Spectral properties of generalized Paley graphs of (qℓ + 1)th powers and applications

We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set Formula: see text and connection set the nonzero Formula: see textth powers in Formula: see text, as well as their complements. We explicitly compute the spectrum and the energy of these graphs. As a consequence, the graphs turn out to be (with trivial exceptions) simple, connected, non-bipartite, integral and strongly regular, of pseudo or negative Latin square type. Using the spectral information we compute several invariants of these graphs. We exhibit infinitely many pairs of integral equienergetic non-isospectral graphs. As applications, on the one hand we solve Waring’s problem over Formula: see text for the exponents Formula: see text, for each Formula: see text and for infinitely many values of Formula: see text and Formula: see text. We obtain that the Waring number Formula: see text or Formula: see text, depending on Formula: see text and Formula: see text, thus solving some open cases. On the other hand, we construct infinite towers of integral Ramanujan graphs in all characteristics. Finally, we give the Ihara zeta functions of these graphs.