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We consider a special class of generalized Paley graphs over finite fields,namely the Cayley graphs with vertex set $\mathbb{F}_{q^m}$ and connection setthe nonzero $(q^\ell+1)$-th powers in $\mathbb{F}_{q^m}$, as well as theircomplements. We explicitly compute the spectrum and the energy of these graphs.As a consequence, the graphs turn out to be (with trivial exceptions) simple,connected, non-bipartite, integral and strongly regular, of pseudo or negativeLatin square type. By using the spectral information we compute severalinvariants of these graphs. We exhibit infinitely many pairs of equienergeticnon-isospectral graphs. As applications, on the one hand we solve Waring'sproblem over $\mathbb{F}_q^m$ for the exponents $q^\ell+1$, for each $q$ andfor infinitely many values of $\ell$ and $m$. We obtain that the Waring'snumber $g(q^\ell+1,q^m)=1$ or $2$, depending on $m$ and $\ell$, thus solvingsome open cases. On the other hand, we construct infinite towers of Ramanujangraphs in all characteristics. Finally, we give the Ihara zeta functions ofthese graphs.

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