On regular graphs equienergetic with their complements

Ricardo A. Podestá, Denis E. Videla

Published 2020-10-12Version 1

We give necessary and sufficient conditions on the parameters of a regular graph $\Gamma$ such that $E(\Gamma)=E(\bar \Gamma)$. Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs or $C_4$. Next, for the family of strongly regular graphs $\Gamma$ we characterize all possible parameters $srg(n,k,e,d)$ such that $E(\Gamma) = E(\bar \Gamma)$. 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 $C(2)$, $C(3)$ and $C(5)$ in Cameron's hierarchy. Finally, we consider unitary Cayley graphs over rings $G_R=X(R,R^*)$. We showed that if $R$ is a finite artinian ring with an even number of local factors, then $G_R$ is equienergetic with its complement if and only if $R=\mathbb{F}_{q} \times \mathbb{F}_{q'}$ is the product of 2 finite fields.