SEMINAR: Groups and Combinatorics Seminar: Cheryl Praeger, 4pm May 17 in Weatherburn LT

Speaker: Cheryl E Praeger (University of Western Australia)

Title: Limited geodesic transitivity for finite regular graphs

Time and place: 4pm Friday 17 May 2019, Weatherburn LT

Abstract: Joint work with Wei Jin.

For vertex transitive graphs, transitivity on t-arcs, t-geodesics, or distance t vertex pairs, for t leq s, all give symmetry measures of the graph in balls of radius s about a vertex. If the graph has girth g, and s leq g/2, then the sets of t-arcs and t-geodesics are the same for each t leq s, and so the conditions of s-arc transitivity and s-geodesic transitivity are equivalent. The next cases where s= (g+1)/2 and s=(g+2)/2 are interesting. There are s-geodesic transitive examples that are not s-arc transitive. Those which have s=2 and g=3 are collinearity graphs of point-line incidence geometries. However there is no nice general description for the cases where s= 3 and g is 4 or 5. Our approach has required us to classify, as a bye product, all 2-arc transitive strongly regular graphs, and to examine their normal covers. We have lots to describe, as well as open problems to pose.