Just to keep the references close to the question, here are two computations of the spectra of a Cayley graph:

* Lovász, L. Spectra of graphs with transitive groups. Period. Math. Hungar.  6  (1975), no. 2, 191--195. [MR0398886](https://www.ams.org/mathscinet-getitem?mr=MR0398886)

* Babai, L. Spectra of Cayley graphs. J. Combin. Theory Ser. B  27  (1979), no. 2, 180--189. [MR0546860](https://www.ams.org/mathscinet-getitem?mr=MR0546860)

(The prevalence of people named *László* in this list is interesting. It reminds me of a little story I recently got from Wikipedia while hunting for a reference on the Higman-Sims group: the extraordinary fact that two people named 'Higman' discovered the same sporadic simple group!)