Distance distribution and intersection numbers in finite Cayley graphs

Let G be a finite group and let C be a class of conjugate elements which generates G. Then we may define a graph structure on G by joining two vertices v and u in G by an edge if and only if v belongs to the set uC or u belongs to vC. This is the Cayley associated on G for generating set C.

There is an fascinating interplay between the group structure and the combinatorics of Cayley graphs associated to the group. For instance, how can one best determine the elements at distance r from a given element u, one might say the `sphere' of radius r around u, and how do such spheres intersect? When such questions are considered for families of groups interesting invariants appear and connections to combinatorial enumeration problems. In this project we want to study such problems for standard series of groups, such as symmetric groups, general linear groups and distinguished conjugacy classes such as Coxeter generators. Further information about this topic can be found in [1] and [2].

Minimum entry requirements: 2:1 in mathematics

This project is open to any applicants (home, EU or Overseas) who have their own funding.