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If you are looking for copies of some of my recent talks go here.

Locally s-arc transitive graphs

A graph is s-arc transitive if its automorphism group is transitive on the set of s-arcs in the graph. Such graphs have a high degree of symmetry. Their study goes back to Tutte (1947,1959) who showed that for graphs of valency three, s<6. For graphs with arbitrary valency, Weiss (1981) showed that s<8. Amazingly, Tutte's methods were quite ingenious and involved elementary arguments whereas Weiss required the classification of finite simple groups.

A graph is locally s-arc transitive if for every vertex, the vertex stabiliser is transitive on the set of s-arcs emerging from that vertex. If the graph is vertex transitive then it is also s-arc transitive and so s<8. However, if the graph is not vertex transitive then it is bipartite and the classical generalised 8-gons are examples where s=9. This is despite the fact that the generalised 8-gons are not regular!

I have been involved in a global action analysis of locally s-arc transitive graphs with Cai Heng Li and Cheryl Praeger which has allowed us to obtain certain classifications and to find some new examples with large values of s.

In a project with Alice Devillers, Cai Heng and Cheryl we have begun to look at locally s-distance transitive graphs. These are graphs where the stabiliser of a vertex v is transitive on all vertices at distance s from v.

Transitive decompositions of graphs

A transitive decomposition of a graph is a partition of the arc set such that there is a group automorpisms of the graph which transitively permutes the parts. A homogeneous factorisation is a transitive decomposition where the kernel of the action on the partition is vertex transitive. This concept was first introduced for complete graphs by Li and Praeger to generalise the notion of vertex transitive self-complementary graphs, which have been widely studied. Homogeneous factorisations are related to many different objects, such as partial linear spaces, locally s-arc transitive graphs and the exceptionality of primitive permutation groups. This has been a joint project with Li, Primoz Potocnik and Praeger.

S3-involution graphs

When investigating transitive decompositions of Johnson graphs (see above) we discovered an interesting tower of graphs involving the group inclusions A5 < PSL(2,11) < M11 < M12. The graph related to A5 is the line graph of the Petersen graph while the graph related to M12 is the Johnson graph J(12,4). The two middle graphs have nice geometric descriptions related to Witt designs. The graphs can be uniformly defined as having vertex set a conjugacy class of involutions in the corresponding group and two involutions are adjacent if they generate an S3-subgroup in a particular conjugacy class. I begun a general investigation of such graphs with Alice Devillers.

Fixed point free elements of prime order

It follows from the Orbit-Counting Lemma that every transitive permutation group has a fixed point free element. Fein, Kantor and Schacher (1981) proved that every transitive permutation group actually has a fixed point free element of prime power order. However, not every transitive permutation group has a fixed point free element of prime order and such groups are called elusive. The polycirculant conjecture (due to Marusic, Jordan and Klin) states that every 2-closed transitive permutation group has a fixed point free element of prime order.

I got involved in this problem as part of my PhD which was under the supervision of Peter Cameron in the School of Mathematical Sciences at Queen Mary, University of London. My thesis was entitled `Fixed point free elements of prime order in permutation groups' and a .ps version can be found here. My main result in this area was to show that the polycirculant conjecture holds for all permutation groups with a transitive minimal normal subgroup. (Such groups have been called innately transitive by John Bamberg.) I have also found many new elusive groups.

Limits of vertex transitive graphs

We say that an infinite sequence of connected vertex-transitive graphs of finite valency converges to a graph &Gamma if, as we progress along the sequence, the graphs agree with &Gamma on balls of increasing radius. Studying limits of sequences of finite vertex-primitive graphs allows us to obtain information about vertex primitive graphs. This was a joint project with Cai Heng Li, Cheryl Praeger, Ákos Seress and Vladimir Trofimov.

Other research

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Last altered September 2009.