Spectral properties of unitary Cayley graphs of finite commutative rings (Q1953318)

Summary: Let \(R\) be a finite commutative ring. The unitary Cayley graph of \(R\), denoted \(G_R\), is the graph with vertex set \(R\) and edge set \(\left\{\{a,b\}:a,b\in R, a-b\in R^\times\right\}\), where \(R^\times\) is the set of units of \(R\). An \(r\)-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than \(\pm r\) is at most \(2\sqrt{r-1}\). In this paper we give a necessary and sufficient condition for \(G_R\) to be Ramanujan, and a necessary and sufficient condition for the complement of \(G_R\) to be Ramanujan. We also determine the energy of the line graph of \(G_R\), and compute the spectral moments of \(G_R\) and its line graph.