Book contents
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
Groups and semisymmetric graphs
Published online by Cambridge University Press: 15 December 2009
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
Summary
Abstract
A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. This paper uses finite groups to construct many infinite families of such graphs.
Introduction
There is an intimate relationship between groups and graphs. For example, any graph X gives rise to its automorphism group A := Aut(X). On the other hand, any group G with generating set S gives rise to its Cayley graph Cay(G, S). The main purpose of this paper is to show how groups can be used to construct examples of semisymmetric graphs, graphs which are regular and edge-transitive but not vertex-transitive. [Relevent definitions are given in Section 2.]
These semisymmetric graphs were first systematically studied in 1967 by J. Folkman [6]. For later works on semisymmetric graphs, the reader is referred to [1, 2, 3, 4, 5, 7, 8, 9].
Since the known semisymmetric graphs are not many and since they have very special symmetry properties, there is a common belief that semisymmetric graphs are rare in number. We do not share this belief since our contruction leads to infinite classes of such graphs.
This paper is organized as follows. First we give the necessary notation, definitions, and concepts needed for this paper, including that of semisymmetric graphs. In particular, we introduce the notion of co-neighbor blocks and noncontractable graphs, and we also give the definition of a bi-lexicographic product of a graph.
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- Chapter
- Information
- Groups St Andrews 2001 in Oxford , pp. 385 - 394Publisher: Cambridge University PressPrint publication year: 2003