Some algebraic properties of a class of integral graphs determined by their spectrum

Let $\mathrm{\Gamma }=(V,E)$ 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 $\mathrm{\Gamma }=Cay({\mathbb{Z}}_n,S)$, where $n=p^m$, ($p$ is a prime integer, $m\in \mathbb{N}$) and $S={{a}\in$\mathbb{Z}$_{n},|,, (a, n)=1}$\mathrm{.}First,weshowthat$\Gamma$ is an integral graph. Also we determine the automorphism group of $\mathrm{\Gamma }$. Moreover, we show that $\mathrm{\Gamma }$ and $K_v▽\mathrm{\Gamma }$ are determined by their spectrum.