Spectral properties of unitary Cayley graphs of finite commutative rings

TL;DR

This paper investigates the spectral properties of unitary Cayley graphs derived from finite commutative rings, providing conditions for Ramanujan properties, and analyzing the energy and spectral moments of related graphs.

Contribution

It establishes necessary and sufficient conditions for these graphs and their complements to be Ramanujan, and computes spectral characteristics of the graphs and their line graphs.

Findings

Characterization of Ramanujan conditions for $G_R$ and its complement

Determination of the energy of the line graph of $G_R$

Calculation of spectral moments of $G_R$ and its line graph

Abstract

Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set ${{a,b}:a,b\in R, a-b\in R^\times}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than $\pm r$ is at most $2\sqrt{r-1}$. In this paper we give a necessary and sufficient condition for $G_R$ to be Ramanujan, and a necessary and sufficient condition for the complement of $G_R$ to be Ramanujan. We also determine the energy of the line graph of $G_R$, and compute the spectral moments of $G_R$ and its line graph.