Spectral properties of Cayley graphs over finite commutative rings

TL;DR

This paper investigates the spectral properties and energy of Cayley graphs over finite commutative rings, providing formulas, conditions for Ramanujan graphs, and insights into their complements.

Contribution

It introduces explicit spectral calculations and Ramanujan conditions for Cayley graphs over finite commutative rings, extending graph spectral theory in algebraic structures.

Findings

Calculated spectra and energies of Cayley graphs over finite rings

Derived conditions for Cayley graphs to be Ramanujan

Analyzed energies of complement graphs

Abstract

Let $R$ be a finite commutative ring with unity and $x$ be a non-zero element of $R$. In this paper, we calculate the spectrum and energy of the Cayley graph ${\rm Cay}(R,xR^{*})$, and also compute the energy of their compliment graph. Further, we give necessary and sufficient condition for Cayley graph ${\rm Cay}(R,xR^{*})$ to be Ramanujan.