Affine primitive symmetric graphs of diameter two

Maria Carmen Amarra, Michael Giudici, Cheryl Praeger

Research output: Contribution to journalArticle

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.
LanguageEnglish
Pages137-165
Number of pages29
JournalArs Mathematica Contemporanea
Volume13
Issue number1
Early online date22 Feb 2017
StatePublished - 2017

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Symmetric Graph
Vector spaces
Subgroup
Graph in graph theory
Dimension of a vector space
Intransitive
Q-integers
Automorphisms
Inclusion
Necessary Conditions
Vertex of a graph
Class

Cite this

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title = "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 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 inone 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.",
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Affine primitive symmetric graphs of diameter two. / Amarra, Maria Carmen; Giudici, Michael; Praeger, Cheryl.

In: Ars Mathematica Contemporanea, Vol. 13, No. 1, 2017, p. 137-165.

Research output: Contribution to journalArticle

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