Affine primitive symmetric graphs of diameter two


  • Carmen Amarra University of the Philippines Diliman, Philippines
  • Michael Giudici The University of Western Australia, Australia
  • Cheryl E. Praeger The University of Western Australia, Australia



Symmetric graphs, Cayley graphs, quasiprimitive permutation groups, linear groups


Let n be a positive integer, q be a prime power, and V be a vector space of dimension n over Fq. Let := 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 Ci, 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.