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].

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

Topics in Algebraic Graph Theory, Encyclopedia of Mathematics and its Applications, Vol 102, 2004, Cambridge University Press

V. Levenshtein, J. Siemons, Error graphs and the reconstruction of elements in groups, Journal of Combinatorial Theory Ser. A, 116 (2009) 795-815