Spectral Properties of Unitary Cayley Graphs of Finite Commutative Rings

Let R be a finite commutative ring. The unitary Cayley graph of R, denoted GR, is the graph with vertex set R and edge set {{a, b} : a, b ∈ R, a− b ∈ R×}, where R× is the set of units of R. An r-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than ±r is at most 2 √ r − 1. In this paper we give a necessary and sufficient condition for GR to be Ramanujan, and a necessary and sufficient condition for the complement of GR to be Ramanujan. We also determine the energy of the line graph of GR, and compute the spectral moments of GR and its line graph.