Algebraic Connectivity of Power Graphs of Finite Cyclic Groups

Abstract

The power graph ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ of ${\displaystyle {\mathbb{Z}}_n}$ for a finite cyclic group ${\displaystyle {\mathbb{Z}}_n}$ is a simple undirected connected graph such that two distinct nodes ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle {\mathbb{Z}}_n}$ are adjacent in ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ if and only if ${\displaystyle x\ne y}$ and ${\displaystyle x^i=y}$ or ${\displaystyle y^i=x}$ for some non-negative integer ${\displaystyle i}$. In this article, we find the Laplacian eigenvalues of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ and show that ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is Laplacian integral (integer algebraic connectivity) if and only if ${\displaystyle n}$ is either the product of two distinct primes or a prime power. That answers a conjecture by Panda, Graphs and Combinatorics, (2019).

1. Introduction

In this paper, we consider only connected, undirected, simple, and finite graphs unless otherwise stated. Let ${\displaystyle G(V,E)}$ be a graph with node (vertex) set ${\displaystyle V=\{z_1,z_2,\dots ,z_n\}}$ and ${\displaystyle E}$ be its edge set. The number ${\displaystyle \left|V\right|}$ is order${\displaystyle n}$ and the number ${\displaystyle \left|E\right|}$ is size ${\displaystyle m}$ of ${\displaystyle G}$. The collection of nodes adjacent to ${\displaystyle u\in V\left(G\right)}$ excluding ${\displaystyle u}$, denoted by ${\displaystyle N_G\left(u\right)}$, is the neighbourhood of ${\displaystyle v}$ (open neighbourhood). The degree ${\displaystyle e_i}$ of ${\displaystyle z_i}$ is ${\displaystyle e_i=\left|N_G\left(z_i\right)\right|}$. A graph ${\displaystyle G}$ is ${\displaystyle r}$-regular, provided ${\displaystyle e_i=r}$ for each ${\displaystyle i=1,\cdots ,n.}$ If ${\displaystyle z_i}$ is adjacent to ${\displaystyle z_j}$, we denote it by ${\displaystyle z_i\sim z_j}$, otherwise ${\displaystyle z_i\nsim z_j.}$ Furthermore, by ${\displaystyle K_n}$ we mean the complete graph and by ${\displaystyle K_{1,n-1}}$ and ${\displaystyle P_n}$ we denote the star and the path graph, respectively. For more terminology and notations, see [1,2].

The adjacency matrix ${\displaystyle A\left(G\right)}$ is an ${\displaystyle n}$-square matrix with ${\displaystyle (0,1)}$ entries such that ${\displaystyle (i,j)}$-entry is ${\displaystyle 1}$ if ${\displaystyle z_i\sim z_j}$, and e ${\displaystyle 0}$ otherwise. The matrix ${\displaystyle A\left(G\right)}$ is a real symmetric matrix, so we index its eigenvalues in the following manner:

$${\zeta }_1\left(G\right)\ge {\zeta }_2\left(G\right)\ge \cdots \ge {\zeta }_n\left(G\right).$$

The collection of ${\displaystyle {\zeta }_i}$’s is the spectrum of ${\displaystyle A\left(G\right)}$ (or spectrum of graph ${\displaystyle G}$). Let ${\displaystyle D\left(G\right)=diag(e_1,e_2,\cdots ,e_n)}$ be the diagonal matrix of node degrees. The matrix ${\displaystyle L\left(G\right)=D\left(G\right)-A\left(G\right)}$ is called the Laplacian matrix of ${\displaystyle G}$ and the set of all its eigenvalues is known as the Laplacian spectrum of ${\displaystyle G}$. As ${\displaystyle L\left(G\right)}$ is a positive semi-definite matrix, its eigenvalues can be indexed as

$${⋌}_1\ge {⋌}_2\ge \cdots \ge {⋌}_{n-1}>{⋌}_n=0,$$

where ${\displaystyle {⋌}_1}$ is the spectral (Laplacian) radius of ${\displaystyle L\left(G\right)}$ and ${\displaystyle {⋌}_{n-1}}$ is the smallest non-zero eigenvalue called the algebraic connectivity of ${\displaystyle G.}$ It is known that ${\displaystyle {⋌}_1\le n}$ with equality if and only if ${\displaystyle G}$ is connected and ${\displaystyle \overline{G}}$ (complement of ${\displaystyle G}$) is not connected [Mohar, [3]]. Additionally, Fiedler [4] showed that ${\displaystyle {⋌}_{n-1}>0}$ if and only if ${\displaystyle G}$ is connected. So, ${\displaystyle {⋌}_{n-1}}$ measures the connectedness of ${\displaystyle G.}$ If ${\displaystyle \ell }$ is the eigenvalue of ${\displaystyle L\left(G\right)}$ repeated ${\displaystyle k\ge 2}$ times, we say ${\displaystyle \ell }$ is the Laplacian eigenvalue with (order) multiplicity ${\displaystyle k}$ and symbolize it by ${\displaystyle {\ell }^{\left[k\right]}}$. More about matrix ${\displaystyle L\left(G\right)}$, including results on ${\displaystyle {⋌}_{n-1}}$, are given in [1,5,6].

The directed power graph (see Kelarev and Quinn [7]) of a semigroup ${\displaystyle S}$ is defined as a directed graph with node set ${\displaystyle S}$ in which two nodes ${\displaystyle x,y\in S}$ are joined by an arc from ${\displaystyle x}$ to ${\displaystyle y}$ if and only if ${\displaystyle x\ne y}$ and ${\displaystyle y^j=x}$ for some positive integer ${\displaystyle j}$. The undirected power graph [8] ${\displaystyle P\left(\mathbb{G}\right)}$ of a group ${\displaystyle \mathbb{G}}$ is defined as an undirected graph with node set as ${\displaystyle \mathbb{G}}$ and two nodes ${\displaystyle x,y\in \mathbb{G}}$ are adjacent if and only if ${\displaystyle x^i=y}$ or ${\displaystyle y^j=x}$ for ${\displaystyle 2\le i,j\le n}$. More about power graphs can be seen in [9,10,11] and the references therein. Power graphs have applications in automata theory [12]; their first survey was carried out in 2013 [13] and more recently in 2021 [14]. The spectra of power graphs is a well-studied topic; the adjacency spectrum of such graphs can be seen in [15], the Laplacian spectrum in [16,17], and the spectrum of power graphs for other matrices can be seen in [18].

A matrix with integer entries having only integer eigenvalues is called an integral matrix. A graph ${\displaystyle G}$ is said to be the Laplacian integral if all eigenvalues of ${\displaystyle L\left(G\right)}$ are integers. We present an integer modulo group of order ${\displaystyle n}$ by ${\displaystyle {\mathbb{Z}}_n}$ and any cyclic group of order ${\displaystyle n}$ is taken as an isomorphic copy of ${\displaystyle {\mathbb{Z}}_n.}$ The results of Martin and Wong [19] additionally inspired us to investigate integer matrices possessing integer eigenvalues. The authors demonstrated that almost all integer matrices have no integer eigenvalues; that is, the probability that a random integer matrix has at least one integer eigenvalue is 0 for all ${\displaystyle n\ge 2}$. Naturally, the following inquiries come up as follows: When does an integer matrix have all integer eigenvalues? In this direction for the study of Laplacian matrices, Panda [17] conjectured the following about power graphs of a finite cyclic group ${\displaystyle {\mathbb{Z}}_n.}$

Conjecture 1

([17]). For ${\displaystyle n\ge 2,}$ the following are equivalent:

(i) 

The algebraic connectivity ${\displaystyle {⋌}_{n-1}}$ of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is an integer.

(ii) 

${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is Laplacian integral.

(iii) 

${\displaystyle n}$ is a prime power or the product of two primes.

We will find the Laplacian eigenvalues of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$, discuss the integral possibility of its algebraic connectivity with the help of interlacing and equitable partitions, and thereby answer the above conjecture in a positive manner. A similar type of problem for the Laplacian integrable eigenvalues of comaximal graphs of commutative rings can be seen in [20].

This is how the remainder of the paper is structured. Section 2 gives some existing results which are used to prove our main results. The Laplacian spectrum of the power graph ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is discussed in Section 3. In order to prove that ${\displaystyle {⋌}_{n-1}}$ is an integer if and only if ${\displaystyle n}$ is either prime or the product of two distinct primes, we demonstrate that ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is Laplacian integral if and only if ${\displaystyle n}$ is either a prime power or the product of two primes. Finally, in the concluding remarks in Section 4, we solve the equivalent form of Conjecture 1 for the distance Laplacian matrix of graphs. We end our article with the conclusion and outline some future work.

2. Preliminary Results

We present a few definitions and established findings in this section, which will be utilized to support our primary results. We determine the Laplacian eigenvalues of the power graphs of ${\displaystyle {\mathbb{Z}}_n}$ and demonstrate the integer Laplacian eigenvalues of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ if and only if ${\displaystyle n=p^m,m\ge 1}$, or ${\displaystyle n=pq}$, where ${\displaystyle p,q}$ are primes.

Consider an ${\displaystyle n\times n}$ matrix

$$M=\left(\begin{array}{ccccc}{A}_{1,1}& A_{1,2}& \cdots & A_{1,s-1}& A_{1,s}\\ A_{2,1}& A_{2,2}& \cdots & A_{2,s-1}& A_{2,s}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ A_{s-1,1}& A_{s-1,2}& \cdots & A_{s-1,s-1}& A_{s-1,s}\\ A_{s,1}& A_{s,2}& \cdots & A_{s,s-1}& A_{s,s}\end{array}\right),$$

whose columns and rows are divided related to a partition ${\displaystyle P=\{L_1,L_2,\cdots ,L_s\}}$ of the initial segment ${\displaystyle \left[n\right]=\{1,\cdots ,n\}.}$ The quotient matrix ${\displaystyle Q={\left(q_{ij}\right)}_{s\times s}}$ (see [1]) is an ${\displaystyle s}$-square matrix such that ${\displaystyle q_{ij}}$-th is the average row/column sum of the block ${\displaystyle A_{ij}}$ of ${\displaystyle M}$. The partition ${\displaystyle P}$ is equitable if each block ${\displaystyle A_{i,j}}$ of ${\displaystyle M}$ has a constant row sum, and in this case the matrix ${\displaystyle Q}$ is called the equitable (regular) quotient matrix.

For two sequences of real numbers ${\displaystyle \{s_1,s_2,\cdots ,s_n\}}$ and ${\displaystyle \{s_1^{\prime },s_2^{\prime },\cdots ,s_m^{\prime }\}}$ with ${\displaystyle n>m}$, the latter interlaces the former if

$$s_i\ge s_i^{\prime }\ge s_{n-m+i}\mathrm{for}i=1,2,\cdots ,m.$$

The interlacing is said to be tight if there exists a positive integer ${\displaystyle k\in [0,m]}$ such that

$$s_i=s_i^{\prime }\mathrm{for}i=1,2,\cdots ,k\mathrm{and}{s}_{n-m+i}=s_i^{\prime }\mathrm{for}k+1\le i\le m.$$

If ${\displaystyle m=n-1}$, then the interlacing becomes

$$s_1\ge s_1^{\prime }\ge s_2\ge s_2^{\prime }\ge \cdots \ge s_{n-1}\ge s_{n-1}^{\prime }\ge s_n.$$

The next result relates the eigenvalues of ${\displaystyle M}$ with that of ${\displaystyle Q.}$

Theorem 1

([1,21]). For a real symmetric ${\displaystyle M}$ of order ${\displaystyle n}$ and its associated quotient matrix ${\displaystyle Q}$ of order ${\displaystyle m}$ with ${\displaystyle (m<n)}$, the following hold:

(i) 

If matrix ${\displaystyle M}$ has a non-equitable partition ${\displaystyle P}$ related to some index set ${\displaystyle \left[n\right]}$, then the eigenvalues of ${\displaystyle Q}$ and ${\displaystyle M}$ satisfy the interlacing condition; that is,

$${\zeta }_i\left(M\right)\ge {\zeta }_i\left(Q\right)\ge {\zeta }_{i+n-m}\left(M\right)\mathit{for}i=1,2,\cdots ,m.$$

(ii) 

If matrix ${\displaystyle M}$ has the partition equitable ${\displaystyle P}$ with some initial segment ${\displaystyle \left[n\right]}$, then we have tight interlacing; that is, each eigenvalue of ${\displaystyle Q}$ is the eigenvalue of ${\displaystyle M.}$

The interlacing between the eigenvalues of a real symmetric matrix and its principal submatrices is provided by the following result.

Theorem 2

(Interlacing result, [1,2]). For a matrix ${\displaystyle M}$ (real symmetric of order ${\displaystyle n}$) with its principal submatrix ${\displaystyle M^{\prime }}$ of order ${\displaystyle m}$, ${\displaystyle (m\le n)}$. The eigenvalues of ${\displaystyle M}$ and ${\displaystyle M^{\prime }}$ interlace in the following relation:

$${\zeta }_{i+n-m}\left(M\right)\le {\zeta }_i\left(M^{\prime }\right)\le {\zeta }_i\left(M\right)\mathit{with}1\le i\le m.$$

Consider graphs ${\displaystyle G_i(U_i,E_i)}$ of order ${\displaystyle n_i}$ with ${\displaystyle i=1,\dots ,n}$. The joined union [22] ${\displaystyle G[G_1,\dots ,G_n]}$ is a graph ${\displaystyle S(W,F)}$ with

$$W=\bigcup _{i=1}^n{U}_i\mathrm{and}F=\bigcup _{i=1}^n{E}_i\cup \bigcup _{\{z_i,z_j\}\in E}{U}_i\times U_j.$$

Equivalently, the joined union of ${\displaystyle G_1,\dots ,G_n}$ is their union with edges from each node in ${\displaystyle G_i}$ to all nodes in ${\displaystyle G_j}$, provided ${\displaystyle z_i}$ and ${\displaystyle z_j}$ are adjacent in ${\displaystyle G.}$ Like the joined ${\displaystyle G_1}$ and ${\displaystyle G_2}$, ${\displaystyle G_1\vee G_2=K_2[G_1,G_2].}$ We note that if each of ${\displaystyle G_i}$ and ${\displaystyle G}$ are complete graphs, then so is ${\displaystyle G[G_1,\dots ,G_n]}$.

With the assumption that each ${\displaystyle G_i}$ is a ${\displaystyle r_i}$-regular graph, we now have a result that provides the Laplacian spectrum of ${\displaystyle G[G_1,G_2,\cdots ,G_n]}$ in terms of the Laplacian spectrum of individual ${\displaystyle G_i}$s and the eigenvalues of the equitable quotient matrix associated with it.

Theorem 3

([23]). For a graph ${\displaystyle G}$ of order ${\displaystyle n\ge 3}$ with ${\displaystyle m}$ edges and let ${\displaystyle G_i}$ be ${\displaystyle r_i}$-regular graphs of order ${\displaystyle n_i}$ with Laplacian eigenvalues ${\displaystyle {\eta }_{i1}\ge {\eta }_{i2}\ge \dots \ge {\eta }_{i{n}_i}}$ with ${\displaystyle i=1,2,\dots ,n}$. The Laplacian spectrum of ${\displaystyle G[G_1,\dots ,G_n]}$ consists of the eigenvalues ${\displaystyle {\alpha }_i+{\eta }_{ik}\left(G_i\right)}$ for ${\displaystyle i=1,\dots ,n}$ and ${\displaystyle k=2,3,\dots ,n_i}$, where ${\displaystyle {\alpha }_i=\sum _{z_j\in N_G\left(z_i\right)}{n}_j}$ is the sum of the cardinality of ${\displaystyle G_j,j\ne i}$, which corresponds to the neighbours of node ${\displaystyle z_i\in G}$. The other ${\displaystyle n}$ Laplacian eigenvalues are the eigenvalues of the quotient matrix given as

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

(1)

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

For a positive integer ${\displaystyle n}$ and let ${\displaystyle n=p_1^{n_1}{p}_2^{n_2}\cdots p_r^{n_r}}$ be its decomposition (canonical), where ${\displaystyle p_1,p_2,\cdots ,p_r}$ are primes and ${\displaystyle n_1,n_2,\cdots ,n_r,r}$ are positive integers. Let ${\displaystyle \tau \left(n\right)}$ (see [24]) be the number of positive factors of ${\displaystyle n.}$ Then

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

The Euler’s function (totient), denoted by ${\displaystyle \varphi \left(n\right)}$ is the number of positive integers less or equal to ${\displaystyle n}$ and relatively prime to it. Furthermore, ${\displaystyle \sum _{d|n}\varphi \left(d\right)=n,}$ where ${\displaystyle d|n}$ denotes ${\displaystyle d}$ divides n.

An integer ${\displaystyle \delta }$ is a proper divisor of ${\displaystyle n}$ if ${\displaystyle \delta |n}$ and ${\displaystyle \delta \notin \{1,n\}.}$ Let ${\displaystyle {\delta }_1,{\delta }_2,\cdots ,{\delta }_t}$ be the proper divisors (distinct) of ${\displaystyle n.}$ Let ${\displaystyle {\Delta }_n}$ be a simple graph with node set ${\displaystyle \{{\delta }_1,{\delta }_2,\cdots ,{\delta }_t\}}$ in which two distinct nodes are adjacent if and only if ${\displaystyle {\delta }_i|{\delta }_j}$ for ${\displaystyle 1\le i<j\le t}$. The graph ${\displaystyle {\Delta }_n}$ is a kind of proper divisor graph. Clearly, the size of ${\displaystyle {\Delta }_n}$ is given by ${\displaystyle |V\left({\Delta }_n\right)|=\prod _{i=1}^r(n_i+1)-2.}$

3. Laplacian Eigenvalues of the Power Graph of Finite Cyclic Groups

In this section, we will discuss the Laplacian eigenvalues of the power graph of ${\displaystyle {\mathbb{Z}}_n.}$ Our first result gives the properties of graph ${\displaystyle {\Delta }_n.}$

Theorem 4.

Let ${\displaystyle {\Delta }_n}$ be the proper divisor graph. Then the following conclusions hold:

(i) 

For prime ${\displaystyle n}$, ${\displaystyle {\Delta }_n}$ is an empty graph.

(ii) 

For ${\displaystyle n=p^2}$ with prime ${\displaystyle p}$, ${\displaystyle {\Delta }_n\cong K_1.}$

(iii) 

For ${\displaystyle n=pq}$, with primes ${\displaystyle p}$ and ${\displaystyle q}$${\displaystyle (p<q)}$, ${\displaystyle {\Delta }_n\cong {\overline{K}}_2.}$

(iv) 

If ${\displaystyle n\notin \{p,p^2,pq\}}$, then ${\displaystyle {\Delta }_n}$ is connected.

Proof. 

(i) For ${\displaystyle n=p}$, ${\displaystyle {\Delta }_n}$ has no nodes and is empty by default. (ii) For ${\displaystyle n=p^2}$, where ${\displaystyle p}$ is prime. Then ${\displaystyle p}$ is the only divisor of ${\displaystyle n}$ and ${\displaystyle {\Delta }_n}$ is ${\displaystyle K_1.}$

(iii) For ${\displaystyle n=pq,}$ where ${\displaystyle p}$ and ${\displaystyle q(p<q)}$ are primes. Then both ${\displaystyle p}$ and ${\displaystyle q}$ are the proper divisors of ${\displaystyle n}$ and ${\displaystyle p}$ does not divide ${\displaystyle q,}$ so ${\displaystyle {\Delta }_n\cong K_1\cup K_1}$.

(iv) For other cases, consider two arbitrary proper divisors of ${\displaystyle n}$, say, ${\displaystyle {\delta }_i<{\delta }_j,}$ and let their greatest common divisor be ${\displaystyle \delta }$. If ${\displaystyle \delta =1}$ and ${\displaystyle {\delta }_i{\delta }_j=n}$, then either ${\displaystyle {\delta }_i}$ or ${\displaystyle {\delta }_j}$ are composite, since ${\displaystyle {\delta }_i}$ and ${\displaystyle {\delta }_j}$ cannot both be primes. Without loss of generality, assume that ${\displaystyle {\delta }_j=p_1{p}_2,}$ where ${\displaystyle p_1}$ and ${\displaystyle p_2}$ are arbitrary and ${\displaystyle p_1\sim {\delta }_j\sim p_2}$. Thus, both ${\displaystyle p_1{\delta }_i}$ and ${\displaystyle p_2{\delta }_i}$ are in ${\displaystyle {\Delta }_n}$ and ${\displaystyle p_1{\delta }_i\sim {\delta }_i\sim p_2{\delta }_i}$. Thus, ${\displaystyle {\delta }_j\sim p_2\sim p_2{\delta }_i\sim {\delta }_i}$ and ${\displaystyle {\delta }_j\sim p_1\sim p_1{\delta }_i\sim {\delta }_i}$, so in both cases ${\displaystyle {\delta }_i}$ is connected to ${\displaystyle {\delta }_j.}$ Again, ${\displaystyle {\delta }_i{\delta }_j={\delta }_k\ne n}$, where ${\displaystyle {\delta }_k}$ is a different node of ${\displaystyle {\Delta }_n}$. In this case, ${\displaystyle {\delta }_i}$ is adjacent to ${\displaystyle {\delta }_k}$ and further, ${\displaystyle {\delta }_j}$ is adjacent to ${\displaystyle {\delta }_k}$. This implies that ${\displaystyle {\delta }_i\sim {\delta }_k\sim {\delta }_j}$. Finally, if ${\displaystyle \delta ={\delta }_l\ne 1}$, ${\displaystyle {\delta }_l|{\delta }_i}$ and ${\displaystyle {\delta }_l|{\delta }_j}$, then ${\displaystyle {\delta }_i\sim {\delta }_l\sim {\delta }_j.}$ ${\displaystyle \square }$

For prime power ${\displaystyle n}$, the following fact shows that ${\displaystyle {\Delta }_n}$ is a complete graph.

Proposition 1.

If ${\displaystyle n=p^m,}$ where ${\displaystyle m\ge 3}$ is a positive integer, then ${\displaystyle {\Delta }_n}$ is a complete graph of order ${\displaystyle m-1}$.

Proof. 

Since ${\displaystyle \{p^1,p^2,\cdots ,p^{m-1}\}}$ is the proper divisor set of ${\displaystyle n,}$ the result follows from the fact that any distinct powers of ${\displaystyle p}$, say, ${\displaystyle p^l}$ and ${\displaystyle p^k}$, necessarily ${\displaystyle p^l}$ divides ${\displaystyle p^k}$ or ${\displaystyle p^k}$ divides ${\displaystyle p^l}$. ${\displaystyle \square }$

The power graph ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ of ${\displaystyle {\mathbb{Z}}_n}$ (cyclic group) can be expressed as the joined union of cliques, as demonstrated by the following result.

Theorem 5

([25]). If ${\displaystyle {\mathbb{Z}}_n}$ is a finite cyclic group, then the power graph of ${\displaystyle \mathbb{Z}}$ cab be expressed as

$$P\left({\mathbb{Z}}_n\right)=K_{\varphi \left(n\right)+1}\vee {\Delta }_n[K_{\varphi \left(e_1\right)},K_{\varphi \left(e_2\right)},\cdots ,K_{\varphi \left(e_t\right)}].$$

The immediate consequence of Theorem 4, Corollary 1, and Theorem 5 gives an alternative proof that ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is complete if and only if ${\displaystyle n}$ is either prime or a prime power.

Corollary 1.

The power graph ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is the complete graph if and only if ${\displaystyle n}$ is either prime or a prime power.

Proof. 

If ${\displaystyle n=p}$ is prime, then ${\displaystyle {\Delta }_n}$ is an empty graph and ${\displaystyle P\left({\mathbb{Z}}_n\right)\cong K_{\varphi \left(p\right)+1}=K_{p-1+1}=K_p.}$ If ${\displaystyle n=p^m}$, ${\displaystyle m\ge 2}$ is a positive integer and ${\displaystyle p}$ is a prime power, then by Corollary 1, ${\displaystyle {\Delta }_n}$ is a complete graph of order ${\displaystyle m-1}$ and by Theorem 5, ${\displaystyle {\Delta }_{p^m}[K_{\varphi \left(p\right)},K_{\varphi \left(p^2\right)},\cdots ,K_{\varphi \left(p^{m-1}\right)}]\cong K_{p^{m-1}-1}}$, since ${\displaystyle \sum _{i=1}^k\varphi \left(p^k\right)=p^k-1.}$ Thus, ${\displaystyle P\left({\mathbb{Z}}_n\right)\cong K_{\varphi \left(p^m\right)+1}\vee K_{p^{m-1}-1}=K_{p^m}.}$ Lastly, if ${\displaystyle n}$ is the product of more than three distinct primes, then clearly their primes do not divide each other; hence, ${\displaystyle {\Delta }_n}$ is not complete. Therefore, ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is not complete when ${\displaystyle n}$ is other than a prime power. ${\displaystyle \square }$

By applying Theorems 3 and 5, we will compute the Laplacian spectrum of ${\displaystyle P\left(G\right)}$ in terms of the eigenvalues of ${\displaystyle L\left(K_n\right)}$ and the eigenvalues of its quotient (equitable) matrix. We recall that the eigenvalues of ${\displaystyle L\left(K_n\right)}$ are ${\displaystyle \left\{n^{[n-1]},0\right\}.}$

The following result gives the Laplacian eigenvalues of the power graph of ${\displaystyle {\mathbb{Z}}_n.}$ Although different techniques were used by different authors to obtain the Laplacian spectrum, complete information was not presented there.

Theorem 6.

The spectrum of ${\displaystyle L\left(P\left({\mathbb{Z}}_n\right)\right)}$ comprises the eigenvalue ${\displaystyle n}$ with multiplicity order ${\displaystyle \varphi \left(n\right)}$, the eigenvalues ${\displaystyle {\beta }_i+\varphi \left(e_i\right)}$ with multiplicity ${\displaystyle \varphi \left(e_i\right)-1,i=1,2,\cdots ,t}$, and the eigenvalues of matrix (equitable quotient) given below as follows:

$$M=\left(\begin{array}{cccccc}n-1-\varphi \left(n\right)& -\varphi \left(e_1\right)& -\varphi \left(e_2\right)& \cdots & -\varphi \left(e_{t-1}\right)& -\varphi \left(e_t\right)\\ -\left(\varphi \right(n)+1)& {\beta }_1& {\theta }_{12}& \cdots & {\theta }_{1(t-1)}& {\theta }_{1t}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ -\left(\varphi \right(n)+1)& {\theta }_{(t-1)1}& {\theta }_{(t-1)2}& \cdots & {\beta }_{t-1}& {\theta }_{(t-1)t}\\ -\left(\varphi \right(n)+1)& {\theta }_{t1}& {\theta }_{t2}& \cdots & {\theta }_{t(t-1)}& {\beta }_t\end{array}\right),$$

(2)

where ${\displaystyle {\beta }_i=\sum _{e_j\in N_{{\Delta }_n}\left(e_i\right)}\varphi \left(e_j\right)+\varphi \left(n\right)+1}$, and ${\displaystyle {\theta }_{ij}=\left\{\begin{array}{cc}-\varphi \left(e_j\right)\hfill & \mathit{if}{e}_i|e_j,\hfill \\ 0\hfill & \mathit{otherwise},\hfill \end{array}\right.}$ for ${\displaystyle i=1,\cdots ,t.}$

Proof. 

Let ${\displaystyle {\mathbb{Z}}_n}$ be a finite cyclic group of order ${\displaystyle n}$. By the structure of the power graph, the identity and the ${\displaystyle \varphi \left(n\right)}$ generators of ${\displaystyle {\mathbb{Z}}_n}$ form the clique and are adjacent to every other node of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$. So, by Theorem 5, the structure of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is

$$\begin{array}{cc}\hfill P\left({\mathbb{Z}}_n\right)=& K_{\varphi \left(n\right)+1}\vee {\Delta }_n[K_{\varphi \left(e_1\right)},K_{\varphi \left(e_2\right)},\cdots ,K_{\varphi \left(e_t\right)}]\hfill \\ \hfill =& H[K_{\varphi \left(n\right)+1},K_{\varphi \left(e_1\right)},K_{\varphi \left(e_2\right)},\cdots ,K_{\varphi \left(e_t\right)}],\hfill \end{array}$$

where ${\displaystyle H=K_1\vee {\Delta }_n}$. Now, with the notations in Theorem 3, we have

$${\alpha }_1=\varphi \left(e_1\right)+\varphi \left(e_2\right)+\cdots +\varphi \left(e_t\right)=\sum _{\begin{array}{c}{e}_i|n\\ d\ne 1,n\end{array}}\varphi \left(e_i\right)=\sum _{d|n}\varphi \left(d\right)-\varphi \left(n\right)-1=n-\varphi \left(n\right)-1.$$

Thus,

$${\alpha }_1+{\eta }_{ik}\left(K_{\varphi \left(n\right)+1}\right)=n-\varphi \left(n\right)-1+(\varphi \left(n\right)+1)=n$$

is a Laplacian eigenvalue of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ with multiplicity ${\displaystyle \varphi \left(n\right)}$. Similarly, ${\displaystyle {\alpha }_k=\varphi \left(n\right)+1+\sum _{e_j\in N_{{\Delta }_n}\left(e_i\right)}\varphi \left(e_j\right)={\beta }_{k-1}}$ for ${\displaystyle k=2,3,\cdots ,t+1}$ and ${\displaystyle \varphi \left(e_j\right)+{\beta }_j}$ are Laplacian eigenvalues of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ with multiplicity ${\displaystyle \varphi \left(e_j\right)-1}$ for ${\displaystyle j=1,2,\cdots ,t.}$ The remaining ${\displaystyle t+1}$ eigenvalues of ${\displaystyle L\left(P\left({\mathbb{Z}}_n\right)\right)}$ are the eigenvalues of the matrix ${\displaystyle M}$ given in (2). ${\displaystyle \square }$

The immediate consequence of the above result proves that ${\displaystyle n}$ and ${\displaystyle 0}$ are always eigenvalues of ${\displaystyle M}$ in (1).

Corollary 2.

Let ${\displaystyle M}$ be given as in (1). Then ${\displaystyle n}$ and ${\displaystyle 0}$ are eigenvalues of ${\displaystyle M}$. Moreover, the multiplicity of the Laplacian eigenvalue ${\displaystyle n}$ of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$${\displaystyle \varphi \left(n\right)+1}$.

Proof. 

The matrix given in (1) can be written in block form as

$$\left(\begin{array}{cccccc}n-1-\varphi \left(n\right)& -\varphi \left(e_1\right)& -\varphi \left(e_2\right)& \cdots & -\varphi \left(e_{t-1}\right)& -\varphi \left(e_t\right)\\ -\left(\varphi \right(n)+1)& {\beta }_1& {\theta }_{12}& \cdots & {\theta }_{1(t-1)}& {\theta }_{1t}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ -\left(\varphi \right(n)+1)& {\theta }_{(t-1)1}& {\theta }_{(t-1)2}& \cdots & {\beta }_{t-1}& {\theta }_{(t-1)t}\\ -\left(\varphi \right(n)+1)& {\theta }_{t1}& {\theta }_{t2}& \cdots & {\theta }_{t(t-1)}& {\beta }_t\end{array}\right),$$

(3)

where ${\displaystyle {\beta }_i=\sum _{e_j\in N_{{\Delta }_n}\left(e_i\right)}\varphi \left(e_j\right)+\varphi \left(n\right)+1}$ and ${\displaystyle {\theta }_{ij}=\left\{\begin{array}{cc}-\varphi \left(e_j\right)\hfill & \mathrm{if}{e}_i|e_j,\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.}$ for ${\displaystyle i=1,\cdots ,t.}$ Using the fact that ${\displaystyle \sum _{d|n}\varphi \left(d\right)=n,}$ the quotient matrix of (3) is

$$\left(\begin{array}{cc}n-1-\varphi \left(n\right)& -(n-1-\varphi (n\left)\right)\\ -\left(\varphi \right(n)+1)& {\displaystyle \frac{1}{t}\sum _{i=1}^t{s}_i}\end{array}\right),$$

(4)

where for ${\displaystyle k=1,2,\cdots ,t}$, we have

$${\displaystyle s_i={\beta }_i+\sum _{\begin{array}{c}i\ne j\\ j=1\end{array}}^t{\theta }_{kj}={\beta }_i+\sum _{e_j\in N_{{\Delta }_n}\left(e_i\right)}\varphi \left(e_j\right)=\varphi \left(n\right)+1}$$

for each ${\displaystyle i=1,2,\cdots ,t}$. Thus, the matrix in (4) is an equitable quotient matrix of (3) and its eigenvalues are ${\displaystyle 0}$ and ${\displaystyle n.}$ Therefore, in view of Theorem 6, ${\displaystyle n}$ is the eigenvalue of ${\displaystyle L\left(P\left({\mathbb{Z}}_n\right)\right)}$ with multiplicity exactly ${\displaystyle \varphi \left(n\right)+1.}$ ${\displaystyle \square }$

From Theorem 6 and Corollary 2, among the ${\displaystyle n}$ eigenvalues of ${\displaystyle L\left(P\left({\mathbb{Z}}_n\right)\right)}$, we see that ${\displaystyle n-(t-1)}$ of them are non-negative integers. The other ${\displaystyle t-1}$ eigenvalues of ${\displaystyle L\left(P\left({\mathbb{Z}}_n\right)\right)}$ are the eigenvalues (different from ${\displaystyle 0}$ and ${\displaystyle n}$) of the matrix in (2).

Another important consequence of Theorem 6 is stated in the next result.

Corollary 3.

For the power graph ${\displaystyle P\left({\mathbb{Z}}_n\right)}$, we have

$${\eta }_k\left(P\left({\mathbb{Z}}_n\right)\right)\ge \varphi \left(e_i\right)+\varphi \left(n\right)+1,$$

(5)

for ${\displaystyle k=1,2,\cdots ,\varphi \left(e_i\right)-1}$ and ${\displaystyle i=1,2,\cdots ,t.}$ Equality holds if and only if ${\displaystyle {\Delta }_n}$ is disconnected; that is, ${\displaystyle n}$ is either prime or the product of two primes.

Proof. 

From the proof of Theorem 6, with ${\displaystyle k=1,2,\cdots ,\varphi \left(e_i\right)-1}$ and ${\displaystyle i=1,2,\cdots ,t,}$ we have

$$\begin{array}{cc}\hfill {\eta }_k\left(P\left({\mathbb{Z}}_n\right)\right)=& {\beta }_i+\varphi \left(e_i\right)\hfill \\ \hfill =& \varphi \left(e_i\right)+\varphi \left(n\right)+1+\sum _{e_j\in N_{{\Delta }_n}\left(e_i\right)}\varphi \left(e_j\right)\hfill \\ \hfill \ge & \varphi \left(e_i\right)+\varphi \left(n\right)+1,\hfill \end{array}$$

since ${\displaystyle \sum _{e_j\in N_{{\Delta }_n}\left(e_i\right)}\varphi \left(e_j\right)\ge 0.}$ The equality holds in (5) if and only if ${\displaystyle \sum _{e_j\in N_{{\Delta }_n}\left(e_i\right)}\varphi \left(e_j\right)=0;}$ that is, each of ${\displaystyle e_j}$ is an isolated node and has no neighbours in ${\displaystyle {\Delta }_n}$. So, by Theorem 4, equality holds if and only if ${\displaystyle n}$ is either prime or the product of two primes. ${\displaystyle \square }$

Next, we state another consequence of Theorem 6.

Corollary 4.

Let ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ be the power graph of ${\displaystyle {\mathbb{Z}}_n}$. Then the following hold:

(i) 

If ${\displaystyle n}$ is a prime power, then the spectrum of ${\displaystyle L\left(P\left({\mathbb{Z}}_n\right)\right)}$ is

$$\left\{0,n^{[n-1]}\right\}.$$

(ii) 

If ${\displaystyle n=pq,}$ and ${\displaystyle p,q(p<q)}$ are primes, then the spectrum of ${\displaystyle L\left(P\left({\mathbb{Z}}_n\right)\right)}$ is

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

Proof. 

(i) By Corollary 1, ${\displaystyle P\left({\mathbb{Z}}_n\right)\cong K_n}$ if and only if ${\displaystyle n}$ is a prime power (also see [8]), its Laplacian spectrum is ${\displaystyle 0}$, and ${\displaystyle n}$ has a multiplicity of ${\displaystyle n-1.}$

(ii) Now, for ${\displaystyle n=pq}$ with primes ${\displaystyle q>q}$, the divisors (proper) of ${\displaystyle n}$ are ${\displaystyle p}$ and ${\displaystyle q}$ and it follows that ${\displaystyle {\Delta }_{pq}\cong {\overline{K}}_2}$. By Theorem 5, the power graph of ${\displaystyle {\mathbb{Z}}_{pq}}$ is

$$P\left({\mathbb{Z}}_{pq}\right)=P_3[K_{\varphi \left(p\right)},K_{\varphi \left(pq\right)},K_{\varphi \left(q\right)}]=K_{pq-p-q+2}\vee \left(K_{p-1}\cup K_{q-1}\right).$$

By Theorem 6, the distance Laplacian spectrum of ${\displaystyle P\left({\mathbb{Z}}_{pq}\right)}$ consists of the eigenvalue ${\displaystyle p-1+q-1+pq-p-q+2=n}$ with multiplicity ${\displaystyle pq-p-q+1}$, the eigenvalue ${\displaystyle pq-p-q+2+p-1=pq-q+1}$ with multiplicity ${\displaystyle p-2}$, the eigenvalue ${\displaystyle pq-p-q+2+q-1=pq-p+1}$ with multiplicity ${\displaystyle q-2}$, and the eigenvalues of the matrix given below as follows:

$$\left(\begin{array}{ccc}p+q-2& -(p-1)& -(q-1)\\ -(pq-p-q+2)& pq-p-q+2& 0\\ -(pq-p-q+2)& 0& pq-p-q+2\end{array}\right).$$

The eigenvalues of the above matrix are ${\displaystyle 0,pq,}$ and ${\displaystyle pq-p-q+2}$. ${\displaystyle \square }$

Corollary 5.

Let ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ be the power graph of ${\displaystyle {\mathbb{Z}}_n}$. Then, we have the following:

(i) 

If ${\displaystyle n=p^{2}q}$ with primes ${\displaystyle q>p}$, then the spectrum of ${\displaystyle L\left(P\left({\mathbb{Z}}_n\right)\right)}$ consists of the eigenvalues

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

together with the non-zero eigenvalues of matrix (6).

(ii) 

If ${\displaystyle n=p{q}^2,}$ where ${\displaystyle p,q(p<q)}$ are primes, then the eigenvalues of ${\displaystyle L\left(P\left({\mathbb{Z}}_n\right)\right)}$ are

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

and the zeros of the following polynomial

$$\begin{array}{cc}\hfill x(& x-n\left)\right(x^3+x^2(-2+p-2q+pq+2q^2-3p{q}^2)+x(5q-5pq+p^{2}q-3q^2+6p{q}^2\hfill \\ \hfill & -2p^2{q}^2-2q^3+5p{q}^3-2p^2{q}^3+q^4-4p{q}^4+3p^2{q}^4)-q+pq+p{q}^2-p^2{q}^2+2q^3\hfill \\ \hfill & -8p{q}^3+6p^2{q}^3-p^3{q}^3-q^4+4p{q}^4-4p^2{q}^4+p^3{q}^4+2p{q}^5-3p^2{q}^5+p^3{q}^5-p{q}^6\hfill \\ \hfill & +2p^2{q}^6-p^3{q}^6).\hfill \end{array}$$

(iii) 

If ${\displaystyle n=pqr}$ and ${\displaystyle p,q,r(p<q<r)}$ are primes, then the eigenvalues of ${\displaystyle L\left(P\left({\mathbb{Z}}_n\right)\right)}$ are as follows: ${\displaystyle n}$ with multiplicity ${\displaystyle \varphi \left(n\right)}$, ${\displaystyle n-qr+1}$ with multiplicity ${\displaystyle p-2}$, ${\displaystyle n-pr+p-qr+q+r-1}$ with multiplicity ${\displaystyle \varphi \left(pq\right)-1}$, ${\displaystyle n-pr+1}$ with multiplicity ${\displaystyle q-2}$, ${\displaystyle n-pq-pr+p+q+r-1}$ with multiplicity ${\displaystyle \varphi \left(qr\right)-1}$, ${\displaystyle n-pq+1}$ with multiplicity ${\displaystyle r-2,}$${\displaystyle n-pq+p-qr+q+r-1}$ with multiplicity ${\displaystyle \varphi \left(pr\right)-1}$, and the eigenvalues of the following matrix:

$$\left(\begin{array}{ccccccc}{a}_1& -(p-1)& -(p-1)(q-1)& -(q-1)& -(q-1)(r-1)& -(r-1)& -(p-1)(r-1)\\ -\left(\varphi \right(n)+)& a_2& -(p-1)(q-1)& 0& 0& 0& -(p-1)(r-1)\\ -\left(\varphi \right(n)+1)& -(p-1)& a_3& -(q-1)& 0& 0& 0\\ -\left(\varphi \right(n)+1)& 0& -(p-1)(q-1)& a_4& -(q-1)(r-1)& 0& 0\\ -\left(\varphi \right(n)+1)& 0& 0& -(q-1)& a_5& -(r-1)& 0\\ -\left(\varphi \right(n)+1)& 0& 0& 0& -(q-1)(r-1)& a_6& -(p-1)(r-1)\\ -\left(\varphi \right(n)+1)& -(p-1)& 0& 0& 0& -(r-1)& a_7\end{array}\right),$$

where ${\displaystyle a_1=pq+pr-p+qr-q-r,a_2=pqr-p-qr+2,a_3=pqr-pq-pr+2p-qr+2q+r-2,a_4=pqr-pr-q+2,a_5=pqr-pq-pr+p-qr+2q+2r-2,a_6=pqr-pq-r+2}$, and ${\displaystyle a_7=pqr-pq-pr+2p-qr+q+2r-2.}$

Proof. 

(i) For ${\displaystyle n=p^{2}q}$, the proper divisors of ${\displaystyle n}$ are ${\displaystyle p,p^2,q}$, and ${\displaystyle pq.}$ So, by definition, ${\displaystyle {\Delta }_n}$ is the path ${\displaystyle P_4}$: ${\displaystyle p^2\sim p\sim pq\sim q}$ and the power graph of ${\displaystyle {\mathbb{Z}}_n}$ is

$$\begin{array}{cc}\hfill P\left({\mathbb{Z}}_n\right)\cong & K_{\varphi \left(p^{2}q\right)+1}\vee P_4[K_{\varphi \left(p^2\right)},K_{\varphi \left(p\right)},K_{\varphi \left(pq\right)},K_{\varphi \left(q\right)}]\hfill \\ \hfill =& K_{p(p-1)(q-1)+1}\vee P_4[K_{p(p-1)},K_{p-1},K_{(p-1)(q-1)},K_{q-1}].\hfill \end{array}$$

By Theorem 6, ${\displaystyle n}$ is the Laplacian eigenvalue of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ with multiplicity ${\displaystyle p(p-1)(q-1).}$ Also,

$$\begin{array}{cc}\hfill {\beta }_1=& p(p-1)(q-1)+1+p-1=p(pq-p-q+2)\hfill \\ \hfill {\beta }_2=& p(p-1)(q-1)+1+p(p-1)+(p-1)(q-1)=p^{2}q-p-q+2\hfill \\ \hfill {\beta }_3=& p(p-1)(q-1)+1+p-1+q-1=p^{2}q-p^2-pq+2p+q-1\hfill \\ \hfill {\beta }_4=& p(p-1)(q-1)+1+(p-1)(q-1)=p^{2}q-p^2-q+2.\hfill \end{array}$$

Thus, the eigenvalues of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ are

$$\begin{array}{cc}\hfill {\beta }_1+p(p-1)=& p(pq-q+1)\mathrm{with}\mathrm{multiplicity}p(p-1)-1,\hfill \\ \hfill {\beta }_2+p-1=& p^{2}q-q+1\mathrm{with}\mathrm{multiplicity}p-2,\hfill \\ \hfill {\beta }_3+(p-1)(q-1)=& p(pq-p+1)\mathrm{with}\mathrm{multiplicity}(p-1)(q-1)-1,\hfill \\ \hfill {\beta }_4+q-1=& p^{2}q-p^2+1\mathrm{with}\mathrm{multiplicity}q-2.\hfill \end{array}$$

The remaining five distance Laplacian eigenvalues of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ are the eigenvalues of the following matrix:

$$\left(\begin{array}{ccccc}p{q}^2-p(p-1)(q-1)-1& -p(p-1)& -(p-1)& -(p-1)(q-1)& -(q-1)\\ -\left(p\right(p-1\left)\right(q-1)+1)& {\beta }_1& -(p-1)& 0& 0\\ -\left(p\right(p-1\left)\right(q-1)+1)& -p(p-1)& {\beta }_2& -(p-1)(q-1)& 0\\ -\left(p\right(p-1\left)\right(q-1)+1)& 0& -(p-1)& {\beta }_3& -(q-1)\\ -\left(p\right(p-1\left)\right(q-1)+1)& 0& 0& -(p-1)(q-1)& {\beta }_4\end{array}\right).$$

(6)

The characteristic polynomial of (6) is

$$\begin{array}{cc}\hfill x(x-& n\left)\right(x^3+x^2(-2-2p+2p^2+q+pq-3p^{2}q)+x(5p-3p^2-2p^3+p^4-5pq\hfill \\ \hfill & +6p^{2}q+5p^{3}q-4p^{4}q+p{q}^2-2p^2{q}^2-2p^3{q}^2+3p^4{q}^2)-p+2p^3-p^4+pq\hfill \\ \hfill & +p^{2}q-8p^{3}q+4p^{4}q+2p^{5}q-p^{6}q-p^2{q}^2+6p^3{q}^2-4p^4{q}^2-3p^5{q}^2+2p^6{q}^2\hfill \\ \hfill & -p^3{q}^3+p^4{q}^3+p^5{q}^3-p^6{q}^3).\hfill \end{array}$$

The other parts, (ii) and (iii), can be similarly proved. ${\displaystyle \square }$

Next, we have a captivating result that characterizes values of ${\displaystyle n}$ for which the algebraic connectivity of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is an integer and provides an upper bound for it.

Theorem 7.

Let ${\displaystyle {⋌}_{n-1}}$ be the algebraic connectivity of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$. Then ${\displaystyle {⋌}_{n-1}}$ is an integer if and only if ${\displaystyle n}$ is either a prime or the product of two primes.

Proof. 

From Theorem 6, ${\displaystyle n-(t+1)}$ the Laplacian eigenvalues of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ are integers. The other ${\displaystyle t+1}$ Laplacian eigenvalues of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ are the eigenvalues of the following block matrix:

$${\left(\begin{array}{cccccc}n-1-\varphi \left(n\right)& -\varphi \left(e_1\right)& -\varphi \left(e_2\right)& \cdots & -\varphi \left(e_{t-1}\right)& -\varphi \left(e_t\right)\\ -\left(\varphi \right(n)+1)& {\beta }_1& {\theta }_{12}& \cdots & {\theta }_{1(t-1)}& {\theta }_{1t}\\ -\left(\varphi \right(n)+1)& {\theta }_{21}& {\beta }_2& \cdots & {\theta }_{2(t-1)}& {\theta }_{2t}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ -\left(\varphi \right(n)+1)& {\theta }_{(t-1)1}& {\theta }_{(t-1)2}& \cdots & {\beta }_{t-1}& {\theta }_{(t-1)t}\\ -\left(\varphi \right(n)+1)& {\theta }_{t1}& {\theta }_{t2}& \cdots & {\theta }_{t(t-1)}& {\beta }_t\end{array}\right)}_{(t+1)\times (t+1)},$$

(7)

where ${\displaystyle {\beta }_i=\sum _{e_j\in N_{{\Delta }_n}\left(e_i\right)}\varphi \left(e_j\right)+\varphi \left(n\right)+1}$, and ${\displaystyle {\theta }_{ij}=\left\{\begin{array}{cc}-\varphi \left(e_j\right)\hfill & \mathrm{if}{e}_i|e_j,\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.}$ for ${\displaystyle i=2,\cdots ,t.}$ The quotient matrix of the block matrix in (7) is

$$Q=\left(\begin{array}{ccc}n-\varphi \left(n\right)-1& -\varphi \left(e_1\right)& -(n-1-\varphi \left(n\right)-\varphi \left(e_1\right))\\ -\left(\varphi \right(n)+1)& R_1+\varphi \left(n\right)+1& -R_1\\ -\left(\varphi \right(n)+1)& -R_2& R\end{array}\right),$$

(8)

where ${\displaystyle R_1=\sum _{j=2}^t{\theta }_{1j}=\sum _{e_j\in N_{{\Delta }_n}\left(e_1\right)}\varphi \left(e_j\right),R_2=\frac{{\displaystyle \sum _{i=2}^t{\theta }_{i1}}}{t-1}=\frac{{\displaystyle \sum _{e_i\in N_{{\Delta }_n}\left(e_1\right)}}\varphi \left(e_i\right)}{t-1}}$, and ${\displaystyle R=\frac{1}{t-1}\sum _{i=1}^{t-1}{l}_i,l_i}$ is the ${\displaystyle i}$-th row sum of the third diagonal block of the matrix in (7). The eigenvalues of the matrix in (8) are

$$0,n,1+R_1+R_2+\varphi \left(n\right).$$

By the equitable partition condition of Theorem 1 and the interlacing property of Theorem 2, we have

$${⋌}_{n-1}\left(P\left({\mathbb{Z}}_n\right)\right)\le {⋌}_2\left(Q\right)=1+R_1+R_2+\varphi \left(n\right).$$

(9)

Again, by Theorem 1, equality holds in (9) if and only if ${\displaystyle l_1=l_2=\cdots =l_{t-1}=l}$ and ${\displaystyle {\theta }_{21}={\theta }_{31}=\cdots ={\theta }_{t1}=\theta .}$ That is, in the case of equality in (9), the block matrix in (7) must be equitable. So, for equitable partitions, with ${\displaystyle R_2=\theta }$, we obtain

$${⋌}_{n-1}\left(P\left({\mathbb{Z}}_n\right)\right)=1+R_1+\theta +\varphi \left(n\right)$$

and, in this case, ${\displaystyle 1+R_1+\theta +\varphi \left(n\right)}$ is an integer since ${\displaystyle R_1,\theta }$, and ${\displaystyle \varphi \left(n\right)}$ are integers and so is their sum. Now, we characterize values of ${\displaystyle n}$ for which ${\displaystyle Q}$ is an equitable quotient matrix of the matrix in (7). If ${\displaystyle n=p^m}$, where ${\displaystyle m}$ is a positive integer and ${\displaystyle p}$ is prime, then ${\displaystyle P\left({\mathbb{Z}}_n\right)\cong K_n}$ and ${\displaystyle K_n}$ is Laplacian integral with ${\displaystyle {⋌}_{n-1}=n}$. If ${\displaystyle n=pq}$, where ${\displaystyle p}$ and ${\displaystyle q(p<q)}$ are primes, then by Theorem 4, ${\displaystyle {\Delta }_n\cong {\overline{K}}_2}$ and ${\displaystyle Q}$ takes the form

$$\left(\begin{array}{ccc}p+q-2& -(p-1)& -(q-1)\\ -(pq-p-q+2)& pq-p-q+2& 0\\ -(pq-p-q+2)& 0& pq-p-q+2\end{array}\right).$$

So, ${\displaystyle R_1=R_2=0}$ and ${\displaystyle {⋌}_{n-1}\left(P\left({\mathbb{Z}}_n\right)\right)=1+\varphi \left(n\right)}$. By Theorem 5, ${\displaystyle P\left({\mathbb{Z}}_n\right)=K_{\varphi \left(n\right)+1}\vee {\Delta }_n[{\varphi }_1\left(e_1\right),\cdots ,\varphi \left(e_t\right)]}$ is the combination of two graphs and ${\displaystyle {\Delta }_n[{\varphi }_1\left(e_1\right),\cdots ,\varphi \left(e_t\right)]}$ is the joined union of cliques in ${\displaystyle P\left({\mathbb{Z}}_n\right).}$ So, when ${\displaystyle n}$ is neither a prime power nor the product of two distinct primes, then (8) is an equitable quotient matrix of (7) if and only if ${\displaystyle {\Delta }_n}$ is a clique and, in this case, ${\displaystyle {\Delta }_n[{\varphi }_1\left(e_1\right),\cdots ,\varphi \left(e_t\right)]}$ is also a clique. So, ${\displaystyle l_1=l_2=\cdots =l_{t-1}=l}$ and ${\displaystyle {\theta }_{21}={\theta }_{31}=\cdots ={\theta }_{t1}=\theta }$ and ${\displaystyle {⋌}_{n-1}}$ is an integer. However, we show that this cannot happen. Suppose the prime power factorization of ${\displaystyle n}$ is ${\displaystyle n=p_1^{n_1}{p}_2^{n_2}\cdots p_t^{n_t}}$, where ${\displaystyle t,n_1,n_2,\cdots ,n_t}$ are positive integers and ${\displaystyle p_1<p_2<\cdots <p_t}$ are distinct prime numbers. Then by the definition of ${\displaystyle {\Delta }_n}$, at least ${\displaystyle p_i}$ does not divide ${\displaystyle p_j}$ for each ${\displaystyle i,j=1,2,\cdots ,t}$ and ${\displaystyle i\ne j.}$ Therefore, it follows that ${\displaystyle {\Delta }_n}$ is not complete and hence ${\displaystyle Q}$ cannot be an equitable quotient matrix of (7) when ${\displaystyle n}$ is neither the product of two distinct primes nor a prime power, and at least one ${\displaystyle l_i}$ is not equal to ${\displaystyle l_j}$ and one ${\displaystyle {\theta }_{i1}}$ is not equal to ${\displaystyle {\theta }_{j1}}$. Consequently, ${\displaystyle {⋌}_{n-1}}$ cannot be an integer when ${\displaystyle n}$ is neither a prime power nor the product of two distinct primes. ${\displaystyle \square }$

From the proof of the above result, we have the following consequence.

Corollary 6.

Let ${\displaystyle {⋌}_{n-1}}$ be the algebraic connectivity of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ with ${\displaystyle R_1=\sum _{e_j\in N_{{\Delta }_n}\left(e_1\right)}\varphi \left(e_j\right),R_2=\frac{1}{t-1}\sum _{e_i\in N_{{\Delta }_n}\left(e_1\right)}\varphi \left(e_i\right)}$, and ${\displaystyle R=\frac{1}{t-1}\sum _{i=1}^{t-1}{l}_i,}$ where ${\displaystyle l_i}$ is the ${\displaystyle i}$-th row sum of the third diagonal block of matrix (7). Then

$${⋌}_{n-1}\left(P\left({\mathbb{Z}}_n\right)\right)\le 1+R_1+R_2+\varphi \left(n\right),$$

(10)

with equality if and only if ${\displaystyle n}$ is the product of two distinct primes.

From Theorem 2, its following consequences, and Theorem 7, we obtain the following result which settles Conjecture 1.

Theorem 8.

Let ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ be the power graph of ${\displaystyle {\mathbb{Z}}_n,n\ge 2}$. Then ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is Laplacian integral if and only if ${\displaystyle n}$ is either a prime power or the product of two distinct primes.

4. Concluding Remark

In a connected graph ${\displaystyle G}$, the distance${\displaystyle d(u,v)}$ between two nodes ${\displaystyle u\ne v\in V\left(G\right)}$ is the length of the shortest path connecting ${\displaystyle u}$ and ${\displaystyle v}$. The diameter of ${\displaystyle G}$ is the maximum distance between any two nodes of ${\displaystyle G.}$ The distance matrix${\displaystyle D\left(G\right)}$ of ${\displaystyle G}$ is defined as ${\displaystyle D\left(G\right)={\left(d(u,v)\right)}_{u,v\in V\left(G\right)}}$. The transmission${\displaystyle T{r}_G\left(v\right)}$ of a node ${\displaystyle v}$ is the sum of the distances from ${\displaystyle v}$ to all other nodes in ${\displaystyle G}$; that is, ${\displaystyle T{r}_G\left(v\right)=\sum _{u\in V\left(G\right)}d(u,v).}$ For any node ${\displaystyle z_i\in V\left(G\right)}$, the transmission ${\displaystyle T{r}_G\left(z_i\right)}$ is called the transmission degree, presented by ${\displaystyle T{r}_i}$ and the sequence ${\displaystyle \{T{r}_1,T{r}_2,\dots ,T{r}_n\}}$ is called the transmission degree sequence of ${\displaystyle G}$. Let ${\displaystyle Tr\left(G\right)=diag(T{r}_1,T{r}_2,\dots ,T{r}_n)}$ be the diagonal matrix of node transmissions of ${\displaystyle G}$. The matrix ${\displaystyle D^L\left(G\right)=Tr\left(G\right)-D\left(G\right)}$ is called the distance Laplacian [26] matrix of ${\displaystyle G}$. The distance Laplacian eigenvalues of ${\displaystyle G}$ are the zeros of ${\displaystyle D^L\left(G\right)}$ and are indexed from largest to smallest ${\displaystyle {\partial }_1^L\ge {\partial }_2^L\ge \cdots \ge {\partial }_{n-1}^L\ge {\partial }_n^L=0.}$ More about ${\displaystyle D^L\left(G\right)}$ matrix (distance Laplacian) can be found in [26,27].

The following result relates Laplacian and distance Laplacian eigenvalues for graphs of diameter two.

Lemma 1

([26]). Let ${\displaystyle G}$ be a connected graph on ${\displaystyle n}$ nodes with diameter at most two and let ${\displaystyle {⋌}_1\ge {⋌}_2\ge \cdots \ge {⋌}_{n-1}\ge {⋌}_n=0}$ be the Laplacian spectrum of ${\displaystyle G}$. Then the distance Laplacian spectrum of ${\displaystyle G}$ is ${\displaystyle {\partial }_1^L=2n-{⋌}_{n-1}\ge {\partial }_2^L=2n-{⋌}_{n-2}\ge \cdots \ge {\partial }_{n-1}^L=2n-{⋌}_1>{\partial }_n^L=0.}$

Keeping in view Lemma 1, the equivalent form of Conjecture 1 for the distance Laplacian matrix of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is given below.

Conjecture 2

([28]). For an integer ${\displaystyle n\ge 2}$, the following statements are equivalent:

(i) 

${\displaystyle {\partial }_1^L}$ of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is an integer.

(ii) 

${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is distance Laplacian integral.

(iii) 

${\displaystyle n}$ is either a prime power or the product of two distinct primes.

By Conjecture 1 (Theorems 7 and 8) and Lemma 1, we have the following positive answer for Conjecture 2.

Theorem 9.

Let ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ be the power graph of ${\displaystyle {\mathbb{Z}}_n,n\ge 2}$. Then ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ is distance Laplacian integral if and only if ${\displaystyle n}$ is either a prime power or the product of two distinct primes.

5. Conclusions

The article proves that the Laplacian and the distance Laplacian spectrum of the power graph of ${\displaystyle {\mathbb{Z}}_n}$ is integral provided ${\displaystyle n}$ is the product of two distinct primes. It still remains open to find sharp bounds for the eigenvalues of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$ when ${\displaystyle n}$ is other than the product of two primes. In addition, it would be interesting to find the sum of the largest Laplacian eigenvalues (Kay Fan norm) of ${\displaystyle P\left({\mathbb{Z}}_n\right)}$, the ratio of the largest and the smallest Laplacian eigenvalues, and applications of Laplacian spectra in the field of groups.