Spectral properties of generalized Paley graphs

Articulo

Resumen *

We study the spectrum of generalized Paley graphs $G(k,q)=Cay(f_q,R_k)$, undirected or not, with $R_k={x^k:xin f_q^*}$ where $q=p^m$ with $p$ prime and $kmid q-1$. We first show that the eigenvalues of $G(k,q)$ are given by the Gaussian periods $eta_{i}^{(k,q)}$ with $0le ile k-1$. Then, we explicitly compute the spectrum of $G(k,q)$ with $1le k le 4$ and of $G(5,q)$ for $pequiv 1pmod 5$ and $5mid m$. Also, we characterize those GP-graphs having integral spectrum, showing that $G(k,q)$ 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 $G(k,q)$ with $1le k le 4$ or where $(k,q)$ is a semiprimitive pair. Información suministrada por el agente en SIGEVA