Spectral properties of generalized Paley graphs

We study the spectrum of generalized Paley graphs Gamma(k, q) = Cay(F-q, R-k), undirected or not, with R-k = {x(k) : x is an element of F-q(& lowast;)} where q = p(m) with p prime and k |q - 1. We first show that the eigenvalues of Gamma(k, q) are given by the Gaussian periods eta((k,q))( i) with 0 <= i <= k - 1. Then, we explicitly compute the spectrum of Gamma(k, q) with 1 <= k <= 4 and of Gamma(5, q) for p equivalent to 1 (mod 5) and 5 m. Also, we characterize those GP-graphs having integral spectrum, showing that Gamma(k, q) is integral if and only if p divides (q-1)/(p-1). Next, we focus on the family of semiprimitive GP-graphs. We show that they are integral strongly regular graphs (of pseudo-Latin square type). Finally, we characterize all integral Ramanujan graphs Gamma(k, q) with 1 <= k <= 4 or where (k, q) is a semiprimitive pair.