Unitary Cayley graphs of matrix rings over finite commutative rings

Abstract

Let R be a finite commutative ring and n a positive integer. In this paper, we study the unitary Cayley graph CMn(R) of the matrix ring over R. If F is a field, we use the additive characters of Mn(F) to determine three eigenvalues of CMn(F) and use them to analyze strong regularity and hyperenergetic graphs. We find conditions on R and n such that CMn(R) is strongly regular. Without explicitly having the spectrum of the graph, we can show that CMn(R) is hyperenergetic and characterize R and n such that CMn(R) is Ramanujan. Moreover, we compute the clique number, the chromatic number and the independence number of the graph.