There appears to be much interest in the subject of the spectra of Cayley graphs. Indeed it appears that the spectra are highly related or may be computed through the irreducible representations of the underlying group in the graph, but I cannot seem to find a general formula for such a computation. For instance an old paper on the spectra of Cayley graphs http://www.sciencedirect.com/science/article/pii/0095895679900790 seems to compute the sum of eigenvalues.
In particular, I am interested in computing the largest and smallest eigenvalues of a particular Cayley graph of the symmetric group with a symmetric but not normal generating set. Is the above the only tool to do something like this? I am struggling to find anything more.