On Finite Affine 2-Arc Transitive Graphs

A 2-arc in a graph Γ is a sequence (α, β, γ) of three vertices of Γ such that {α, β} and {β, γ} are edges of Γ and α ≠ γ. A graph Γ is said to be 2-arc transitive if its automorphism group is transitive on the set of 2-arcs of Γ. Furhtermore, a graph Γ is said to be affine if there is a vector space N, and a subgroup G of the automorphism group of Γ, such that N ⩽ G ⩽ AGL(N) (where AGL(N) is the group of all affine transformations of N, and N is identified with the subgroup of translations) with N acting regularly on the vertex set of Γ and G acting 2-arc-transitively on Γ. This paper gives a classification of all primitive affine 2-arc transitive graphs, and all finite ʹbi-primitiveʹ affine 2-arc transitive graphs (that is, affine bipartite 2-arc transitive graphs such that the stabilizer of the bipartition of the vertices is primitive on each part of the bipartition).