Generalized Paley graphs have real spectra exactly when undirected; directed versions with three eigenvalues are only oriented Paley graphs or their unions; weak Waring numbers reduce to classical Waring numbers over finite fields.
Which graphs have integral spectra? Graphs and Combin.\@, Proc.\@ Capital Conf.\@, Washington D.C.\@ 1973, Lect.\@ Notes Math.\@ 406 (1974) 45--51
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The nature of the spectrum of generalized Paley graphs and weak Waring numbers over finite fields
Generalized Paley graphs have real spectra exactly when undirected; directed versions with three eigenvalues are only oriented Paley graphs or their unions; weak Waring numbers reduce to classical Waring numbers over finite fields.