Last week I was at the 8th Slovenian Conference on Graph Theory. This was the latest in what is commonly known as the ‘Bled conference’ but this year was in Kranjska Gora. This meant that the conference excursion was to Lake Bled. It was a very enjoyable conference with lots of interesting talks and it was good to catch up with lots of people. I was one of the plenary speakers and my talk was entitled ‘Bounding the number of automorphisms of a graph’. This surveyed the recent work on the Weiss conjecture and its generalisation the PSV conjecture. It also discussed my recent work with Luke Morgan on the PSV conjecture for semiprimitive groups with a nilpotent regular normal subgroup. More details can be found on my slides.
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 here, here, 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 graph-restrictive, if there is a constant such that for any vertex-transitive graph with group of automorphisms such that the stabiliser in of a vertex induces on the set of neighbours of , then the order of is bounded above by . 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.