Quadratic unitary Cayley graphs of finite commutative rings

The purpose of this paper is to study spectral properties of a family of Cayley graphs on finite commutative rings. Let R be such a ring and R× be its set of units. Let QR={u2:u∈R×} and TR=QR∪(−QR). We define the quadratic unitary Cayley graph of R, denoted by GR, to be the Cayley graph on the additive group of R with respect to TR; that is, GR has vertex set R such that x,y∈R are adjacent if and only if x−y∈TR. It is well known that any finite commutative ring R can be decomposed as R=R1×R2×⋯×Rs, where each Ri is a local ring with maximal ideal Mi. Let R0 be a local ring with maximal ideal M0 such that |R0|/|M0|≡3(mod4). We determine the spectra of GR and GR0×R under the condition that |Ri|/|Mi|≡1(mod4) for 1≤i≤s. We compute the energies and spectral moments of such quadratic unitary Cayley graphs, and determine when such a graph is hyperenergetic or Ramanujan.