On regular graphs equienergetic with their complements

We give necessary and sufficient conditions on the parameters of a regular graph $\Gamma$ (with or without loops) such that $E(\Gamma)=E(\overline \Gamma)$. 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 ($\Gamma = K_3 \square K_2$ or $Q_3$). Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs $Cr(n)$ or $C_4$. Next, for the family of strongly regular graphs $\Gamma$ we characterize all possible parameters $srg(n,k,e,d)$ such that $E(\Gamma) = E(\overline \Gamma)$. 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 $\mathcal{C}(2)$, $\mathcal{C}(3)$ and $\mathcal{C}(5)$ in Cameron's hierarchy (the cases $\mathcal{C}(1)$ and $\mathcal{C}(4)$ are still open). Finally, we consider unitary Cayley graphs over rings $G_R=X(R,R^*)$. We show that if $R$ is a finite Artinian ring with an even number of local factors, then $G_R$ is complementary equienergetic if and only if $R=\mathbb{F}_q \times \mathbb{F}_{q'}$ is the product of 2 finite fields.