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Vertex‐transitive graphs: Symmetric graphs of prime valency
Author(s) -
Lorimer Peter
Publication year - 1984
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190080107
Subject(s) - valency , combinatorics , mathematics , vertex (graph theory) , bipartite graph , edge transitive graph , discrete mathematics , transitive relation , prime (order theory) , graph , graph power , line graph , philosophy , linguistics
Let G be a group acting symmetrically on a graph Σ, let G 1 be a subgroup of G minimal among those that act symmetrically on Σ, and let G 2 be a subgroup of G 1 maximal among those normal subgroups of G 1 which contain no member except 1 which fixes a vertex of Σ. The most precise result of this paper is that if Σ has prime valency p , then either Σ is a bipartite graph or G 2 acts regularly on Σ or G 1 | G 2 is a simple group which acts symmetrically on a graph of valency p which can be constructed from Σ and does not have more vertices than Σ. The results on vertex‐transitive groups necessary to establish results like this are also included.