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-x be its set of units. Let Q(R) = {u(2) : u is an element of R-x} and T-R = Q(R)U(-Q(R)). We define the quadratic unitary Cayley graph of R, denoted by G(R), to be the Cayley graph on the additive group of R with respect to T-R; that is, G(R) has vertex set R such that x, y is an element of R are adjacent if and only if x-y is an element of T-R. It is well known that any finite commutative ring R can be decomposed as R= R-1 x R-2 x ... x R-s, where each R-i is a local ring with maximal ideal M-i. Let R-0 be a local ring with maximal ideal M-0 such that vertical bar R-0 vertical bar / vertical bar M-0 vertical bar 3 (mod 4). We determine the spectra of G(R) and G(R0) x R under the condition that vertical bar R-i vertical bar/ vertical bar M-i vertical bar equivalent to (mod 4) 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. (C) 2015 Elsevier Inc. All rights reserved.