Algebraic methods in graph theory and finite geometries

The program involves a happy interplay of three areas of mathematics: first, group actions on combinatorial objects (semiregular elements of permutation groups, lifts of automorphisms, various transitivity conditions on graphs), second, a study of purely graph-theoretic concepts (distance -transitivity, hamiltonicity), and third, structural properties of certain objects in finite geometries. Much of this program will be devoted to half-arc-transitive group actions on graphs, in particular, graphs of valency 4. Our main interest lies in the study of structural properties and classification problems for such graphs. Getting an understanding of the structure of the corresponding vertex stabilizers will be one of our goals in that respect. Furthermore, 2-arc-transitive Cayley graphs of abelian and dihedral groups will also be dealt with. The open problem of existence of semiregular elements in 2-closed transitive permutation groups will also be touched upon. Last but not least, distance-regular graphs, hamiltonian properties of graphs and arcs in projective planes are also going to be considered. A monograph on transiitve group actions on graphs is a long-term goal of the research group involved in this program.