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 ⋊ 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 ΓSp (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 ≤ ΓL (n, q) with i ∈ {2, 4, 8}, and from G 0 ≤ ΓSp (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.

Quotient-Complete Arc-Transitive Latin Square Graphs from Groups

“…If H denotes the set of n symbols, then, by labelling the rows and columns by elements of H, each cell in L can be represented by an ordered triple (h 1 , h 2 , h 3 ) of elements of H, where h 1 denotes the row label, h 2 the column label, and h 3 the symbol contained in the cell. An autoparatopism of L is an element of Sym(H) ≀ Sym(3), in its product action on H 3 , which preserves L setwise; an autotopism of L is an autoparatopism which belongs in Sym(H) 3 . Thus an autotopism is an ordered triple of permutations acting on the set of row labels, the set of column labels, and the set of symbols, while an autoparatopism consists of an autotopism followed by a permutation of the three coordinates.…”

“…Thus an autotopism is an ordered triple of permutations acting on the set of row labels, the set of column labels, and the set of symbols, while an autoparatopism consists of an autotopism followed by a permutation of the three coordinates. In particular, we denote any autoparatopism of L by [(σ 1 , σ 2 , σ 3 ), γ], where (σ 1 , σ 2 , σ 3 ) ∈ Sym(H) 3 and γ ∈ Sym(3), with action given by…”

“…Throughout this section we assume that H is a finite group, Γ = LSG(H) is the latin square graph of H, and G is the autoparatopism group of the Cayley table of H. Recall from Subsection 2.2 that G ∼ = H 2 ⋊ Aut(H) Sym (3). Let λ a and ρ b be as in (2), and let T be the autotopism subgroup of Γ, that is, T = {(λ a , ρ b , λ a ρ b ) a, b ∈ H}.…”

“…This paper is part of a general study of arc-transitive, diameter 2 graphs carried out in [1,2,3]. The family F of all such graphs is analyzed using normal quotient reduction, and it is shown in [1,Theorem 2.2] that any graph in F has a normal quotient graph Γ with automorphism group G such that Γ is either G-vertex-quasiprimitive (i.e., all nontrivial normal subgroups of G are transitive on vertices) or G-quotient-complete.…”

“…The family F of all such graphs is analyzed using normal quotient reduction, and it is shown in [1,Theorem 2.2] that any graph in F has a normal quotient graph Γ with automorphism group G such that Γ is either G-vertex-quasiprimitive (i.e., all nontrivial normal subgroups of G are transitive on vertices) or G-quotient-complete. A subclass of vertex-quasiprimitive graphs were studied in [2] and [3]. Quotient-complete graphs were studied in [1], in which it was shown that a significant parameter of quotient-complete graphs is the number k of distinct nontrivial, complete normal quotients.…”