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

Abstract

Let $\mathrm{\Gamma }=\left(V,E\right)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $\mathrm{\Gamma }$ are integers, then we say that $\mathrm{\Gamma }$ is an integral graph. A graph $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }=\text{Cay}\left({\mathbb{Z}}_n,S\right)$ , where $n=p^m$ ( $p$ is a prime integer and $m\in \mathrm{\mathbb{N}}$ ) and $S=\left\{a\in {\mathbb{Z}}_n|\left(a,n\right)=1\right\}$ . First, we show that $\mathrm{\Gamma }$ is an integral graph. Also, we determine the automorphism group of $\mathrm{\Gamma }$ . Moreover, we show that $\mathrm{\Gamma }$ and $K_v\mathrm{▽}\mathrm{\Gamma }$ are determined by their spectrum.