An Interesting Property of a Class of Circulant Graphs

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 matrixdetermines 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.