### Affine primitive symmetric graphs of diameter two

#### Abstract

Let *n* be a positive integer, *q* be a prime power, and *V* be a vector space of dimension *n* over *F*_{q}. Let *G *:= *V *rtimes* **G*_{0}, where *G*_{0} is an irreducible subgroup of *G**L*(*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 *G*_{0} is a subgroup of either Γ L(n,q) or Γ Sp(*n*, *q*) and is maximal in one of the Aschbacher classes *C*_{i}, where *i* ∈ {2, 4, 5, 6, 7, 8}. We are able to determine all graphs Γ which arise from *G*_{0} ≤ Γ L(*n*, *q*) with *i* ∈ {2, 4, 8}, and from *G*_{0} ≤ 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.

#### Keywords

ISSN: 1855-3974

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