Weinberg bounds over nonspherical graphs

Abstract

Let Aut(G) and E(G) denote the automorphism group and the edge set of a graph G, respectively. Weinberg's Theorem states that 4 is a constant sharp upper bound on the ratio \Aut(G)\/\E(G)\ over planar (or spherical) 3-connected graphs G. We have obtained various analogues of this theorem for nonspherical graphs, introducing two Weinberg-type bounds for an arbitrary closed surface Sigma, namely: W-P(Sigma) and W-T (Sigma) (=) (def) (G) (sup) \Aut(G)\/\E(G)\, where supremum is taken over the polyhedral graphs G with respect to C for W-P(Sigma) and over the graphs G triangulating Sigma for W-T (Sigma). We have proved that Weinberg bounds are finite for any surface; in particular: W-P = W-T = 48 for the projective plane, and W-T = 240 for the torus. We have also proved that the original Weinberg bound of 4 holds over the graphs G triangulating the projective plane with at least 8 vertices and, in general, for the graphs of sufficiently large order triangulating a fixed closed surface Sigma. (C) 2000 John Wiley & Sons, Inc.