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 Σ, namely: where supremum is taken over the polyhedral graphs G with respect to Σ for WP(Σ) and over the graphs G triangulating Σ for WT(Σ). We have proved that Weinberg bounds are finite for any surface; in particular: WP = WT = 48 for the projective plane, and WT = 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 Σ. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 220–236, 2000
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