On unitary Cayley graphs of matrix rings

Abstract

Let Fq be a finite field of order q and let R be a finite commutative local ring which is not a field. Recently, three (resp. four) distinct eigenvalues of the unitary Cayley graph CMn(Fq) (resp. CMn(R)) have been determined in Rattanakangwanwong and Meemark (2020) [20]. In this paper, completely explicit closed formulas for all the eigenvalues of CMn(Fq) and CMn(R) are obtained by using a new approach. As applications, the energy, the Kirchhoff index and the number of spanning trees of CMn(Fq) and CMn(R) are derived, respectively.