Generalized Paley graphs equienergetic with their complements

Abstract

We consider generalized Paley graphs , generalized Paley sum graphs , and their corresponding complements and , for k = 3, 4. Denote by either or . We compute the spectra of and and from them we obtain the spectra of and also. Then we show that, in the non-semiprimitive case, the spectrum of and with p prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs and for any , respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of such that and are equienergetic for k = 3, 4. In a previous work we have classified all bipartite regular graphs and all strongly regular graphs which are complementary equienergetic, i.e. and are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs which are neither bipartite nor strongly regular.