Abstract

In this talk we will focus on groups acting on graphs with a good amount of symmetry, such as vertex transitivity. Several conjectures have connected global and local properties of the graphs in this class. In particular, “global” can refer to the number of automorphisms of a graph, and local then refers to certain conditions placed on the local action, that is, the action induced by a vertex-stabiliser on the neighbours of the vertex it fixes. A conjecture of Weiss from 1978 asserts that under mild conditions on this local action the number of automorphisms of a connected vertex-transitive graph should be bounded by a function of the valency. This conjecture is still very much open. I will report on recent progress on an expanded version of the conjecture which uses tools from group theory developed for the classification of the finite simple groups.