Mathematics Events

Location

Jones Hall, Room 301
200 Ukrop Way
Williamsburg, VA 23185Map this location

ABSTRACT: Given a finite graph $\Gamma$ with $n$ vertices, can we obtain useful bounds on the number of automorphisms of $\Gamma$? If $\Gamma$ is vertex transitive then the Orbit-Stabiliser Theorem implies that $|\mathrm{Aut}(\Gamma)|=n|G_v|$, where $G_v$ is the stabiliser in $G$ of the vertex $v$. Thus the question becomes one about $|G_v|$. When $\Gamma$ is a arc-transitive graph of valency three then Tutte (1947, 1959) showed that $|G_v|$ divides $48$. This work has been the starting point for detailed studies about the automorphism groups of graphs of valency three and I will give a survey of some of this work. For higher valencies, Weiss conjectured that if the graph is locally primitive of valency $d$ then there is some function $f$ such that $|G_v|\leqslant f(d)$. Verret recently proposed that the correct way to view the result of Tutte and the conjecture of Weiss is in terms of graph-restrictive permutation groups. This subsequently lead to the Poto\v{c}nik-Spiga-Verret Conjecture. I will outline this viewpoint and discuss some recent results towards the conjecture.

Contact

Eric Swartz