Journal of Mathematics
Volume 2017 (2017), Article ID 6454736, 4 pages
https://doi.org/10.1155/2017/6454736

Research Article

An Interesting Property of a Class of Circulant Graphs

Seyed Morteza Mirafzal and Ali Zafari

Department of Mathematics, Lorestan University, Khoramabad, Iran

Correspondence should be addressed to Ali Zafari; zafari.ali@fs.lu.ac.ir

Received 22 July 2016; Accepted 22 January 2017; Published 27 February 2017

Academic Editor: Michel Bauer

Copyright © 2017 Seyed Morteza Mirafzal and Ali Zafari. 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.

Abstract.

Suppose that and are two Cayley graphs on the cyclic additive group , where is an even integer, , , and are the inverse-closed subsets of . In this paper, it is shown that is a distance-transitive graph, and, by this fact, we determine the adjacency matrix spectrum of . Finally, we show that if and is an even integer, then the adjacency matrix spectrum of is , , , (we write multiplicities as exponents).

1. Introduction

In this paper, graph always means a simple connected graph with vertices (without loops, multiple edges, and isolated vertices), where is the vertex set and is the edge set. Graph is called a vertex-transitive graph, if, for any , there is some in , the automorphism group of , such that . Let be a graph, the complement of is the graph whose vertex set is and whose edges are the pairs of nonadjacent vertices of . It is well known that, for any graph , [1]. If is a connected graph and denotes the distance in between the vertices and , then, for any automorphism in , we have

Let be a set and be a group; then, writing to denote the set of all functions from into , we can turn into a group by defining a product:

where the product on the right is in . Since is finite then the group is isomorphic to (a direct product of copies of ) via the isomorphism . Let and be groups and suppose that acts on the nonempty set . Then, the wreath product of by with respect to this action is defined to be the semidirect product where acts on the group via

We denote this group by . Consider the wreath product . If acts on a set , then we can define an action of on by

where [2].

Let be a group and a subgroup of and . The Schreier coset graph on generated by is the graph with the set of left cosets of , and there is an edge for each coset and each . If is inverse-closed, then is an undirected multigraph (possibly with loops). Note that if is the identity element of , then is the Cayley graph on generated by . It is well known that every Cayley graph is vertex-transitive [3].

Let be a graph with automorphism group . Say that is symmetric graph if, for all vertices of such that and are adjacent, also, and are adjacent, and there is an automorphism in such that and . We say that is distance-transitive if, for all vertices of such that , there is an automorphism in satisfying and [3]. It is clear that hierarchy of the conditions is

Eigenvalues of an undirected graph are the eigenvalues of an arbitrary adjacency matrix of . Harary and Schwenk [4] defined to be integral, if all of its eigenvalues are integers. For a survey of integral graphs, see [5]. In [6], the number of integral graphs on vertices is estimated. Known characterizations of integral graphs are restricted to certain graph classes; see [7].

In this paper, suppose and are two Cayley graphs on the cyclic additive group , where is an even integer, , , and are the inverse-closed subsets of . One of our goals in this paper is to obtain all eigenvalues of the Cayley graph . First, we determine the group automorphism of and we show that is a distance transitive graph; also, by this fact, we determine the adjacency matrix spectrum of . Finally, according to these facts, we show that if and is an even integer, then the adjacency matrix spectrum of is (we write multiplicities as exponents).

2. Definitions and Preliminaries

Definition 1 (see [3, 8]). For any vertex of a connected graph , one defines

where 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 , one has , and , The numbers , and , where

is the number of neighbours of in for , are called the intersection numbers of . Clearly , , and .

Remark 2 (see [3]). It is clear that if is distance-transitive graph, then is distance-regular.

Lemma 3 (see [3]). A connected graph with diameter and automorphism group is distance-transitive if and only if it is vertex-transitive and the vertex-stabilizer is transitive on the set , for each , and .

Theorem 4 (see [8]). Let be a distance-regular graph which the valency of each vertex as , with diameter , adjacency matrix , and intersection array, is

Then, the tridiagonal matrix

determines all the eigenvalues of .

Theorem 5 (see [9]). Let be a field and let be a commutative subring of , the set of all matrices over . Let , then .

Theorem 6 (see [10]). Let be a graph such that contains components . If, for any , , then .

2.1. Main Results

Proposition 7. Let be the Cayley graph on the cyclic group , where is the inverse-closed subset of . Then , where

Proof. Let be the vertex set of . By assumption, the size of the every independent set of vertices in is , because is a vertex-transitive graph and the size of every clique in graph is . Therefore, for any , there is exactly , such that . Hence, if , then two vertices and are adjacent in the complement of , so contains components such that , where is the complete graph of vertices. Therefore, . Hence, by Theorem 6, .

Proposition 8. Let be the Cayley graph on the cyclic group , where is an even integer and is the inverse-closed subset of ; 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 .
(a) If , then and . Therefore, two vertices and are adjacent in the complement of , also two vertices and are adjacent in the complement of . So contains components such that . Therefore ; hence we may assume , so and .
(b) If , then, by Lemma 3, it is sufficient to show that vertex-stabilizer is transitive on set for every and every , because is a vertex-transitive graph. In this case, let be the vertex set of and . Consider the vertex in , then , , and . Let be the group that is generated by all elements of sets and , say . It is clear that is a subgroup of , so the group is a subgroup of such that transitive on the set for each . Note that if , then, we can show that vertex-stabilizer is transitive on the set for each , because is a vertex-transitive graph.

Proposition 9. Let be the Cayley graph on the cyclic group , where is an even integer and is the inverse-closed subset of ; then is an integral graph.

Proof. By Remark 2, it is clear that is distance-regular, because is a distance-transitive graph. Let be the vertex set of . Consider the vertex in ; then , , and . Let be in such that ; then and ; hence and, by Definition 1, . 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 So the intersection array of is Therefore, by Theorem 4, the tridiagonal matrix

determines all the eigenvalues of . It is clear that all the eigenvalues of are , and their multiplicities are , respectively. So is an integral graph.

Conclusion 10. Let be the Cayley graph on the cyclic group as before with the adjacency matrix , and characteristic polynomial then is

Proof. It is easy to show that the adjacency matrix , where is matrix; hence, by Proposition 9 and Theorem 5,

Proposition 11. Let be the Cayley graph on the cyclic group as before with the adjacency matrix and characteristic polynomial . If and is an even integer, then

Proof. It is easy to show that the adjacency matrix , where is matrix; hence, by Conclusion 10 and Theorem 5,

Conclusion 12. Let be the Cayley graph on the cyclic group , where is an even integer, , and is the inverse-closed subset of . If and is an even integer, then the adjacency matrix spectrum of is , , , .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

C. Godsil and G. Royle, Algebraic Graph Theory, vol. 207 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2001.

J. D. Dixon and B. Mortimer, “Permutation groups,” Mathematical Proceedings of the Cambridge Philosophical Society, 1996.

N. Biggs, Algebraic graph theory, Cambridge Mathematical Library, Cambridge University Press, Cambridge, Second edition, 1993.

F. Harary and A. Schwenk, “Which graphs have integral spectra?” in Graphs and Combinatorics, vol. 406 of Lecture Notes in Mathematics, pp. 45–51, Springer, 1974.

K. Balinska, D. Cvetković, Z. Rodosavljević, S. Simić, and D. Stevanović, “A survey on integral graphs,” Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat, vol. 13, pp. 42–65, 2002.

O. Ahmadi, N. Alon, I. F. Blake, and I. E. Shparlinski, “Graphs with integral spectrum,” Linear Algebra and its Applications, vol. 430, no. 1, pp. 547–552, 2009.

A. Abdollahi and E. Vatandoost, “Which Cayley graphs are integral?” Electronic Journal of Combinatorics, vol. 16, no. 1, article R122, pp. 1–17, 2009.

A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs, vol. 18, Springer, Berlin, Germany, 1989.

J. R. Silvester, “Determinants of block matrices,” The Mathematical Gazette, vol. 84, pp. 460–467, 2000.

L. W. Beineke and R. J. Wilson, Eds., Topics in Algebraic Graph Theory, vol. 102 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 2004.