{\phi\left(p^{m+2}\right)}\cup G_{2}$ is

[TABLE]

Again using Theorem 4.2, it can be calculated that the Laplacian characteristic polynomial of $G_{3}$ is

[TABLE]

Continuing in this way, we finally get that

[TABLE]

Since $\Gamma(\mathbb{Z}_{p^{2m}})=G_{m}$, we have $\Theta\left(\Gamma\left(\mathbb{Z}_{p^{2m}}\right),x\right)=\Theta\left(G_{m},x\right)$. Then the result follows from the above using the fact that $\underset{i=1}{\overset{r}{\sum}}\phi\left(p^{i}\right)=p^{r}-1$ for any positive integer $r$.

(ii) Here $n=p^{2m+1}$ and the proper divisors of $n$ are $p,p^{2},\ldots,p^{2m}$. As in (i), we shall express the graph $\Upsilon_{p^{2m+1}}$ as the join and union of certain graphs. The vertex $p^{i}$, $1\leq i\leq 2m$, of $\Upsilon_{p^{2m+1}}$ is adjacent to the vertex $p^{j}$ for every $j\geq 2m+1-i$. Define the graphs $X_{1},X_{2},\ldots,X_{m}$ recursively as given below:

[TABLE]

Then $X_{m}$ is precisely the graph $\Upsilon_{p^{2m+1}}$. Now define the graphs $Y_{1},Y_{2},\ldots,Y_{m}$ recursively as given below:

[TABLE]

As in (i), it can be seen that $\Gamma\left(\mathbb{Z}_{p^{2m+1}}\right)$ is precisely the graph $Y_{m}$. Using Theorem 4.2, we get

[TABLE]

Starting with $\Theta\left(Y_{1},x\right)$ and applying the argument as in (i), we can calculate the Laplacian characteristic polynomials of $Y_{2},Y_{3},\ldots,Y_{m}$ and get the required result. ∎

As a consequence of Proposition 4.1 and Theorem 4.3, we have the following.

Corollary 4.4**.**

If $p$ is a prime and $t\geq 2$, then $\Gamma\left(\mathbb{Z}_{p^{t}}\right)$ is Lapacian integral and so all the eigenvalues of $\textbf{L}\left(\Upsilon_{p^{t}}\right)$ are integers.

Corollary 4.5**.**

Let $n=p^{t}$ for some prime $p$ and positive integer $t$ with $n\neq 4$. Then $\lambda(\Gamma(\mathbb{Z}_{p^{t}}))=|\Gamma(\mathbb{Z}_{p^{t}})|$.

Proof.

We have $\left|\Gamma\left(\mathbb{Z}_{p^{t}}\right)\right|=p^{t}-\phi(p^{t})-1=p^{t-1}-1$. From Theorem 4.3, we get that $\lambda(\Gamma(\mathbb{Z}_{p^{t}}))=p^{t-1}-1$ and so the corollary follows. ∎

5 Algebraic connectivity and Laplacian spectral radius of $\Gamma(\mathbb{Z}_{n})$

In this section, we shall study the algebraic connectivity and the Laplacian spectral radius of $\Gamma(\mathbb{Z}_{n})$. We recall two well-known bounds for the Laplacian spectral radius of a graph.

Theorem 5.1**.**

[7]** If $G$ is a graph on $m$ vertices, then $\lambda(G)\leq m$. Further, equality holds if and only if $\overline{G}$ is disconnected if and only if $G$ is the join of two graphs.

The above theorem follows from the relation $\lambda(G)=m-\mu(\overline{G})$ and the fact that $\overline{G}$ is disconnected if and only if $G$ is the join of two graphs. The following result was proved in [13, Theorem 2.3].

Theorem 5.2**.**

[13]** Let $G$ be a connected graph on $m$ vertices with maximal degree $\Delta(G)$. Then $\lambda(G)\geq\Delta(G)+1$, and equality holds if and only if $\Delta(G)=m-1$.

The following proposition characterizes the values of $n$ for which the complement graph of $\Gamma(\mathbb{Z}_{n})$ is disconnected. Note that if $n=4$, then $\Gamma(\mathbb{Z}_{4})=\overline{\Gamma(\mathbb{Z}_{4})}=\{2\}$ is a singleton.

Proposition 5.3**.**

$\overline{\Gamma(\mathbb{Z}_{n})}$* is disconnected if and only if $n$ is a product of two distinct primes or $n$ is a prime power with $n\neq 4$.*

Proof.

If $n=pq$ for distinct primes $p$ and $q$, then $\Gamma(\mathbb{Z}_{n})=K_{\phi(p),\phi(q)}$, see Example 3.4(i). If $n=p^{2}$ for some prime $p\geq 3$, then $\Gamma(\mathbb{Z}_{n})=\Gamma(A_{p})=K_{\phi(p)}$ by Corollary 2.5(i) and it contains at least two vertices. If $n=p^{t}$ for some prime $p$ with $t\geq 3$, then the vertex $p^{t-1}$ is adjacent to all other vertices of $\Gamma(\mathbb{Z}_{n})$. In all the three cases, it follows that $\overline{\Gamma(\mathbb{Z}_{n})}$ is disconnected.

Conversely, let $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{r}^{n_{r}}$, where $r$, $n_{1},n_{2},\ldots,n_{r}$ are positive integers and $p_{1},p_{2},\ldots,p_{r}$ are distinct primes. Suppose that $r\geq 2$ and that $n_{1}>1$ or $n_{2}>1$ if $r=2$. We show that $\overline{\Gamma(\mathbb{Z}_{n})}$ is connected.

The vertices $p_{i}$ and $p_{j}$ are not adjacent in $\Upsilon_{n}$ for $1\leq i\neq j\leq k$. So the vertices $p_{1},p_{2},\ldots,p_{r}$ form a clique in $\overline{\Upsilon_{n}}$.

Let $v$ be vertex of $\overline{\Upsilon_{n}}$ different from $p_{1},p_{2},\ldots,p_{r}$. There exists $i\in\{1,2,\ldots,r\}$ such that $p_{i}^{t}$ divides $v$, but $p_{i}^{t+1}$ does not divide $v$ for some $t$ with $0\leq t<n_{i}$. Then, for $j\in\{1,2,\ldots,r\}\setminus\{i\}$, $v$ and $p_{j}$ are not adjacent in $\Upsilon_{n}$ as $n$ does not divide $vp_{j}$ and so $v$ and $p_{j}$ are adjacent in $\overline{\Upsilon_{n}}$. It follows that $\overline{\Upsilon_{n}}$ is connected. If $d_{1},d_{2},\ldots,d_{k}$ are the proposer divisors of $n$, then $\Gamma(\mathbb{Z}_{n})=\Upsilon_{n}\left[\Gamma(A_{d_{1}}),\Gamma(A_{d_{2}}),\ldots,\Gamma(A_{d_{k}})\right]$ implies that $\overline{\Gamma(\mathbb{Z}_{n})}=\overline{\Upsilon_{n}}\left[\overline{\Gamma(A_{d_{1}})},\overline{\Gamma(A_{d_{2}})},\ldots,\overline{\Gamma(A_{d_{k}})}\right]$. As $k\geq 2$, Lemma 2.2 implies that $\overline{\Gamma(\mathbb{Z}_{n})}$ is connected. ∎

The following proposition characterizes the values of $n$ for which equality holds in Theorem 5.1 when $G=\Gamma(\mathbb{Z}_{n})$.

Proposition 5.4**.**

$\lambda(\Gamma(\mathbb{Z}_{n}))=|\Gamma(\mathbb{Z}_{n})|$* if and only if $n$ is a product of two distinct primes or $n$ is a prime power with $n\neq 4$.*

Proof.

If $n$ is not a product of two distinct primes nor a prime power, then $\overline{\Gamma(\mathbb{Z}_{n})}$ is connected by Proposition 5.3. In this case, $\lambda(\Gamma(\mathbb{Z}_{n}))<|\Gamma(\mathbb{Z}_{n})|$ by Theorem 5.1. If $n=4$, then $\lambda(\Gamma(\mathbb{Z}_{4}))=0<1=|\Gamma(\mathbb{Z}_{4})|$.

If $n$ is a prime power with $n\neq 4$, then $\lambda(\Gamma(\mathbb{Z}_{n}))=|\Gamma(\mathbb{Z}_{n})|$ by Corollary 4.5. Assume that $n=pq$ for two distinct primes $p$ and $q$. Then $|\Gamma(\mathbb{Z}_{pq})|=pq-\phi(pq)-1=p+q-2$. From Example 3.4(i), we have $\lambda(\Gamma(\mathbb{Z}_{pq}))=p+q-2$ and so $\lambda(\Gamma(\mathbb{Z}_{pq}))=|\Gamma(\mathbb{Z}_{pq})|$. ∎

The following theorem was proved in [8, Theorem 3.2], which determines the vertex connectivity $\kappa(\Gamma(\mathbb{Z}_{n}))$ of $\Gamma(\mathbb{Z}_{n})$.

Theorem 5.5**.**

[8]** Let $p$ be the smallest prime divisor of $n$ and let $\delta\left(\Gamma\left(\mathbb{Z}_{n}\right)\right)$ denote the minimal degree of $\Gamma\left(\mathbb{Z}_{n}\right)$. Then the following hold:

1. (i)

If $n$ is divisible by at least two distinct primes, then $\kappa\left(\Gamma\left(\mathbb{Z}_{n}\right)\right)=\delta\left(\Gamma\left(\mathbb{Z}_{n}\right)\right)=p-1$ and the vertex $p$ has minimal degree. 2. (ii)

Let $n=p^{t}$ with $t\geq 2$. Then $\kappa\left(\Gamma\left(\mathbb{Z}_{n}\right)\right)=\delta\left(\Gamma\left(\mathbb{Z}_{n}\right)\right)=p-2$ if $t=2$, and $\kappa\left(\Gamma\left(\mathbb{Z}_{n}\right)\right)=\delta\left(\Gamma\left(\mathbb{Z}_{n}\right)\right)=p-1$ if $t>2$. In both cases, the vertex $p$ has minimal degree.

We shall use Theorem 5.5 along with the following result of Krikland et al. [9, Theorem 2.1] to characterize the values of $n$ for which vertex connectivity and algebraic connectivity of $\Gamma(\mathbb{Z}_{n})$ are equal.

Theorem 5.6**.**

[9]** Let $G$ be a noncomplete connected graph on $m$ vertices. Then $\kappa(G)=\mu(G)$ if and only if $G$ can be written as $G_{1}\vee G_{2}$, where $G_{1}$ is a disconnected graph on $m-\kappa(G)$ vertices and $G_{2}$ is a graph on $\kappa(G)$ vertices with $\mu(G_{2})\geq 2\kappa(G)-m$.

Proposition 5.7**.**

$\mu(\Gamma(\mathbb{Z}_{n}))=\kappa(\Gamma(\mathbb{Z}_{n}))$* if and only if $n$ is product of two distinct primes or $n=p^{t}$ for some prime $p$ and integer $t\geq 3$.*

Proof.

We have $\mu(\Gamma(\mathbb{Z}_{n}))\leq\kappa(\Gamma(\mathbb{Z}_{n}))$ if and only if $\Gamma(\mathbb{Z}_{n})$ is not a complete graph, that is, if and only if $n$ is not the square of a prime by Corollary 2.9.

If $n$ is not a product of two distinct primes nor a prime power, then $\overline{\Gamma(\mathbb{Z}_{n})}$ is connected by Proposition 5.3 and so $\Gamma(\mathbb{Z}_{n})$ is not a join of two graphs. Since $\Gamma(\mathbb{Z}_{n})$ is noncomplete and connected, Theorem 5.6 implies that $\mu(\Gamma(\mathbb{Z}_{n}))<\kappa(\Gamma(\mathbb{Z}_{n}))$.

If $n=pq$ for some primes $p<q$, then $\kappa(\Gamma(\mathbb{Z}_{pq}))=p-1$ by Theorem 5.5(i). From Example 3.4(i), we have $\mu(\Gamma(\mathbb{Z}_{pq}))=p-1$ and so $\mu(\Gamma(\mathbb{Z}_{pq}))=\kappa(\Gamma(\mathbb{Z}_{pq}))$.

If $n=p^{t}$ for some prime $p$ and positive integer $t\geq 3$, then $\mu(\Gamma(\mathbb{Z}_{p^{t}}))=p-1=\kappa(\Gamma(\mathbb{Z}_{p^{t}}))$ by Theorems 4.3(ii), 4.3(iii) and 5.5(ii). ∎

Theorem 5.8**.**

The following hold:

1. (i)

If $n$ is not a prime power nor a product of two distinct primes, then $\mu(\Gamma(\mathbb{Z}_{n}))$ is the second smallest eigenvalue of $\textbf{L}(\Upsilon_{n})$. 2. (ii)

If $n$ is not a prime power, then $\lambda(\Gamma(\mathbb{Z}_{n}))$ is the largest eigenvalue of $\textbf{L}(\Upsilon_{n})$.

Proof.

By Theorem 3.3, the Laplacian spectrum of $\Gamma(\mathbb{Z}_{n})$ is given by

[TABLE]

where $d_{1},d_{2},\cdots,d_{k}$ are the proper divisors of $n$ and $M_{d_{j}}$ is defined in (2) for $1\leq j\leq k$.

(i) Let $p$ be the smallest prime divisor of $n$. Since $n$ is not a product of two distinct primes nor a prime power, Theorem 5.5(i) and Proposition 5.7 give that

[TABLE]

Let $\alpha$ be the minimum of the Laplacian eigenvalues of $\Gamma(\mathbb{Z}_{n})$ which are contained in

[TABLE]

Then

[TABLE]

where the minimum is taken over all $j$ for which $\Gamma(A_{d_{j}})$ is not a singleton. The connectedness of $\Upsilon_{n}$ (Lemma 2.6) implies that $M_{d_{j}}\geq p-1$ for $1\leq j\leq k$ and hence $\alpha\geq p-1$. Then (3) implies that $\mu(\Gamma(\mathbb{Z}_{n}))$ must be an eigenvalue of $\textbf{L}(\Upsilon_{n})$. Since [math] is an eigenvalue of $\textbf{L}(\Upsilon_{n})$, it follows that $\mu(\Gamma(\mathbb{Z}_{n}))$ is the second smallest eigenvalue of $\textbf{L}(\Upsilon_{n})$.

(ii) If $n$ is a product of two distinct primes, then the result follows from Example 3.4(i). Assume that $n$ is not a prime power nor a product of two distinct primes. Then $\overline{\Gamma(\mathbb{Z}_{n})}$ is connected by Proposition 5.3. It follows from Theorems 5.1 and 5.2 that

[TABLE]

where $\Delta(\Gamma(\mathbb{Z}_{n}))$ is the maximal degree in $\Gamma(\mathbb{Z}_{n})$. Let $\beta$ be the maximum of the Laplacian eigenvalues of $\Gamma(\mathbb{Z}_{n})$ which are contained in $\underset{j=1}{\overset{k}{\bigcup}}\left(M_{d_{j}}+\left(\sigma_{L}\left(\Gamma\left(A_{d_{j}}\right)\right)\setminus\{0\}\right)\right)$. Then

[TABLE]

where the maximum is taken over all $j$ for which $\Gamma(A_{d_{j}})$ is not a singleton.

Let $v$ be a vertex of $\Gamma(\mathbb{Z}_{n})$. Then $v\in A_{d_{j}}$ for some $j\in\{1,2,\ldots,k\}$. By Corollary 2.5(i), $\Gamma\left(A_{d_{j}}\right)$ is $K_{\phi\left(\frac{n}{d_{j}}\right)}$ or $\overline{K}_{\phi\left(\frac{n}{d_{j}}\right)}$. If $\Gamma\left(A_{d_{j}}\right)=K_{\phi\left(\frac{n}{d_{j}}\right)}$, then

[TABLE]

If $\Gamma\left(A_{d_{j}}\right)=\overline{K}_{\phi\left(\frac{n}{d_{j}}\right)}$, then

[TABLE]

Thus $\Delta\left(\Gamma\left(\mathbb{Z}_{n}\right)\right)+1=\max\{\deg(v)+1:v\in V\left(\Gamma\left(\mathbb{Z}_{n}\right)\right)\}\geq M_{d_{j}}+\lambda\left(\Gamma\left(A_{d_{j}}\right)\right)$ for $1\leq j\leq k$ and hence $\Delta\left(\Gamma\left(\mathbb{Z}_{n}\right)\right)+1\geq\beta$. Then (4) gives that

[TABLE]

and it follows that $\lambda(\Gamma(\mathbb{Z}_{n}))$ is the largest eigenvalue of $\textbf{L}(\Upsilon_{n})$. ∎

From the proof of the above theorem, the following corollary follows.

Corollary 5.9**.**

Let $d_{1},d_{2},\cdots,d_{k}$ be the proper divisors of $n$. Then the following hold:

1. (i)

If $n$ is not a prime power nor a product of two distinct primes, then $\mu(\Gamma(\mathbb{Z}_{n}))$ is not contained in $\left(M_{d_{j}}+\left(\sigma_{L}\left(\Gamma\left(A_{d_{j}}\right)\right)\setminus\{0\}\right)\right)$ for $1\leq i\leq k$. 2. (ii)

If $n$ is not a prime power, then $\lambda(\Gamma(\mathbb{Z}_{n}))$ is not contained in $\left(M_{d_{j}}+\left(\sigma_{L}\left(\Gamma\left(A_{d_{j}}\right)\right)\setminus\{0\}\right)\right)$ for $1\leq i\leq k$.