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 GR=X(R,R⁎), where R is a finite commutative ring with identity. We show that under mild conditions, {GR,GR+} are pairs of integral equienergetic non-isospectral graphs (generically connected and non-bipartite). Then, we obtain conditions such that {GR,G¯R} are equienergetic non-isospectral graphs. Finally, we characterize all integral equienergetic non-isospectral triples {GR,GR+,G¯R} such that all the graphs are also Ramanujan.