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 $Q_R={u^2: uin R^times}$ and $T_R=Q_Rcup(-Q_R)$. We define the quadratic un
itary 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 $x-yin T_R$. It is well known that any finite commutative ring $R$ can be decomposed as $R=R_1times R_2timescdotstimes R_s$, 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 $mathcal{G}_{R_0times R}$ 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.
