Affine primitive symmetric graphs of diameter two - Amarra, Carmen (Ars mathematica contemporanea, 2017)
Let ?$n$? be a positive integer, ?$q$? be a prime power, and ?$V$? be a vector space of dimension ?$n$? over ?$\mathbb{F}_{q}$?. Let ?$G := V \rtimes G_{0}$?, where ?$G_{0}$? 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 ?$G_{0}$? is a subgroup of either ?$ΓL(n,q)$? or ?$Γ Sp(n, q)$? and is maximal in one of the Aschbacher classes ?$\mathcal{C}_{i}$?, where ?$i \in \{2, 4, 5, 6, 7, 8\}$?. We are able to determine all graphs ?$Γ$? which arise from ?$G_{0}≤ ΓL(n,q)$? with ?$i \in \{2, 4, 8\}$?, and from ?$G_{0} ≤ Γ Sp(n, q)$? with ?$i \in \{2, 8\}$?. 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: Cayleyevi grafi; linearne grupe; permutacijske grupe; simetrični grafi