Spectral properties of generalized Paley graphs of (q^{\ell}+1)-th powers and applications

PODESTÁ, RICARDO A.; DENIS E. VIDELA

Discrete Mathematics, Algorithms and Applications

World Scientific Publishing Co. Pte Ltd

Año: 2023

1793-8309

We consider a special class of generalized Paley graphs over finite fields, namelythe Cayley graphs with vertex set Fqm and connection set the nonzero (qℓ + 1)-th powers inFqm, as well as their complements. We explicitly compute the spectrum and the energy of thesegraphs. 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. By usingthe spectral information we compute several invariants of these graphs. We exhibit infinitelymany pairs of equienergetic non-isospectral graphs. As applications, on the one hand we solveWaring’s problem over Fqm for the exponents qℓ + 1, for each q and for infinitely many valuesof ℓ and m. We obtain that the Waring’s number g(qℓ + 1, qm) = 1 or 2, depending on m andℓ, thus solving some open cases. On the other hand, we construct infinite towers of Ramanujangraphs in all characteristics. Finally, we give the Ihara zeta functions of these graphs.