Quadratic unitary Cayley graphs of finite commutative rings

Abstract

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 $|R_0|/|M_0|\equiv 3(\mathrm{mod}4)$. We determine the spectra of ${\mathcal{G}}_R$ and ${\mathcal{G}}_{R_0\times R}$ under the condition that $|R_i|/|M_i|\equiv 1(\mathrm{mod}4)$ 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.

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