Symmetric diameter two graphs with affine-type vertex-quasiprimitive automorphism group
We study the class of G-symmetric graphs I" with diameter 2, where G is an affine-type quasiprimitive group on the vertex set of I". These graphs arise from normal quotient analysis as basic graphs in the class of symmetric diameter 2 graphs. It is known that , where V is a finite-dimensional vector space over a finite field and G (0) is an irreducible subgroup of GL (V), and I" is a Cayley graph on V. In particular, we consider the case where for some prime p and G (0) is maximal in GL (d, p), with G (0) belonging to the Aschbacher classes , and . For , we determine all diameter 2 graphs which arise. For 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.
cayley graphs symmetric graphs quasiprimitive permutation groups linear groups