Symmetric graphs of diameter two

A graph Γ is G-symmetric if it admits an arc-transitive subgroup G of automorphisms, and has diameter 2 if it is not a complete graph (that is, it has at least one pair of nonadjacent vertices) and if any two nonadjacent vertices have a common neighbour. Using normal quotient analysis, the study of G-symmetric diameter 2 graphs can be reduced to the following cases: (i) All nontrivial G-normal quotient graphs of Γ are complete graphs. (ii) All nontrivial normal subgroups of G act transitively on the vertex set of Γ. We consider in detail the pairs (Γ,G) that satisfy (i) where Γ may have diameter greater than two, as well as those that satisfy (ii) where Γ has diameter 2 and G is maximal in the symmetric group of the vertex set of Γ subject to being non-2-transitive. For the first case, we show that if Γ has at least three nontrivial normal quotients, then G corresponds to a finite transitive linear group H and Γ can be constructed from the natural vector space of H. We classify all connected graphs arising from groups H which are not subgroups of a one-dimensional affine group, and identify those which have diameter greater than two. For the second case, the group G is given by C. E. Praeger's classification of quasiprimitive permutation groups, and we focus on the subcase where G is of affine type. Such groups G correspond to irreducible subgroups of the general linear group which, in turn, have been classified by M. Aschbacher. Moreover, a uniform construction for Γ is known, so it only remains to determine which graphs have diameter 2. Using a case-by-case analysis, we are able to classify all diameter 2 graphs for some of the Aschbacher classes; in the others we determine bounds on certain parameters in order to have diameter 2, which reduce the number of unresolved cases.