Quadratic unitary Cayley graphs of finite commutative rings

TL;DR

This paper investigates the spectral properties of quadratic unitary Cayley graphs constructed from finite commutative rings, providing explicit spectra, energy, and conditions for special graph properties like hyperenergetic and Ramanujan.

Contribution

It determines the spectra and spectral characteristics of quadratic unitary Cayley graphs for finite commutative rings, extending understanding of their spectral graph theory.

Findings

Spectra of $\

Energy and spectral moments computed for these graphs

Conditions identified for hyperenergetic and Ramanujan properties

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 $|R_0|/|M_0| \equiv 3\,(\mod\,4)$. 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.