English

Affine primitive symmetric graphs of diameter two

Combinatorics 2016-01-29 v1

Abstract

Let nn be a positive integer, qq be a prime power, and VV be a vector space of dimension nn over Fq\mathbb{F}_q. Let G:=VG0G := V \rtimes G_0, where G0G_0 is an irreducible subgroup of GL(V){\rm 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\Gamma that admit such a group GG as an arc-transitive, vertex-quasiprimitive subgroup of automorphisms. In particular, we consider those graphs for which G0G_0 is a subgroup of either ΓL(n,q){\rm \Gamma L}(n,q) or ΓSp(n,q){\rm \Gamma Sp}(n,q) and is maximal in one of the Aschbacher classes Ci\mathcal{C}_i, where i{2,4,5,6,7,8}i \in \{2,4,5,6,7,8\}. We are able to determine all graphs Γ\Gamma which arise from G0ΓL(n,q)G_0 \leq {\rm \Gamma L}(n,q) with i{2,4,8}i \in \{2,4,8\}, and from G0ΓSp(n,q)G_0 \leq {\rm \Gamma Sp}(n,q) with i{2,8}i \in \{2,8\}. For the remaining classes we give necessary conditions in order for Γ\Gamma to have diameter two, and in some special subcases determine all GG-symmetric diameter two graphs.

Keywords

Cite

@article{arxiv.1601.07663,
  title  = {Affine primitive symmetric graphs of diameter two},
  author = {Carmen Amarra and Michael Giudici and Cheryl E. Praeger},
  journal= {arXiv preprint arXiv:1601.07663},
  year   = {2016}
}
R2 v1 2026-06-22T12:38:21.957Z