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