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

Liu, Jia-Bao; Mirafzal, S. Morteza; Zafari, Ali

doi:https://doi.org/10.1155/2021/6632206

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Research Article | Open Access

Volume 2021 | Article ID 6632206 | https://doi.org/10.1155/2021/6632206

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

Jia-Bao Liu,¹S. Morteza Mirafzal,²and Ali Zafari³

Academic Editor: Ghulam Mustafa

Received28 Dec 2020

Revised27 Jan 2021

Accepted05 Feb 2021

Published17 Feb 2021

Abstract

Let be a graph. If all the eigenvalues of the adjacency matrix of the graph are integers, then we say that is an integral graph. A graph 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 , where ( is a prime integer and ) and . First, we show that is an integral graph. Also, we determine the automorphism group of . Moreover, we show that and are determined by their spectrum.

1. Introduction

The graphs in this paper are simple, undirected, and connected. We always assume that denotes the complement graph of . The eigenvalues of a graph are the eigenvalues of the adjacency matrix of . The spectrum of is the list of the eigenvalues of the adjacency matrix of together with their multiplicities, and it is denoted by Spec; see [1]. If all the eigenvalues of the adjacency matrix of the graph are integers, then we say that is an integral graph. The 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. There are good surveys in this area; see [3]. For more results depending on the integral graphs and their applications in engineering networks, see [4–6]. For any vertex of a connected graph , we denote the set of vertices of at distance from by . Then, we havewhere denotes the distance in between the vertices and and is a nonnegative integer not exceeding , the diameter of . It is clear that , and is partitioned into the disjoint subsets , for each in . The graph is called distance regular with diameter and intersection array if it is regular of valency and, for any two vertices and in at distance , we have , , and . The intersection numbers , and satisfy , where is the number of neighbours of in . Let be a finite group and let be a subset of such that it is closed under taking inverses and does not contain the identity. A Cayley graph is the graph whose vertex set and edge set are defined as follows:

It is well known that if is a distance regular graph with valency , diameter , adjacency matrix , and intersection arraythen the tridiagonal matrix,determines all the eigenvalues of [7]. Note that the concept of distance regular graphs dates 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 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 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 , the complete graph and the cycle , graph , and their complements; see [10–12]. For a graph , let and be, respectively, the adjacency matrix and Laplacian matrix of , where is the diagonal matrix of vertex degrees with 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 , denoted by , where ( is a prime integer and ) and . It is easy to check that is an inverse closed subset in the group and . Thus, is a simple graph. This 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 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 is determined by its adjacency spectrum as well as its Laplacian spectrum, where is the complete graph on vertices.

2. Definitions and Preliminaries

Definition 1. (see [7, 21]). Let be a graph with automorphism group . We say that is a vertex transitive graph if, for all vertices of , there is an automorphism in satisfying . Also, we say that is distance transitive graph if, for all vertices of such that , there is an automorphism in satisfying and .

Theorem 1. (see [22]). Let be a graph such that it contains components . If, for any , we have , then , where the wreath product is defined.

Definition 2. (see [23]). Let denote the disjoint union of graphs and . The join is the graph obtained from by joining every vertex of with every vertex of . A multicone graph is defined to be the join of a clique and a regular graph.

Theorem 2. (see [9]). If is a distance regular graph with diameter and girth satisfying one of the following properties, then every graph cospectral with is also distance regular, with the same parameters as :(i)(ii) and is bipartite

Proposition 1. (see [9]). For regular graphs, being (or not ) is equivalent for the adjacency matrix, the adjacency matrix of the complement, and the Laplacian matrix.

Proposition 2 (see [9]). The following graph and its complement, which have at most four eigenvalues, are regular graphs:(i)The disjoint union of copies of a strongly regular graph.

Theorem 3. (see [24]). Let and be two graphs with the Laplacian spectrum and , respectively. Then, the Laplacian spectrum of is .

Theorem 4. (see [1]). Let be a graph on vertices. Then, is a Laplacian eigenvalue of if and only if is the join of two graphs.

Lemma 1. (see [1]). A connected graph has exactly one positive eigenvalue if and only if it is a complete multipartite graph.

3. Main Results

Theorem 5. Let be the Cayley graph on the cyclic group , where ( is a prime integer and ) and . Then,where .

Proof. Let be the vertex set of . Note that if , then the result immediately follows. Because, in this case, , where is the complete graph on vertices, in the sequel, we assume that . Let be the subgroup of the group of order . It is clear that and every coset of represent an independent set in the graph . In fact, if is a coset of in the group such that , then and are coprime and hence we have . It follows that every coset of is a clique of order in the complement of the graph . Thus, contains disjoint components such that (), where is the complete graph on vertices. It follows that . Hence, by Theorem 1, . On the other hand, it is well known that, for any graph , ; see [1].

Proposition 3. Let be the Cayley graph on the cyclic group , where ( is a prime integer and ) and . Then is a distance transitive graph.

Proof. Suppose that are vertices of such that , where is a nonnegative integer not exceeding , the diameter of . So or 2, since we now have the diameter of as . In the following cases, we show that is a distance transitive graph. Case 1. If , then and . Therefore, two vertices and are adjacent in the complement of ; also two vertices and are adjacent in the complement of . By Theorem 5, we know that contains components such that, for any , . Therefore, . If , then lie in a clique of graph , and hence we may assume that , so and . If and , then lie in a clique of graph , say ; also lie in a clique of graph , say , where or . Hence, we may assume that . Thus, and . Case 2. If , then we can show that there is an automorphism in such that and .

Proposition 4. Let be the Cayley graph on the cyclic group , where ( is a prime integer and ) and . Then is an integral graph.

Proof. It is well known that if is a distance transitive graph, then is also distance regular; see [21]. Now, let be the vertex set of . Consider the vertex in ; then , , and . Let be the vertex in such that ; then and . Hence, , and, by definition of distance regularity of graph, we have . Also, if in and , then two vertices are adjacent in , so , and . Hence, , , and . Finally, if in and , then two vertices are not adjacent in , so ; hence, and . Thus, the intersection array of is . Therefore, the tridiagonal matrix,determines all the eigenvalues of . It is clear that all the eigenvalues of are , and their multiplicities are , respectively. Thus, is an integral graph.

Corollary 1. Let be the Cayley graph on the cyclic group , where ( is a prime integer and ) and . Then the adjacency spectrum of is .

Theorem 6. Let be the Cayley graph on the cyclic group , where ( is a prime integer and ) and . Then is a graph with respect to its adjacency spectrum.

Proof. We know that if is even prime integer, then is isomorphic to the bipartite graph , and hence the result immediately follows.
Now, let be an odd prime integer; then, is not bipartite graph. In particular, , because the diameter of is and the girth of is . Hence, by Theorem 2, every graph cospectral with is also distance regular, with the same parameters as . Because by Proposition 3 we know that is a distance regular graph, is a graph with respect to its adjacency spectra. Because, by Proposition 2, contains disjoint union of copies of the strongly regular graph in addition to the graph and its complement, which have at most four eigenvalues.

Proposition 5. Let be a graph cospectral with the multicone graph with respect to its adjacency matrix spectrum, where , which is defined as before. Then is a bidegreed graph. Also,where and .

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

Theorem 7. Consider the multicone graph , where , which is defined as before. Then is with respect to its adjacency matrix spectrum.

Proof. In the following, we proceed by induction on the number of vertices in . Let have one vertex and let be a graph cospectral with the multicone graph with respect to its adjacency matrix spectrum. By Proposition 5, it is easy to see that has one vertex of degree , say . Hence, if , then . Because, by Theorem 6, we know that is graph with respect to its adjacency matrix spectrum, . We assume inductively that this claim holds for ; that is, if is a graph cospectral with the multicone graph with respect to its adjacency matrix spectrum, then . We show that the claim is true for ; that is, if is a graph cospectral with the multicone graph with respect to its adjacency matrix spectrum, then . It is obvious that has one vertex and edges more than . On the other hand, by Proposition 5, we know that has vertices of degree and vertices of degree , and also has vertices of degree and vertices of degree . So, we must have . Now, by assuming induction, we conclude that and complete the proof.

Theorem 8. Consider the complement of multicone graph with respect to its adjacency spectrum, where , which is defined as before. Then, is a graph.

Proof. By Theorem 5, we know that contains components such that (). So . In addition, the adjacency matrix spectrum of isAlso, the adjacency matrix spectrum of is . Thus, the adjacency matrix spectrum of isOn the other hand, it is not hard to see that . Let be a graph cospectral with the complement of multicone graph with respect to its adjacency spectrum; then,It is easy to prove that cannot be regular, since regularity of a graph can be determined by its spectrum. Also, we show that is disconnected graph. Suppose to the contrary that is connected; hence, by Lemma 1, is complete multipartite graph, contradicting the adjacency spectrum of . Thus, is disconnected graph. Therefore, we conclude that is with respect to its adjacency spectrum.

Proposition 6. Consider the multicone graph , where , which is defined as before. Then is with respect to its Laplacian spectrum.

Proof. By Theorem 3, the Laplacian matrix spectrum of isWe proceed by induction on the number of vertices in . If , there is nothing to prove. We assume inductively that this claim holds for ; that is, if , then , where is a graph cospectral with the multicone graph with respect to its Laplacian spectrum. We show that the claim is true for ; that is, ifthen , where is a graph cospectral with the multicone graph with respect to its Laplacian spectrum. By Theorem 4, we know that and are join of two graphs, because and are eigenvalues of and , respectively. In addition, has one vertex of degree more than , say ; hence, , and, by assuming induction, . Thus, it can be concluded that .

4. Conclusion

In this paper, we computed the adjacency spectrum of a class of integral graphs, denoted by , where ( is a prime integer and ) and . Indeed, by using the theory of distance regular graphs, it is shown that the adjacency spectrum of is , where the superscripts give the multiplicities of eigenvalues with multiplicity greater than one. Moreover, it is shown that the Cayley graph and 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

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

Acknowledgments

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.

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Copyright © 2021 Jia-Bao Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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