Cheeger constants of Platonic graphs

The Platonic graphs πn arise in several contexts, most simply as a quotient of certain Cayley graphs associated to the projective special linear groups. We show that when n = p is prime, πn can be viewed as a complete multigraph in which each vertex is itself a wheel on n + 1 vertices. We prove a similar structure theorem for the case of an arbitrary prime power. These theorems are then used to obtain new upper bounds on the Cheeger constants of these graphs. These results lead immediatley to similar results for Cayley graphs of the group PSL(2,Zn).