# Symmetries of graphs and networks

I have just spent the last week at the 2nd International worskshop on Symmetries of Graphs and Networks which was held in Rogla, Slovenia. The first conference in this series, which I also attended, was held in Banff in November 2008. Rogla is located on top of a mountain in north east Slovenia and is at about 1500m above sea level, thus keeping what seems to be a tradition of holding such conferences at altitude. Overall the conference went very well and the organisers should be congratulated for their great efforts.

As well as being a ski lodge in the winter, the hotel here is also used for training camps by many professional and junior sporting teams from Slovenia and nearby. In this spirit the conference included a Slovenia vs The Rest of the World basketball game (which the Slovenians won), a 6 a side soccer tournament and a 4x400m relay. I was the goalkeeper in Marston Conder’s soccer team which by one measure (of course the one that we preferred) won the tournament.

On mathematical matters the conference had 6 keynote talks. Ted Dobson spoke on Normal Cayley Graphs. These are Cayley graphs of a group with joining set such that the full automorphism group is where is the group of all automorphisms of that fix setwise. The talk included many interesting open problems and a few conjectures, including one that almost all Cayley graphs whose automorphism group is not as small as possible (that is, only or twice as large as ) are normal Cayley graphs.

Misha Klin gave an interesting talk on strongly regular graphs with no triangles. There are currently only 7 known examples and all are subgraphs of the Higman-Sims graph. The talk included a description of how Dale Mesner discovered the Higman-Sims graph in his PhD thesis in 1956 and proved its uniqueness in an unpublished manuscript in 1964, before Higman and Sims discovered it in 1968 along with the Higman-Sims sporadic simple group.

Marston Conder gave an interesting talk on regular abstract polytopes with few flags. He has proved that for rank at least 4 the usual simplicial complex is not the nontrivial regular polytope with the least number of flags and for at least 9 the smallest examples come from a new infinite family of polytopes.

The other keynote talks were by Sasha Ivanov on Majorana Theory; Paul Terwilliger who outlined his classification of tridiagonal pairs, and Cai Heng Li from Perth who spoke on edge-transitive graphs and outlined the global analysis of such graphs including locally quasiprimitive graphs.

I gave a talk on locally distance transitive graphs outlining the work I am doing with Alice Devillers, Cai Heng Li and Cheryl Praeger. The slides are available here. I understand that the slides of the other participants will be posted on the conference website.

What did Misha Klin say about SRGs with no triangles? Is anything more known about the problem… perhaps we should do a blog post describing it.

From what I gathered from the talk there was not much new stuff about the actual problem. Most of the talk was spent outlining Mesner’s construction of Higman-Sims and discussing material from Mesner’s thesis and a related manuscript.