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 (Formula presented.). 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 ((Formula presented.) or (Formula presented.)). Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs (Formula presented.) or (Formula presented.). Next, for the family of strongly regular graphs Γ we characterize all possible parameters (Formula presented.) such that (Formula presented.). 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 (Formula presented.), (Formula presented.) and (Formula presented.) in Cameron's hierarchy (the cases (Formula presented.) in the non-bipartite case and (Formula presented.) are still open). Finally, we consider unitary Cayley graphs over rings (Formula presented.). We show that if R is a finite Artinian ring with an even number of local factors, then (Formula presented.) is complementary equienergetic if and only if (Formula presented.) is the product of 2 finite fields.