Affine primitive symmetric graphs of diameter two

Abstract

Let n be a positive integer, q be a prime power, and V be a vector space of dimension n over F_q. Let G := V rtimes G₀, where G₀ is an irreducible subgroup of GL(V) which is maximal by inclusion with respect to being intransitive on the set of nonzero vectors. We are interested in the class of all diameter two graphs Γ  that admit such a group G as an arc-transitive, vertex-quasiprimitive subgroup of automorphisms. In particular, we consider those graphs for which G₀ is a subgroup of either Γ L(n,q) or Γ Sp(n, q) and is maximal in one of the Aschbacher classes C_i, where i ∈ {2, 4, 5, 6, 7, 8}. We are able to determine all graphs Γ  which arise from G₀ ≤ Γ L(n, q) with i ∈ {2, 4, 8}, and from G₀ ≤ Γ Sp(n, q) with i ∈ {2, 8}. For the remaining classes we give necessary conditions in order for Γ  to have diameter two, and in some special subcases determine all G-symmetric diameter two graphs.