Some algebraic properties of a class of integral graphs determined by their spectrum

TL;DR

This paper studies the algebraic properties of a specific class of integral Cayley graphs on cyclic groups of prime power order, showing they are determined by their spectrum and analyzing their automorphism groups.

Contribution

It proves that certain Cayley graphs on cyclic groups of prime power order are integral, determines their automorphism groups, and shows they are uniquely identified by their spectrum.

Findings

The Cayley graph $ ext{Cay}(bZ_{n}, S)$ is integral for $n=p^m$.

The automorphism group of the graph is explicitly determined.

The graph and its join with the complete graph are uniquely identified by their spectrum.

Abstract

Let $\Gamma=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $\Gamma$ are integers, then we say that $\Gamma$ is an integral graph. A graph $\Gamma$ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph $\Gamma=Cay(\mathbb{Z}_{n}, S)$, where $n=p^m$, ($p$ is a prime integer, $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\, (a, n)=1\}$. First, we show that $\Gamma$ is an integral graph. Also we determine the automorphism group of $\Gamma$. Moreover, we show that $\Gamma$ and $K_v \bigtriangledown\Gamma$ are determined by their spectrum.

Equations

$${\Gamma }_r(v)=\{u\in V(\Gamma )|d(u,v)=r\},$$

$$V(\Gamma )=G;E(\Gamma )=\{\{x,y\}|x^{-1}y\in H\}.$$

$$\{b_0,b_1,...,b_{d-1};c_1,c_2,...,c_d\}.$$

$$Aut(\Gamma )\cong Sym(p^{m-1})w{r}_{I}Sym(p),$$

$$Spec(\Pi )=\{0^{(n-p)},{(-p^{m-1})}^{(p-1)},-1^{(v-1)},(\frac{M+\sqrt{M^2+4N}}{2}),(\frac{M-\sqrt{M^2+4N}}{2})\},$$

$$\{(p^{m-1}-1{)}^{(p)},-1^{(p^m-p)}\}.$$

$$\{(p^{m-1}-1{)}^{(p)},-1^{(p^m-p)},0^{(v)}\}.$$

$$Spec(\Pi )=Spec(\overline{K_v▽\Gamma })=\{(p^{m-1}-1{)}^{(p)},-1^{(p^m-p)},0^{(v)}\}.$$

$$\{(n+v{)}^{(p+v-1)},(n+v-p^{m-1}{)}^{(n-p)},0\}.$$

$$Spec(L(\Pi ))=Spec(L(K_{v+1}▽\Gamma ))$$

$$=\{(n+v+1{)}^{(p+v)},(n+v+1-p^{m-1}{)}^{(n-p)},0\},$$

Full text

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1 Introduction

The graphs in this paper are simple, undirected and connected. We always assume that $\overline{\Gamma}$ denotes the complement graph of $\Gamma$. The eigenvalues of a graph $\Gamma$ are the eigenvalues of the adjacency matrix of $\Gamma$. The spectrum of $\Gamma$ is the list of the eigenvalues of the adjacency matrix of $\Gamma$ together with their multiplicities, and it is denoted by Spec$(\Gamma)$, see  [1]. If all the eigenvalues of the adjacency matrix of the graph $\Gamma$ are integers, then we say that $\Gamma$ is an integral graph. The notion of integral graphs was first introduced by F. Harary and A. J. Schwenk in 1974, see  [2]. In general, the problem of characterizing integral graphs seems to be very difficult. There are good surveys in this area, see  [3]. For more results depend on the integral graphs and their applications in engineering networks see  [4, 5, 6]. For any vertex $v$ of a connected graph $\Gamma$, we denote the set of vertices of $\Gamma$ at distance $r$ from $\Gamma$ by $\Gamma_{r}(v)$. Then we have,

[TABLE]

where $d(u,v)$ denotes the distance in $\Gamma$ between the vertices $u$ and $v$, and $r$ is a non negative integer not exceeding $d$, the diameter of $\Gamma$. It is clear that $\Gamma_{0}(v)=\{v\}$, and $V(\Gamma)$ is partitioned into the disjoint subsets $\Gamma_{0}(v),...,\Gamma_{d}(v)$, for each $v$ in $V(\Gamma)$. The graph $\Gamma$ is called distance regular with diameter $d$ and intersection array $\{b_{0},...,b_{d-1};c_{1},...,c_{d}\}$ if it is regular of valency $k$ and, for any two vertices $u$ and $v$ in $\Gamma$ at distance $r$ we have $|\Gamma_{r+1}(v)\cap\Gamma_{1}(u)|=b_{r}$, $(0\leq r\leq d-1)$ and $|\Gamma_{r-1}(v)\cap\Gamma_{1}(u)|=c_{r}$ $(1\leq r\leq d)$. The intersection numbers $c_{r},b_{r}$ and $a_{r}$ satisfy $a_{r}=k-b_{r}-c_{r}\,\ \,\,\,\,(0\leq r\leq d),$ where $a_{r}$ is the number of neighbours of $u$ in $\Gamma_{r}(v)$. Let $G$ be a finite group and $H$ a subset of $G$ such that it is closed under taking inverses and does not contain the identity. A Cayley graph $\Gamma=Cay(G,H)$ is the graph whose vertex set and edge set are defined as follows:

[TABLE]

It is well known that if $\Gamma$ is a distance regular graph with valency $k$, diameter $d$, adjacency matrix $A$, and intersection array

[TABLE]

then, the tridiagonal $(d+1)\times(d+1)$ matrix,

$B=\begin{bmatrix}a_{0}&b_{0}&0&0&...&\\ c_{1}&a_{1}&b_{1}&0&...&\\ 0&c_{2}&a_{2}&b_{2}&&\\ &&&...&\\ &&&&\\ &&&&c_{d-2}&a_{d-2}&b_{d-2}&0\\ &&&...&0&c_{d-1}&a_{d-1}&b_{d-1}\\ &&&...&0&0&c_{d}&a_{d}\\ \end{bmatrix},$

determines all the eigenvalues of $\Gamma$  [7]. Note that, the concept of distance regular graphs date back to the 1960s. They were defined by Biggs, see  [8], and their basic theory was developed by him and others. Distance regular graphs of diameter $2$ are just the connected strongly regular graphs. The theory of distance regular graphs has connections to many parts of graph theory such as, design theory, coding theory, geometry, and group theory. Two graphs with the same spectrum are called cospectral. It is not hard to see that the spectrum of a graph does not determine its isomorphism class. The authors in  [9] proposed the question: which graphs are determined by their spectrum? It seems hard to prove a graph to be determined by its spectrum. Up to now, only a few classes of graphs are proved to be determined by their spectrum, such as: the path $P_{n}$, the complete graph $K_{n}$ and the cycle $C_{n}$, graph $Z_{n}$ and their complements, see  [10, 11, 12]. For a graph $\Gamma$, let $A(\Gamma)$ and $L(\Gamma)=D(\Gamma)-A(\Gamma)$ be respectively the adjacency matrix and Laplacian matrix of $\Gamma$, where $D(\Gamma)$ is the diagonal matrix of vertex degrees with $\{d_{1},d_{2},...,d_{n}\}$ as diagonal entries. Laplacian spectrum and their applications are involved indiverse theoretical problems on complex networks  [13, 14]. Many results have been devoted to studying Laplacian spectrum for complex networks  [15, 16]. Calculating the Laplacian spectrum of networks has many applications in lots of aspects, such as the topological structures and dynamical processes  [17]. Algebraic properties of various classes of Cayley graphs have been studied by various authors, see  [18, 19]. In this paper, we want to study some algebraic properties of a class of Cayley graphs constructed on the cyclic additive group $\mathbb{Z}_{n}$, denoted by $\Gamma=Cay(\mathbb{Z}_{n},S)$, where $n=p^{m}$, ($p$ is a prime integer, $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\,(a,n)=1\}$. It is easy to check that $S$ is an inverse closed subset in the group $\mathbb{Z}_{n}$ and $0\notin S$. Thus, $\Gamma$ is a simple graph. This class of graphs is a espacial subclass of graphs, which are investigated from some other aspects by Basić and Ilić  [20]. Using the theory of distance regular graphs, we show that the adjacency spectrum of $\Gamma$ is $\{n-p^{m-1},0^{(n-p)},(-p^{m-1})^{(p-1)}\}$, where the superscripts give the multiplicities of eigenvalues with multiplicity greater than one. Finally, we show that any graph cospectral with the multicone graph $K_{v}\bigtriangledown\Gamma$ is determined by their adjacency spectrum as well as their Laplacian spectrum, where $K_{v}$ is the complete graph on $v$ vertices.

2 Definitions And Preliminaries

Definition 2.1**.**

[7, 21]** Let $\Gamma$ be a graph with automorphism group $Aut(\Gamma)$. We say that $\Gamma$ is a vertex transitive graph if for all vertices $x,y$ of $\Gamma$, there is an automorphism $\theta$ in $Aut(\Gamma)$ satisfying $\theta(x)=y$. Also, we say that $\Gamma$ is distance transitive graph if, for all vertices $u,v,x,y$ of $\Gamma$ such that $d(u,v)=d(x,y)$, there is an automorphism $\theta$ in $Aut(\Gamma)$ satisfying $\theta(u)=x$ and $\theta(v)=y$

Theorem 2.1**.**

[22]** Let $\Gamma$ be a graph such that contains $k$ components $\Gamma_{1},...,\Gamma_{k}$. If for any $i\in I=\{1,...,k\}$, we have $\Gamma_{i}\cong\Gamma_{1}$ then $Aut(\Gamma)\cong Aut(\Gamma_{1})wr_{I}Sym(k)$, where the wreath product is defined.

Definition 2.2**.**

[23]** Let $\Gamma_{1}\cup\Gamma_{2}$ denote the disjoint union of graphs $\Gamma_{1}$ and $\Gamma_{2}$. The join $\Gamma_{1}\bigtriangledown\Gamma_{2}$ is the graph obtained from $\Gamma_{1}\cup\Gamma_{2}$ by joining every vertex of $\Gamma_{1}$ with every vertex of $\Gamma_{2}$. A multicone graph is defined to be the join of a clique and a regular graph.

Theorem 2.2**.**

[9]** If $\Gamma$ is a distance regular graph with diameter $d$ and girth $g$ satisfying one of the following properties, then every graph cospectral with $\Gamma$ is also distance regular, with the same parameters as $\Gamma$:

• (i)

$g\geq 2d-1$,

• (ii)

$g\geq 2d-2$* and $\Gamma$ is bipartite.*

Proposition 2.1**.**

[9]** For regular graphs, being $DS$ (or not $DS$) is equivalent for the adjacency matrix, the adjacency matrix of the complement, and the Laplacian matrix.

Proposition 2.2**.**

[9]** The following graph and its complement, which have at most four eigenvalues, are regular $DS$ graphs:

• (i)

The disjoint union of $k$ copies of a strongly regular $DS$ graph.

Theorem 2.3**.**

[24]** Let $\Gamma_{1}$ and $\Gamma_{2}$ be two graphs with the Laplacian spectrum $\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{n}$ and $\mu_{1}\geq\mu_{2}\geq...\geq\mu_{m}$, respectively. Then, the Laplacian spectrum of $\Gamma_{1}\bigtriangledown\Gamma_{2}$, is $n+m,m+\lambda_{1},m+\lambda_{2},...,m+\lambda_{n-1},n+\mu_{1},n+\mu_{2},...,n+\mu_{m-1},0$.

Theorem 2.4**.**

[1]** Let $\Gamma$ be a graph on $n$ vertices. Then, $n$ is a Laplacian eigenvalue of $\Gamma$ if and only if $\Gamma$ is the join of two graphs.

Lemma 2.1**.**

[1]** A connected graph $\Gamma$ has exactly one positive eigenvalue if and only if it is a complete multipartite graph.

3 Main results

Theorem 3.1**.**

Let $\Gamma=Cay(\mathbb{Z}_{n},S)$ be the Cayley graph on the cyclic group $\mathbb{Z}_{n}$, where $n=p^{m}$, ($p$ is a prime integer and $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\,\text{(a, n)=1}\}$. Then,

[TABLE]

where $I=\{1,2,...,p\}.$

Proof.

Let $V(\Gamma)=\{1,...,n\}$ be the vertex set of $\Gamma$. Note that if $m=1$, then the result immediately follows. Because in this case, $\Gamma\cong K_{p}$, where $K_{p}$ is the complete graph on $p$ vertices. Hence in the sequel, we assume that $m\geq 2$. Let $T=\langle p\rangle=\{kp\ |\ 0\leq k\leq p^{m-1}-1\}$ be the subgroup of the group $\mathbb{Z}_{n}$ of order $p^{m-1}$. It is clear that $T$ and every co-set of $T$ is an independent set in the graph $\Gamma$. In fact if $T+a$ is a co-set of $T$ in the group $\mathbb{Z}_{n}$ such that $T\cap T+a=\emptyset$, then $a$ and $p$ are co-prime and hence we have $a\in S$. This follows that every co-set of $T$ is a clique of order $p^{m-1}$ in the complement of the graph $\Gamma$. Thus, $\overline{\Gamma}$ contains of $p$ disjoint components $\Gamma_{1},\Gamma_{2},...,\Gamma_{p}$ such that $\Gamma_{i}\cong K_{p^{m-1}}$ ($1\leq i\leq p$), where $K_{p^{m-1}}$ is the complete graph on $p^{m-1}$ vertices. This follows that $\overline{\Gamma}\cong pK_{p^{m-1}}$. Hence, by Theorem 2.1, $Aut(\overline{\Gamma})\cong Aut(K_{p^{m-1}})wr_{I}Sym{(p)}=Sym(p^{m-1})wr_{I}Sym{(p)}$. On the other hand, it is well known that for any graph $\Gamma$, $Aut(\Gamma)=Aut(\overline{\Gamma})$, see  [1]. ∎

Proposition 3.1**.**

Let $\Gamma=Cay(\mathbb{Z}_{n},S)$ be the Cayley graph on the cyclic group $\mathbb{Z}_{n}$, where $n=p^{m}$, ($p$ is a prime integer and $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\,\text{(a, n)=1}\}$. Then $\Gamma$ is a distance transitive graph.

Proof.

Suppose $u,v,x,y$ are vertices of $\Gamma$ such that $d(u,v)=d(x,y)=r$, where $r$ is a non negative integer not exceeding $d$, the diameter of $\Gamma$. So $d(u,v)=d(x,y)=1$ or 2, since we now have the diameter of $\Gamma$ is $d=2$. In the following cases we show that $\Gamma$ is a distance transitive graph.

Case 1. If $d(u,v)=d(x,y)=2$, then $u^{-1}v\notin S$ and $x^{-1}y\notin S$. Therefore, two vertices $u$ and $v$ are adjacent in the complement $\overline{\Gamma}$ of $\Gamma$, also two vertices $x$ and $y$ are adjacent in the complement $\overline{\Gamma}$ of $\Gamma$. By Theorem 3.1, we know that $\overline{\Gamma}$ contains $p$ components $\Gamma_{1},\Gamma_{2},...,\Gamma_{p}$ such that for any $i\in\{1,2,...,p\}$, $\Gamma_{i}\cong K_{p^{m-1}}$. Therefore, $\overline{\Gamma}\cong pK_{p^{m-1}}$. If $u=x$, then $u,v,y$ lie in a clique of graph $\overline{\Gamma}$, and hence we may assume $\theta=(vy)\in Aut(\overline{\Gamma})=Aut({\Gamma})$, so $\theta(u)=x$ and $\theta(v)=y$. If $u\neq x$ and $v\neq y$, then $u,v$ lie in a clique of graph $\overline{\Gamma}$, say $\Gamma_{i}$, also $x,y$ lie in a clique of graph $\overline{\Gamma}$, say $\Gamma_{j}$, where $\Gamma_{i}\neq\Gamma_{j}$ or $\Gamma_{i}=\Gamma_{j}$. Hence we may assume $\theta=(ux)(vy)\in Aut(\overline{\Gamma})=Aut({\Gamma})$. Thus $\theta(u)=x$ and $\theta(v)=y$.

Case 2. If $d(u,v)=d(x,y)=1$, then we can show that there is an automorphism $\theta$ in $Aut(\Gamma)$ such that $\theta(u)=x$ and $\theta(v)=y$. ∎

Proposition 3.2**.**

Let $\Gamma=Cay(\mathbb{Z}_{n},S)$ be the Cayley graph on the cyclic group $\mathbb{Z}_{n}$, where $n=p^{m}$, ($p$ is a prime integer and $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\,\text{(a, n)=1}\}$. Then $\Gamma$ is an integral graph.

Proof.

It is well known that if $\Gamma$ is a distance transitive graph, then $\Gamma$ is also distance regular, see  [21]. Now, let $V(\Gamma)=\{1,2,...,n\}$ be the vertex set of $\Gamma$. Consider the vertex $v=n$ in $V(\Gamma)$, then $\Gamma_{0}(v)=\{n\}$, $\Gamma_{1}(v)=\{a\in V(\Gamma)\,|\,\,(a,n)=1\}$, and $\Gamma_{2}(v)=\{a\in V(\Gamma)\,|\,\,(a,n)\neq 1\}$. Let $u$ be the vertex in $V(\Gamma)$ such that $d(u,v)=0$, then $u=v=n$ and $|\Gamma_{1}(v)\cap\Gamma_{1}(u)|=n-p^{m-1}$. Hence $b_{0}=n-p^{m-1}$, and by Definition of distance regularity of graph we have $a_{0}=(n-p^{m-1})-b_{0}=0$. Also, if $u$ in $V(\Gamma)$ and $d(u,v)=1$, then two vertices $u,v$ are adjacent in $\Gamma$, so $|\Gamma_{0}(v)\cap\Gamma_{1}(u)|=1$, and $|\Gamma_{2}(v)\cap\Gamma_{1}(u)|=p^{m-1}-1$. Hence $c_{1}=1$, $b_{1}=p^{m-1}-1$ and $a_{1}=(n-p^{m-1})-b_{1}-c_{1}=n-2p^{m-1}$. Finally, if $u$ in $V(\Gamma)$ and $d(u,v)=2$, then two vertices $u,v$ are not adjacent in $\Gamma$, so $|\Gamma_{1}(v)\cap\Gamma_{1}(u)|=n-p^{m-1}$, hence $c_{2}=n-p^{m-1}$ and $a_{2}=(n-p^{m-1})-(n-p^{m-1})=0.$ Thus the intersection array of $\Gamma$ is $\{n-p^{m-1},p^{m-1}-1;1,n-p^{m-1}\}.$ Therefore, the tridiagonal $(3)\times(3)$ matrix,

$\begin{bmatrix}a_{0}&b_{0}&0\\ c_{1}&a_{1}&b_{1}\\ 0&c_{2}&a_{2}\\ \end{bmatrix}=\begin{bmatrix}0&n-p^{m-1}&0\\ 1&n-2p^{m-1}&p^{m-1}-1\\ 0&n-p^{m-1}&0\\ \end{bmatrix}$

determines all the eigenvalues of $\Gamma$. It is clear that all the eigenvalues of $\Gamma$ are $n-p^{m-1},0,-p^{m-1}$, and their multiplicities are $1,n-p,p-1$, respectively. Thus $\Gamma$ is an integral graph. ∎

Corollary 3.1**.**

Let $\Gamma=Cay(\mathbb{Z}_{n},S)$ be the Cayley graph on the cyclic group $\mathbb{Z}_{n}$, where $n=p^{m}$, ($p$ is a prime integer and $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\,\text{(a, n)=1}\}$. Then the adjacency spectrum of $\Gamma$ is $\{n-p^{m-1},0^{(n-p)},(-p^{m-1})^{(p-1)}\}$.

Theorem 3.2**.**

Let $\Gamma=Cay(\mathbb{Z}_{n},S)$ be the Cayley graph on the cyclic group $\mathbb{Z}_{n}$, where $n=p^{m}$, ($p$ is a prime integer and $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\,\text{(a, n)=1}\}$. Then $\Gamma$ is a $DS$ graph with respect to its the adjacency spectrum.

Proof.

We know that if $p$ is even prime integer, then $\Gamma$ is isomorphic to the bipartite graph $K_{p^{m-1},p^{m-1}}$, and hence the result immediately follows.

Now, let $p$ is an odd prime integer, then $\Gamma$ is not bipartite graph. In particular, $g\geq 2d-1$, because the diameter of $\Gamma$ is $2$, and the girth of $\Gamma$ is $3$. Hence by Theorem 2.2, every graph cospectral with $\Gamma$ is also distance regular, with the same parameters as $\Gamma$. Because by Proposition 3.1, we know that $\Gamma$ is a distance regular graph. Thus, $\Gamma$ is a $DS$ graph with respect to its the adjacency spectra. Because by Proposition 2.2, $\overline{\Gamma}$ contains disjoint union of $p$ copies of the strongly regular $DS$ graph $K_{p^{m-1}}$, in addition the graph $\Gamma$ and its complement, which have at most four eigenvalues. ∎

Proposition 3.3**.**

Let $\Pi$ be a graph cospectral with the multicone graph $K_{v}\bigtriangledown\Gamma$ with respect to its adjacency matrix spectrum, where $\Gamma=Cay(\mathbb{Z}_{n},S)$ which is defined as before. Then $\Pi$ is a bidegreed graph. Also,

[TABLE]

where $M=v-1+p^{m}-p^{m-1}$ and $N=p^{m}+p^{m-1}v-p^{m-1}$.

Proof.

We can deduce the following from Theorem 2.1.8 in  [25] and Theorem 2.1 in  [26]. ∎

Theorem 3.3**.**

Consider the multicone graph $K_{v}\bigtriangledown\Gamma$, where $\Gamma=Cay(\mathbb{Z}_{n},S)$ which is defined as before. Then $K_{v}\bigtriangledown\Gamma$ is $DS$ with respect to its adjacency matrix spectrum.

Proof.

In the following, we proceed by induction on the number of vertices in $K_{v}$. Let $K_{v}$ have one vertex and $\Pi$ be a graph cospectral with the multicone graph $K_{1}\bigtriangledown\Gamma$ with respect to its adjacency matrix spectrum. By Proposition 3.3, it is easy to see that $\Pi$ has one vertex of degree $p^{m}$, say $j$. Hence, if $Spec(\Pi-j)=Spec(\Gamma)$, then $\Pi-j\cong\Gamma$. Because, by Theorem 3.2, we know that $\Gamma$ is $DS$ graph with respect to its the adjacency matrix spectrum. So $\Pi\cong K_{1}\bigtriangledown\Gamma$. We assume inductively that this claim holds for $K_{v}$ that is, if $\Pi_{1}$ is a graph cospectral with the multicone graph $K_{v}\bigtriangledown\Gamma$ with respect to its adjacency matrix spectrum, then $\Pi_{1}\cong K_{v}\bigtriangledown\Gamma$. We show that the claim is true for $K_{v+1}$ that is, if $\Pi$ is a graph cospectral with the multicone graph $K_{v+1}\bigtriangledown\Gamma$ with respect to its adjacency matrix spectrum, then $\Pi\cong K_{v+1}\bigtriangledown\Gamma$. It is obvious that $\Pi$ has one vertex and $p^{m}+v$ edges more than $\Pi_{1}$. On the other hand, by Proposition 3.3, we know that $\Pi_{1}$ has $v$ vertices of degree $p^{m}+v-1$ and $p^{m}$ vertices of degree $p^{m}-p^{m-1}+v$, and also $\Pi$ has $v+1$ vertices of degree $p^{m}+v$ and $p^{m}$ vertices of degree $p^{m}-p^{m-1}+v+1$. So, we must have $\Pi\cong K_{1}\bigtriangledown\Pi_{1}$. Now, by assume induction we conclude that $\Pi\cong K_{v+1}\bigtriangledown\Gamma$, and complete the proof. ∎

Theorem 3.4**.**

Consider the complement $\overline{K_{v}\bigtriangledown\Gamma}$ of multicone graph $K_{v}\bigtriangledown\Gamma$ with respect to its adjacency spectrum, where $\Gamma=Cay(\mathbb{Z}_{n},S)$ which is defined as before. Then $\overline{K_{v}\bigtriangledown\Gamma}$ is a $DS$ graph.

Proof.

By Theorem 3.1, we know that $\overline{\Gamma}$ contains $p$ components $\Gamma_{1},\Gamma_{2},...,\Gamma_{p}$ such that $\Gamma_{i}\cong K_{p^{m-1}}$ ($1\leq i\leq p$). So $\overline{\Gamma}\cong pK_{p^{m-1}}$. In addition, the adjacency matrix spectrum of $\overline{\Gamma}$ is

[TABLE]

Also, the adjacency matrix spectrum of $\overline{K_{v}}$ is $\{0^{(v)}\}$. Thus the adjacency matrix spectrum of $\overline{\Gamma}\cup\overline{K_{v}}$ is

[TABLE]

On the other hand, it is not hard to see that $\overline{\Gamma}\cup\overline{K_{v}}\cong\overline{K_{v}\bigtriangledown\Gamma}$. Let $\Pi$ be a graph cospectral with the complement $\overline{K_{v}\bigtriangledown\Gamma}$ of multicone graph $K_{v}\bigtriangledown\Gamma$ with respect to its adjacency spectrum, then

[TABLE]

It is easy to prove that $\Pi$ cannot be regular, since regularity of a graph can be determined by its spectrum. Also, we show that $\Pi$ is disconnected graph. Suppose to the contrary $\Pi$ is connected, hence by Lemma 2.1, $\Pi$ is complete multipartite graph, contradicting the adjacency spectrum of $\Pi$. Thus $\Pi$ is disconnected graph. Therefore, we conclude that $\overline{K_{v}\bigtriangledown\Gamma}$ is $DS$ with respect to its adjacency spectrum. ∎

Proposition 3.4**.**

Consider the multicone graph $K_{v}\bigtriangledown\Gamma$, where $\Gamma=Cay(\mathbb{Z}_{n},S)$ which is defined as before. Then $K_{v}\bigtriangledown\Gamma$ is $DS$ with respect to its Laplacian spectrum.

Proof.

By Theorem 2.3, the Laplacian matrix spectrum of $K_{v}\bigtriangledown\Gamma$ is

[TABLE]

We proceed by induction on the number of vertices in $K_{v}$. If $v=1$, there is nothing to prove. We assume inductively that this claim holds for $K_{v}$ that is, if $Spec(L(\Pi_{1}))=Spec(L(K_{v}\bigtriangledown\Gamma))$, then $\Pi_{1}\cong K_{v}\bigtriangledown\Gamma$, where $\Pi_{1}$ is a graph cospectral with the multicone graph $K_{v}\bigtriangledown\Gamma$ with respect to its Laplacian spectrum. We show that the claim is true for $K_{v+1}$ that is, if

[TABLE]

[TABLE]

then $\Pi\cong K_{v+1}\bigtriangledown\Gamma$, where $\Pi$ is a graph cospectral with the multicone graph $K_{v+1}\bigtriangledown\Gamma$ with respect to its Laplacian spectrum. By Theorem 2.4, we know that $\Pi_{1}$ and $\Pi$ are join of two graphs, because $n+v$ and $n+v+1$ are eigenvalues of $\Pi_{1}$ and $\Pi$, respectively. In addition, $\Pi$ has one vertex of degree $n+v$ more than $\Pi_{1}$, say $j$, hence $Spec(L(\Pi-j))\cong Spec(L(K_{v}\bigtriangledown\Gamma))$, and by assume induction $\Pi-j\cong K_{v}\bigtriangledown\Gamma$. Thus it can be concluded $\Pi\cong K_{v+1}\bigtriangledown\Gamma$. ∎

4 Conclusion

In this paper, we computed the adjacency spectrum of a class of integral graphs, denoted by $\Gamma=Cay(\mathbb{Z}_{n},S)$, where $n=p^{m}$, ($p$ is a prime integer, $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\,(a,n)=1\}$. Indeed, by using the theory of distance regular graphs, it is shown that the adjacency spectrum of $\Gamma$ is $\{n-p^{m-1},0^{(n-p)},(-p^{m-1})^{(p-1)}\}$, where the superscripts give the multiplicities of eigenvalues with multiplicity greater than one. Moreover, it is shown that the Cayley graph $\Gamma$ and $K_{v}\bigtriangledown\Gamma$ are determined by their spectrum. Note that, this class of graphs is a espacial subclass of integral circulants, and hence clearly this class not only they are mathematically applicable, but also they are used in the design of engineering networks.

**Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgements

This work was supported in part by Anhui Provincial Natural Science Foundation under Grant 2008085J01 and Natural Science Fund of Education Department of Anhui Province under Grant KJ2020A0478.

**