Spectral properties of Cayley graphs over finite commutative rings

Abstract

Let $R$ be a finite commutative ring with unity and $x$ be a non-zero element of $R$. In this paper, we calculate the spectrum and energy of the Cayley graph $\mathrm{Cay}(R,x{R}^{\ast })$, and also compute the energy of their compliment graph. Further, we give necessary and sufficient condition for Cayley graph $\mathrm{Cay}(R,x{R}^{\ast })$ to be Ramanujan.