## Abstract

Let n be a positive integer, q be a prime power, and V be a vector space of dimension n over Fq. Let G := V o 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 􀀀 that admit such a group G as an arctransitive, vertex-quasiprimitive subgroup of automorphisms. In particular, we consider those graphs for which G0 is a subgroup of either GammaL(n; q) or GammaSp(n; q) and is maximal in

one of the Aschbacher classes Ci, where i in {2; 4; 5; 6; 7; 8}. We are able to determine all graphs Gamma which arise from G0 in GammaL(n; q) with i in { 2; 4; 8}, and from G0 in GammaSp(n; q) with i in { 2; 8 }. For the remaining classes we give necessary conditions in order for Gamma to have diameter two, and in some special subcases determine all G-symmetric diameter two graphs.

one of the Aschbacher classes Ci, where i in {2; 4; 5; 6; 7; 8}. We are able to determine all graphs Gamma which arise from G0 in GammaL(n; q) with i in { 2; 4; 8}, and from G0 in GammaSp(n; q) with i in { 2; 8 }. For the remaining classes we give necessary conditions in order for Gamma to have diameter two, and in some special subcases determine all G-symmetric diameter two graphs.

Original language | English |
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Pages (from-to) | 137-165 |

Number of pages | 29 |

Journal | Ars Mathematica Contemporanea |

Volume | 13 |

Issue number | 1 |

Early online date | 22 Feb 2017 |

Publication status | Published - 2017 |