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 $G_R=X(R,R^*)$, 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.