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 $R^\times$ its set of units. Let $Q_R=\{u^2: u\in R^\times\}$ and $T_R=Q_R\cup(-Q_R)$. We define the quadratic unitary Cayley graph of $R$, denoted by $\mathcal{G}_R$, to be the Cayley graph on the additive group of $R$ with respect to $T_R$; that is, $\mathcal{G}_R$ has vertex set $R$ such that $x, y \in R$ are adjacent if and only if $x-y\in T_R$. It is well known that any finite commutative ring $R$ can be decomposed as $R=R_1\times R_2\times\cdots\times R_s$, 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 $|R_0|/|M_0| \equiv 3\,(\mod\,4)$. We determine the spectra of $\mathcal{G}_R$ and $\mathcal{G}_{R_0\times R}$ under the condition that $|R_i|/|M_i|\equiv 1\,(\mod\,4)$ for $1 \le i \le s$. We compute the energies and spectral moments of such quadratic unitary Cayley graphs, and determine when such a graph is hyperenergetic or Ramanujan.