The spectra of generalized Paley graphs of $q^\ell+1$-th powers and applications

Abstract

We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set Fqm and connection set the nonzero (q ℓ + 1)-th powers in Fqm , as well as their complements. We explicitly compute the spectrum of these graphs. As a consequence, the graphs turn out to be (with trivial exceptions) simple, connected, non-bipartite, integral and strongly regular (of Latin square type in half of the cases). As applications, on the one hand we solve Waring's problem over Fqm for the exponents q ℓ +1, for each q and for infinite values of ℓ and m. We obtain that the Waring's number g(q ℓ + 1, q m ) = 1 or 2, depending on m and ℓ, thus tackling some open cases. On the other hand, we construct infinite towers of Ramanujan graphs in all characteristics.

Spectral properties of generalized Paley graphs and their associated irreducible cyclic codes

“…k ) then we have the disjoint union Γ(k, q) = Γ (1) (k, q) ∪ • • • ∪ Γ (n−1) (k, q) where Γ (j) (k, q) ≃ Γ(k, q) for every 1 ≤ j ≤ n − 1 (same proof as in Lemma 4.2 in [14]).…”

“…There are few cases of known spectrum of GP-graphs. For instance unitary Cayley graphs over rings X(R, R * ), where R is a finite abelian ring and R * is the group of units (see [1], this includes the cases X(Z n , Z * n )) and X(F q m , S ℓ ) with S ℓ = {x q ℓ +1 : x ∈ F * q m } where ℓ | m (see [14], this includes the classical Paley graphs P (q)). We will compute Spec(Γ(k, q)), which includes X(F q m , S ℓ ) since S ℓ = R q ℓ +1 .…”

“…The Ihara zeta function ζ Γ (u) for a regular graph Γ has a determinantial expression in spectral terms. For a GP-graph Γ(k, q), we have (see (8.3) in [14] for details)…”

The Waring's problem over finite fields through generalized Paley graphs

“…(a) If 2 ≤ k < 4 √ q + 1 then g(k, q) = 2 (1977, [17]). [13], for k = p ℓ + 1; 2018, [15], for k = p ℓ + 1). (c) Let p, r be primes such that p is a primitive root modulo r m for some m ∈ N. Then (2.1)…”

A reduction formula for Waring numbers through generalized Paley graphs

“…(c) There are some complicated conditions, depending on the p-adic weight of k and the field size q, assuring g(k, q) = 2 (Moreno-Castro, 2005, see Theorem 3.3 and Corollary 3.4 in [9]). (d) g(k, p 2ℓs ) = 2 for k | p ℓ + 1 and s = 1 (Moreno-Castro, 2008, [10], for k = p ℓ + 1; Podestá-Videla, 2018, [13], for k = p ℓ + 1).…”