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.
Highlights
A symmetric graph is one which admits a subgroup of automorphisms that acts transitively on its arc set; if G is such a subgroup, we say in particular that the graph is G-symmetric
We consider those G-symmetric diameter two graphs where G is a primitive group of affine type, and where the point stabiliser G0 is maximal in the general semilinear group or in the symplectic semisimilarity group
Let V = Fnq for some prime power q and positive integer n, and let G = V G0, where G0 is an irreducible subgroup of the general semilinear group ΓL(n, q) or the symplectic semisimilarity group ΓSp(n, q), and G0 is maximal by inclusion with respect to being intransitive on the set of nonzero vectors in V
Summary
A symmetric graph is one which admits a subgroup of automorphisms that acts transitively on its arc set; if G is such a subgroup, we say in particular that the graph is G-symmetric. In paper [2] we considered the graphs corresponding to the groups G0 which are maximal in their respective classes Ci, for i ≤ 8, and which are intransitive on nonzero vectors. We consider the two cases: (1) where G0 ≤ ΓL(n, q) and G0 does not preserve an alternating form on Fnq , and (2) where G0 ≤ ΓSp(n, q) Note that in this case it is possible for d/n to be not prime, and it follows from the maximality of q that G0 is not contained in a proper C3-subgroup of ΓL(n, q) or ΓSp(n, q), respectively. If Γ is a graph, V (Γ) and E(Γ) are, respectively, its vertex set and edge set
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