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 $R=R_1\times R_2\times\cdots\times R_s$,whereeach$R_i$isalocalringwithmaximalideal$M_i$\mathrm{.}Let$R_0$bealocalringwithmaximalideal$M_0$suchthat$|R_0|/|M_0| \equiv 3,(\mod,4)$\mathrm{.}Wedeterminethespectraof$$\mathcal{G}$R$and$$\mathcal{G}${R_0\times R}$undertheconditionthat$|R_i|/|M_i|\equiv 1,(\mod,4)$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.