On regular graphs equienergetic with their complements

Abstract

We give necessary and sufficient conditions on the parameters of a regular graph Γ (with or without loops) such that . We study complementary equienergetic cubic graphs obtaining classifications up to isomorphisms for connected cubic graphs with single loops (5 non-isospectral pairs) and connected integral cubic graphs without loops ( or ). Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs or . Next, for the family of strongly regular graphs Γ we characterize all possible parameters such that . Furthermore, using this, we prove that a strongly regular graph is equienergetic to its complement if and only if it is either a conference graph or else it is a pseudo Latin square graph (i.e. has OA parameters). We also characterize all complementary equienergetic pairs of graphs of type , and in Cameron's hierarchy (the cases in the non-bipartite case and are still open). Finally, we consider unitary Cayley graphs over rings . We show that if R is a finite Artinian ring with an even number of local factors, then is complementary equienergetic if and only if is the product of 2 finite fields.