Integral equienergetic non-isospectral unitary Cayley graphs

TL;DR

This paper investigates the spectral energy properties of certain Cayley graphs derived from finite rings, establishing conditions for their equienergetic and non-isospectral nature, and characterizing when these graphs are Ramanujan.

Contribution

It introduces new classes of integral equienergetic non-isospectral Cayley graphs from finite rings and characterizes when they are Ramanujan.

Findings

Cayley graphs $X(G,S)$ and $X^+(G,S)$ are equienergetic for abelian groups.

Families of unitary Cayley graphs $G_R$ are shown to be integral, equienergetic, and non-isospectral.

Conditions are provided for these graphs to be Ramanujan.

Abstract

We prove that the Cayley graphs $X(G,S)$ and $X^+(G,S)$ are equienergetic for any abelian group $G$ and any symmetric subset $S$. We then focus on the family of unitary Cayley graphs $G_R=X(R,R^*)$, where $R$ is a finite commutative ring with identity. We show that under mild conditions, $\{G_R, G_R^+\}$ are pairs of integral equienergetic non-isospectral graphs (generically connected and non-bipartite). Then, we obtain conditions such that $\{G_R, \bar G_R\}$ are equienergetic non-isospectral graphs. Finally, we characterize all integral equienergetic non-isospectral triples $\{G_R, G_R^+, \bar G_R \}$ such that all the graphs are also Ramanujan.