  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 set
the nonzero $(q^\ell+1)$-th powers in $\mathbb{F}_{q^m}$, as well as their
complements. 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 negative
Latin square type. By using the spectral information we compute several
invariants of these graphs. We exhibit infinitely many pairs of equienergetic
non-isospectral graphs. As applications, on the one hand we solve Waring's
problem over $\mathbb{F}_q^m$ for the exponents $q^\ell+1$, for each $q$ and
for infinitely many values of $\ell$ and $m$. We obtain that the Waring's
number $g(q^\ell+1,q^m)=1$ or $2$, depending on $m$ and $\ell$, thus solving
some open cases. On the other hand, we construct infinite towers of Ramanujan
graphs in all characteristics. Finally, we give the Ihara zeta functions of
these graphs.
