Spectral properties of generalized Paley graphs

We study the spectrum of generalized Paley graphs $\mathrm{\Gamma }(k,q)=Cay({\mathbb{F}}_q,R_k)$, undirected or not, with $R_k=\{x^k:x\in {\mathbb{F}}_q^{\ast }\}$ where $q=p^m$ with $p$ prime and $k\mid q-1$. We first show that the eigenvalues of $\mathrm{\Gamma }(k,q)$ are given by the Gaussian periods ${\eta }_i^{(k,q)}$ with $0\le i\le k-1$. Then, we explicitly compute the spectrum of $\mathrm{\Gamma }(k,q)$ with $1\le k\le 4$ and of $\mathrm{\Gamma }(5,q)$ for $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}5)$ and $5\mid m$. Also, we characterize those GP-graphs having integral spectrum, showing that $\mathrm{\Gamma }(k,q)$ is integral if and only if $p$ divides $(q-1)\mathrm{/}(p-1)$. 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 $\mathrm{\Gamma }(k,q)$ with $1\le k\le 4$ or where $(k,q)$ is a semiprimitive pair.