Bounding the number of automorphisms of a graph

Michael Giudici
The University of Western Australia



Content: 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 $|\mathrm{Aut}(\Gamma)|=n|G_v|$ and so the question becomes one about $|G_v|$. When $\Gamma$ is a cubic arc-transitive graph then Tutte showed that $|G_v|$ divides $48$. 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.

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