Energy of Cayley graphs for alternating groups

Abstract

Let G be a finite group and S be a subset of G where S does not include the identity of G and is inverse closed. A Cayley graph of a finite group G with respect to the subset S is a graph where the vertices are the elements of G and two vertices a and b in G are adjacent if ab−1 are in the set S. For a simple graph, the energy of a graph can be determined by its eigenvalues. Let Γ be a simple graph. Then by the summation of the absolute values of the eigenvalues of the adjacency matrix of the graph, its energy can be determined. This paper presents the Cayley graphs of alternating groups with respect to the subset S of valency 1 and 2. From the Cayley graphs, the eigenvalues are computed by using some properties of special graphs and then used to compute their energy.