Orbital graphs of primitive permutation groups

The project aims to study certain families of graphs associated with primitive permutation groups, which are the basic building blocks for all groups of permutations. If G is a group acting transitively on a set X, then each orbit of X on the Cartesian product X x X can be regarded as the set of edges of a graph with vertex set X, on which G acts edge-transitively. These are called the orbital graphs associated with G. A well-known criterion of D Higman states that the orbital graphs are all connected if and only if the action of G on X is primitive. There has been recent work on classifying infinite classes of finite primitive groups for which the diameters of all the orbital graphs are bounded by some fixed constant. This work was originally motivated by questions in the area of model theory, a branch of mathematical logic - one reason being that the ultraproduct of the groups in such a bounded class is a well-defined object with many interesting properties. The classification just mentioned is somewhat qualitative - for example, given an explicit bound d, it does not give explicit families of groups for which the orbital diameters are bounded by d. Such explicit results would be extremely interesting, and would give rise to new families of closely related edge-transitive graphs of small diameters. Even for d = 2 or 3, very few such families are known. This project aims to find and classify such families explicitly, and investigate some of their basic properties. It is a new direction at the interface between group theory and algebraic combinatorics, arising from applications in mathematical logic. The planned methodology will be largely algebraic, a mixture of group theory (mainly permutation groups and finite and algebraic simple groups), together with algebraic graph theory. It is aligned to the EPSRC Research Area of Algebra within the Strategic Theme of the Mathematical Sciences.