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

Abstract

Let $Γ=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $Γ$ are integers, then we say that $Γ$ is an integral graph. A graph $Γ$ 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 $Γ=Cay(\mathbb{Z}_{n}, S)$, 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 $Γ$ is an integral graph. Also we determine the automorphism group of $Γ$. Moreover, we show that $Γ$ and $K_v \bigtriangledownΓ$ are determined by their spectrum.