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Vertex-primitive graphs having vertices with almost equal neighbourhoods, and vertex-primitive graphs of valency 5

Banff International Research Station for Mathematical Innovation and Discovery

A graph is vertex-primitive if its automorphism group does not preserve any nontrivial partition of its vertex-set. It is an easy exercise to prove that (apart from some trivial exceptions) a vertex-primitive graph cannot have distinct vertices with equal neighbourhoods. I will discuss some results about vertex-primitive graphs having two vertices with “almost” equal neighbourhoods, and how these results were used to answer a question of Araújo and Cameron about synchronising permutation groups. These results were also the motivation for a recent classification of vertex-primitive graphs of valency 5. (Graphs of valency at most 4 had previously been classified.) I will describe this classification, some of the issues that arose in the proof, and the connection with the previous problem.

Mathematics; Group theory and generalizations; Topological groups, lie groups; Algebraic structures

video/mp4

Author affiliation: University of Western Australia

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61671

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/