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

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 = V × G0, 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 = Fd p for some prime p and G 0 is maximal in GL (d, p), with G 0 belonging to the Aschbacher classes C2, C4, C6, C7 and C8. For G0 ε C6, C 7, we determine all diameter 2 graphs which arise. For G0 ε C6 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. © 2012 Springer Science+Business Media, LLC.