Algebraic Graph Theory Seminar - Gabriel Verret

Title: Vertex-primitive digraphs having vertices with almost equal
neighbourhoods

Abstract:

A permutation group G on V is transitive if for every x, y
in V there exists g in G mapping x to y. The group G is called
primitive if, in addition, it preserves no nontrivial partition of V.
Let X be a vertex-primitive digraph, that is, its automorphism group
acts primitively on its vertex-set. It is not hard to see that, in
this case, X cannot have two distinct vertices with equal
neighbourhoods, unless X is in some sense trivial. I will discuss some
recent results about the case when X has two vertices with “almost”
equal neighbourhoods, and how these results were used to answer a
question of Araújo and Cameron about synchronising groups. (This is
joint work with Pablo Spiga.)