Alice Devillers, Cai Heng Li and Cheryl Praeger and I submitted a paper last week entitled `An infinite family of biquasiprimitive 2-arc transitive cubic graphs‘. In this post I will outline the program into which the paper fits. Some of this was covered in my 33ACCMCC plenary talk.
A s-arc in a graph is an (s+1)-tuple of vertices such that is adjacent to but , that is, it is a path in the graph which cannot immediately go back upon itself. A graph is called s-arc-transitive if acts transitively on the set of s-arcs in . If each vertex has valency at least two then every vertex has an s-arc starting at it and every (s-1)-arc can be extended to an s-arc. Hence if is s-arc-transitive then it is also (s-1)-arc-transitive. In particular, acts transitively on the set of arcs of , that is the set of ordered pairs of adjacent vertices, and on the set of vertices.
The complete graph , for is 2-arc-transitive. However, it is not 3-arc-transitive as it contains two types of 3-arcs: those whose first vertex is equal to its last those and those for which the first vertex is distinct from the last vertex. A cycle is s-arc-transitive for all s.
The study of s-arc-transitive graphs was begun by Tutte [5,6], who showed that for cubic graphs (that is those of valency 3), we have . This bound is met by the Tutte-Coxeter graph which is the point-line incidence graph of the generalised quadrangle . With the aid of the classification of 2-transitive groups and hence using the classification of finite simple groups, Weiss proved that for graphs with valency at least 3, we have . The bound is met by the point-line incidence graphs of the classical generalised hexagons associated with the groups .