Some Algebraic Properties of a Class of Integral Graphs Determined by Their Spectrum

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                  </jats:inline-formula> is an integral graph. Also, we determine the automorphism group of <jats:inline-formula>
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                  </jats:inline-formula> are determined by their spectrum.</jats:p>


Introduction
e graphs in this paper are simple, undirected, and connected. We always assume that Γ denotes the complement graph of Γ. e eigenvalues of a graph Γ are the eigenvalues of the adjacency matrix of Γ. e spectrum of Γ is the list of the eigenvalues of the adjacency matrix of Γ together with their multiplicities, and it is denoted by Spec(Γ); see [1]. If all the eigenvalues of the adjacency matrix of the graph Γ are integers, then we say that Γ is an integral graph. e notion of integral graphs was first introduced by Harary and Schwenk in 1974; see [2]. In general, the problem of characterizing integral graphs seems to be very difficult.
ere are good surveys in this area; see [3]. For more results depending on the integral graphs and their applications in engineering networks, see [4][5][6]. For any vertex v of a connected graph Γ, we denote the set of vertices of Γ at distance r from Γ by Γ r (v). en, we have where d (u, v) denotes the distance in Γ between the vertices u and v and r is a nonnegative integer not exceeding d, the diameter of Γ. It is clear that Γ 0 (v) � v { }, and V(Γ) is partitioned into the disjoint subsets Γ 0 (v), . . . , Γ d (v), for each v in V(Γ). e graph Γ 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 Γ at distance r, we have |Γ r+1 (v) ∩ Γ 1 (u)| � b r , (0 ≤ r ≤ d − 1), e intersection numbers c r , b r , and a r satisfy where a r is the number of neighbours of u in Γ r (v). Let G be a finite group and let H be a subset of G such that it is closed under taking inverses and does not contain the identity. A Cayley graph Γ � Cay(G, H) is the graph whose vertex set and edge set are defined as follows: It is well known that if Γ is a distance regular graph with valency k, diameter d, adjacency matrix A, and intersection array determines all the eigenvalues of Γ [7]. Note that the concept of distance regular graphs dates back to the 1960s. ey 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. e 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. e authors in [9] proposed the following 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 Γ, let A(Γ) and L(Γ) � D(Γ) − A(Γ) be, respectively, the adjacency matrix and Laplacian matrix of Γ, where D(Γ) is the diagonal matrix of vertex degrees with d 1 , d 2 , . . . , d n as diagonal entries. Laplacian spectra and their applications are involved in diverse 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 Z n , denoted by Γ � Cay(Z n , S), where n � p m (p is a prime integer and m ∈ N) and S � a ∈ Z n |(a, n) � 1 . It is easy to check that S is an inverse closed subset in the group Z n and 0 ∉ S. us, Γ is a simple graph. is class of graphs is a special 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 Γ is , 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 ▽Γ is determined by its adjacency spectrum as well as its Laplacian spectrum, where K v is the complete graph on v vertices.

Definitions and Preliminaries
Definition 1 (see [7,21]). Let Γ be a graph with automorphism group Aut(Γ). We say that Γ is a vertex transitive graph if, for all vertices x, y of Γ, there is an automorphism θ in Aut(Γ) satisfying θ(x) � y. Also, we say that Γ is distance Theorem 1 (see [22]). Let Γ be a graph such that it contains k Definition 2 (see [23]). Let Γ 1 ∪ Γ 2 denote the disjoint union of graphs Γ 1 and Γ 2 . e join Γ 1 ▽Γ 2 is the graph obtained from Γ 1 ∪ Γ 2 by joining every vertex of Γ 1 with every vertex of Γ 2 . A multicone graph is defined to be the join of a clique and a regular graph.
Theorem 2 (see [9]). If Γ is a distance regular graph with diameter d and girth g satisfying one of the following properties, then every graph cospectral with Γ is also distance regular, with the same parameters as Γ: Proposition 1 (see [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 (see [9]). e following graph and its complement, which have at most four eigenvalues, are regular DS graphs: (i) e disjoint union of k copies of a strongly regular DS graph.

Main Results
Theorem 5. Let Γ � Cay(Z n , S) be the Cayley graph on the cyclic group Z n , where n � p m (p is a prime integer and m ∈ N) and S � a ∈ Z n |(a, n) � 1 . en, where I � 1, 2, . . . , p .
Proof. Let V(Γ) � 1, . . . , n { } be the vertex set of Γ. Note that if m � 1, then the result immediately follows. Because, in this case, Γ � K p , where K p is the complete graph on p vertices, in the sequel, we assume that m ≥ 2. Let T � 〈p〉 � kp | 0 ≤ k ≤ p m− 1 − 1 be the subgroup of the group Z n of order p m− 1 . It is clear that T and every coset of T represent an independent set in the graph Γ. In fact, if T + a is a coset of T in the group Z n such that T ∩ T + a � ∅, then a and p are coprime and hence we have a ∈ S. It follows that every coset of T is a clique of order p m− 1 in the complement of the graph Γ.
us, Γ contains p disjoint components Hence, by eorem 1, On the other hand, it is well known that, for any graph Γ, Aut(Γ) � Aut(Γ); see [1]. □ Proposition 3. Let Γ � Cay(Z n , S) be the Cayley graph on the cyclic group Z n , where n � p m (p is a prime integer and m ∈ N) and S � a ∈ Z n |(a, n) � 1 . en Γ is a distance transitive graph.
Proof. Suppose that u, v, x, y are vertices of Γ such that d(u, v) � d(x, y) � r, where r is a nonnegative integer not exceeding d, the diameter of Γ. So d(u, v) � d(x, y) � 1 or 2, since we now have the diameter of Γ as d � 2. In the following cases, we show that Γ is a distance transitive graph. Case 1. If d(u, v) � d(x, y) � 2, then u − 1 v ∉ S and x − 1 y ∉ S. erefore, two vertices u and v are adjacent in the complement Γ of Γ; also two vertices x and y are adjacent in the complement Γ of Γ. By eorem 5, we know that Γ contains p components Γ 1 , Γ 2 , . . . , Γ p such that, for any i ∈ 1, 2, . . . , p , Γ i � K p m− 1 . erefore, Γ � pK p m− 1 . If u � x, then u, v, y lie in a clique of graph Γ, and hence we may assume that θ � (vy) ∈ Aut(Γ) � Aut(Γ), so θ(u) � x and θ(v) � y. If u ≠ x and v ≠ y, then u, v lie in a clique of graph Γ, say Γ i ; also x, y lie in a clique of graph Γ, say Γ j , where Γ i ≠ Γ j or Γ i � Γ j . Hence, we may assume that θ � (ux)(vy) ∈ Aut(Γ) � Aut(Γ). us, θ(u) � x and θ(v) � y. Case 2. If d(u, v) � d(x, y) � 1, then we can show that there is an automorphism θ in Aut(Γ) such that θ(u) � x and θ(v) � y. □ Proposition 4. Let Γ � Cay(Z n , S) be the Cayley graph on the cyclic group Z n , where n � p m (p is a prime integer and m ∈ N) and S � a ∈ Z n |(a, n) � 1 . en Γ is an integral graph.
and, by definition of distance regularity of graph, we have determines all the eigenvalues of Γ. It is clear that all the eigenvalues of Γ are n − p m− 1 , 0, − p m− 1 , and their multiplicities are 1, n − p, p − 1, respectively. us, Γ is an integral graph.
□ Corollary 1. Let Γ � Cay(Z n , S) be the Cayley graph on the cyclic group Z n , where n � p m (p is a prime integer and m ∈ N) and S � a ∈ Z n |(a, n) � 1 . en the adjacency spectrum of Γ is n − p m− 1 , 0 (n− p) , (− p m− 1 ) (p− 1) . Theorem 6. Let Γ � Cay(Z n , S) be the Cayley graph on the cyclic group Z n , where n � p m (p is a prime integer and m ∈ N) and S � a ∈ Z n |(a, n) � 1 . en Γ is a DS graph with respect to its adjacency spectrum.
Proof. We know that if p is even prime integer, then Γ is isomorphic to the bipartite graph K p m− 1 ,p m− 1 , and hence the result immediately follows. Now, let p be an odd prime integer; then, Γ is not bipartite graph. In particular, g ≥ 2 d − 1, because the diameter of Γ is 2 and the girth of Γ is 3. Hence, by eorem 2, every graph cospectral with Γ is also distance regular, with the same parameters as Γ. Because by Proposition 3 we know that Γ is a distance regular graph, Γ is a DS graph with respect to its adjacency spectra. Because, by Proposition 2, Γ contains disjoint union of p copies of the strongly regular DS graph K p m− 1 in addition to the graph Γ and its complement, which have at most four eigenvalues. □ Proposition 5. Let Π be a graph cospectral with the multicone graph K v ▽Γ with respect to its adjacency matrix spectrum, where Γ � Cay(Z n , S), which is defined as before.
en Π is a bidegreed graph. Also, where Proof. We can deduce the following from eorem 2.1.8 in [25] and eorem 2.1 in [26]. □ Theorem 7. Consider the multicone graph K v ▽Γ, where Γ � Cay(Z n , S), which is defined as before. en K v ▽Γ 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 let Π be a graph cospectral with the multicone graph K 1 ▽Γ with respect to its adjacency matrix spectrum. By Proposition 5, it is easy to see that Π has one vertex of degree p m , say j.
Because, by eorem 6, we know that Γ is DS graph with respect to its adjacency matrix spectrum, Π � K 1 ▽Γ. We assume inductively that this claim holds for K v ; that is, if Π 1 is a graph cospectral with the multicone graph K v ▽Γ with respect to its adjacency matrix spectrum, then Π 1 � K v ▽Γ. We show that the claim is true for K v+1 ; that is, if Π is a graph cospectral with the multicone graph K v+1 ▽Γ with respect to its adjacency matrix spectrum, then Π � K v+1 ▽Γ. It is obvious that Π has one vertex and p m + v edges more than Π 1 . On the other hand, by Proposition 5, we know that Π 1 has v vertices of degree p m + v − 1 and p m vertices of degree p m − p m− 1 + v, and also Π 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 Π � K 1 ▽Π 1 . Now, by assuming induction, we conclude that Π � K v+1 ▽Γ and complete the proof.
□ Theorem 8. Consider the complement K v ▽Γ of multicone graph K v ▽Γ with respect to its adjacency spectrum, where Γ � Cay(Z n , S), which is defined as before. en, K v ▽Γ is a DS graph.
Proof. By eorem 5, we know that Γ contains p compo- In addition, the adjacency matrix spectrum of Γ is Also, the adjacency matrix spectrum of K v is 0 (v) . us, the adjacency matrix spectrum of Γ ∪ K v is On the other hand, it is not hard to see that Γ ∪ K v � K v ▽Γ. Let Π be a graph cospectral with the complement K v ▽Γ of multicone graph K v ▽Γ with respect to its adjacency spectrum; then, (10) It is easy to prove that Π cannot be regular, since regularity of a graph can be determined by its spectrum. Also, we show that Π is disconnected graph. Suppose to the contrary that Π is connected; hence, by Lemma 1, Π is complete multipartite graph, contradicting the adjacency spectrum of Π. us, Π is disconnected graph. erefore, we conclude that K v ▽Γ is DS with respect to its adjacency spectrum. □ Proposition 6. Consider the multicone graph K v ▽Γ, where Γ � Cay(Z n , S), which is defined as before. en K v ▽Γ is DS with respect to its Laplacian spectrum.
Proof. By eorem 3, the Laplacian matrix spectrum of 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(Π 1 )) � Spec(L(K v ▽Γ)), then Π 1 � K v ▽Γ, where Π 1 is a graph cospectral with the multicone graph K v ▽Γ with respect to its Laplacian spectrum. We show that the claim is true for K v+1 ; that is, if Spec(L(Π)) � Spec L K v+1 ▽Γ � (n + v + 1) (p+v) , n + v + 1 − p m− 1 (n− p) , 0 , then Π � K v+1 ▽Γ, where Π is a graph cospectral with the multicone graph K v+1 ▽Γ with respect to its Laplacian spectrum. By eorem 4, we know that Π 1 and Π are join of two graphs, because n + v and n + v + 1 are eigenvalues of Π 1 and Π, respectively. In addition, Π has one vertex of degree n + v more than Π 1 , say j; hence, Spec(L(Π − j)) � Spec(L(K v ▽Γ)), and, by assuming induction, Π − j � K v ▽Γ.

Conclusion
In this paper, we computed the adjacency spectrum of a class of integral graphs, denoted by Γ � Cay(Z n , S), where n � p m (p is a prime integer and m ∈ N) and S � a ∈ Z n |(a, n) � 1 . Indeed, by using the theory of distance regular graphs, it is shown that the adjacency spectrum of Γ 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 Γ and K v ▽Γ are determined by their spectrum. Note that this class of graphs is a special subclass of integral circulants, and hence clearly not only is this class of graphs mathematically applicable, but also it is used in the design of engineering networks.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.