Symmetric diameter two graphs with affine-type vertex-quasiprimitive automorphism group

Abstract

We study the class of G-symmetric graphs Γ with diameter 2, where G is an affine-type quasiprimitive group on the vertex set of Γ. These graphs arise from normal quotient analysis as basic graphs in the class of symmetric diameter 2 graphs. It is known that \({G \cong V \rtimes G_0}\) , where V is a finite-dimensional vector space over a finite field and G 0 is an irreducible subgroup of GL (V), and Γ is a Cayley graph on V. In particular, we consider the case where \({V = \mathbb {F}_p^d}\) for some prime p and G 0 is maximal in GL (d, p), with G 0 belonging to the Aschbacher classes \({\mathcal {C}_2, \mathcal {C}_4, \mathcal {C}_6, \mathcal {C}_7}\) , and \({\mathcal {C}_8}\) . For \({G_0 \in \mathcal {C}_i, i = 2,4,8}\) , we determine all diameter 2 graphs which arise. For \({G_0 \in \mathcal {C}_6, \mathcal {C}_7}\) we obtain necessary conditions for diameter 2, which restrict the number of unresolved cases to be investigated, and in some special cases determine all diameter 2 graphs.