For q = p, the graph G^{∗}(q) is called the q^{th} Platonic graph. The name comes from the fact that when p = 3 or 5, the graphs G^{∗}(p) correspond to 1-skeletons of two of the Platonic solids, in other words, the tetrahedron and the icosahedron as in [2]. More generally, G^{∗}(p) is the 1-skeleton of a triangulation of the modular curve X(p). In this paper, we determine the spectrum of G^{∗}(q) and thus generalize results of P.E. Gunnells, “Some elementary Ramanujan graphs”.

*with *Dominic Lanphier and Marvin Minei, *Discrete Math. 307 (2007), no. 9-10, 1074–1081. *DOI*: *10.1016/j.disc.2006.07.032 **MR2292536 05C25 (05C50) **