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)