Integral equienergetic non-isospectral unitary Cayley graphs

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 $G_R=X(R,R^{\ast })$, 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.

Small typos corrected from v2 (warning: v2 is different from v1, see the comments in v2)