A new family of 2-arc-transitive graphs

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 (v_0,v_1,\ldots,v_s) of vertices such that v_i is adjacent to v_{i+1} but v_{i+2}\neq v_i, that is, it is a path in the graph which cannot immediately go back upon itself. A graph \Gamma is called s-arc-transitive if \mathrm{Aut}(\Gamma) acts transitively on the set of s-arcs in \Gamma. 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 \Gamma is s-arc-transitive then it is also (s-1)-arc-transitive. In particular, \mathrm{Aut}(\Gamma) acts transitively on the set of arcs of \Gamma, that is the set of ordered pairs of adjacent vertices, and on the set of vertices.

The complete graph K_n, for n\geq 4 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 s\leq 5. This bound is met by the Tutte-Coxeter graph which is the point-line incidence graph of the generalised quadrangle W(3,2). With the aid of the classification of 2-transitive groups and hence using the classification of finite simple groups, Weiss[7] proved that for graphs with valency at least 3, we have s\leq 7. The bound is met by the point-line incidence graphs of the classical generalised hexagons associated with the groups G_2(q).

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