For q = p, the graph G∗(q) is called the qth 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 . 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)