Weinberg bounds over nonspherical graphs

Let   G Aut 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     G E G Aut 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  , namely:         G E G W W G T P Aut sup and def    , where supremum is taken over the polyhedral graphs  G with respect to   for  WP    and over the graphs  G triangulating   for  WT   . 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  .