# Classification of a family of arc-transitive graphs

*by Dr Sanming Zhou*

*Institution:*The University of Melbourne

*Date: Thu 9th August 2007*

*Time: 3:15 PM*

*Location: Elisabeth Murdoch, Theatre A, The University of Melbourne*

*Abstract*: For a positive integer s, an s-arc of a graph is a sequence of s+1 vertices such that two consecutive terms are adjacent and three consecutive terms are distinct. A graph X is called (G, s)-arc transitive, where G is a subgroup of the automorphism group of X, if G is transitive on the set of s-arcs of X. A 1-arc (i.e. an ordered pair of adjacent vertices) is usually called an arc, and a (G,1)-arc transitive graph is called a G-arc transitive graph. For example, the graphs of Platonic polyhedra are arc-transitive.

I will present a classification of a family of finite G-arc transitive graphs X such that G is imprimitive on the vertex set of X and the corresponding quotient graph is a complete (G, 2)-arc transitive graph. Of particular interest arising from this classification are two subfamilies of graphs which admit an arc-transitive action of a projective linear group. The graphs in these subfamilies can be defined in terms of the cross ratio of certain 4-tuples of elements of a finite projective line. Other interesting graphs from the classification include two sporadic arc-transitive graphs arising from each of the Mathieu groups M_(11) and M_(22). I will also mention a combinatorial construction of a class of arc-transitive graphs that motivated the classification above.

*For More Information:* Dr. Lawrence Reeves L.Reeves@ms.unimelb.edu.au