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

For $q=p^m$ with $p$ prime and $k\mid q-1$, we consider the generalized Paley graph $\mathrm{\Gamma }(k,q)=Cay({\mathbb{F}}_q,R_k)$, with $R_k={ x^k : x \in $\mathbb{F}$q^* }$,andtheirreducible$p$-arycycliccode$$\mathcal{C}$(k,q) = {($\text{Tr}${q/p}(\gamma \omega^{ik}){i=0}^{n-1})}{\gamma \in $\mathbb{F}$_q}$,with$\omega$aprimitiveelementof$$\mathbb{F}$_q$and$n=\tfrac{q-1}{k}$\mathrm{.}Wefirstexpressthespectraof$\Gamma(k,q)$ in terms of Gaussian periods. Then, we show that the spectra of $\mathrm{\Gamma }(k,q)$ and $\mathcal{C}(k,q)$ are mutually determined by each other if further $k\mid \tfrac{q-1}{p-1}$\mathrm{.}Wegive$Spec(\Gamma(k,q))$ explicitly for those graphs associated with irreducible 2-weight cyclic codes in the semiprimitive and exceptional cases. We also compute $Spec(\mathrm{\Gamma }(3,q))$ and $Spec(\mathrm{\Gamma }(4,q))$.