14.73 (1999)

Conjecture: There is a function $f$ on the natural numbers such that, if $\Gamma$ is a finite, vertex-transitive, locally-quasiprimitive graph of valency $v$, then the number of automorphisms fixing a given vertex is at most $f(v)$. (By definition, a vertex-transitive graph $\Gamma$ is locally-quasiprimitive if the stabilizer in $\operatorname{Aut}(\Gamma)$ of a vertex $\alpha$ is quasiprimitive (see 14.46) in its action on the set of vertices adjacent to $\alpha$.)

To prove the conjecture above one need only consider the case where $\operatorname{Aut}(\Gamma)$ has the property that every non-trivial normal subgroup has at most two orbits on vertices (C. E. Praeger, Ars Combin., 19 A (1985), 149–163). The analogous conjecture for finite, vertex-transitive, locally-primitive graphs was made by R. Weiss in 1978 and is still open. For non-bipartite graphs, there is a “reduction” of Weiss’ conjecture to the case where the automorphism group is almost simple (see 14.46 for definition) (M. Conder, C. H. Li, C. E. Praeger, Proc. Edinburgh Math. Soc. (2), 43, no. 1 (2000), 129–138).