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

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=Cayℤn,S, where n=pm (p is a prime integer and m∈ℕ) and S=a∈ℤn|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 Kv▽Γ are determined by their spectrum.