Algebraic Graph Theory in Rogla

This week I have been in Rogla, Slovenia for an Algebraic Graph Theory Summer School organised by Dragan Marušič’s group at the University of Primorska. It has consisted of six minicourses on Group actions, Association schemes, Graph covers, Magma, Hamiltonicity of vertex transitive graphs, and Maps. Slides from the courses are available at the conference webpage. The week has been a great success with the organisers doing a great job.

Cheryl, John and I delivered the one on Group actions and our slides can be found herehere, and here. John spoke first on some permutation group basics, Cheryl  then outlined the normal quotient method for edge-transitive graphs and the O’Nan-Scott Theorem, while I showed how it is all applied to 2-arc transitive and locally 2-arc transitive graphs.

Another event at the summer school was Gabriel Verret defending his PhD thesis. He was a student at the University of Ljubljana under Primož Potočnik. Gabriel has some very impressive results related to the Weiss Conjecture. The conjecture is

given a vertex-transitive graph for which the stabiliser of a vertex acts primitively on the set of neighbours, the order of the vertex stabiliser is bounded above by a function of the valency.

When the graph has valency three, Tutte showed that the order of the stabiliser is at most 48. Tony Gardiner has also proved the conjecture in the valency 4 case. The conjecture has also been proved by Trofimov in the 2-transitive case.

Gabriel calls a transitive permutation group L graph-restrictive, if there is a constant c(L) such that for any vertex-transitive graph \Gamma with group of automorphisms G such that the stabiliser in G of a vertex v induces L on the set of neighbours of v, then the order of G_v is bounded above by c(L). The Weiss conjecture is then that primitive groups are graph-restrictive.  Cheryl has also conjectured that quasiprimitive groups are graph-restrictive.

Gabriel had many results in his thesis on which groups are graph-restrictive. Subsequently, with Pablo Spiga and Primož Potočnik, he has conjectured that a transitive group is graph-restrictive if and only if it is semiprimitive. (A transitive group is semiprimitive is all normal subgroups are either transitive or semiregular.)  They have proved the  conjecture in one direction by showing that graph-restrictive implies semiprimitive.

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