Semi-transitive graphs

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi-transitive and semi-transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi-transitive spider-graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi-transitive graph (if it has more than one) is a bi-transitive graph. We show how the alternets can be used to understand the structure of a semi-transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi-transitive graphs of degree 4 begun by Marušič and Praeger. This classification shows that nearly all such graphs are spider-graphs.

Original languageEnglish (US)
Pages (from-to)1-27
Number of pages27
JournalJournal of Graph Theory
Volume45
Issue number1
DOIs
StatePublished - Jan 2004

Fingerprint

Graph in graph theory
Spiders
Arrowhead
Symmetry Group
Directed Graph
Term

Keywords

  • Alternet
  • Graph automorphism
  • Graph symmetry
  • Semi-transitive graph

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Semi-transitive graphs. / Wilson, Stephen E.

In: Journal of Graph Theory, Vol. 45, No. 1, 01.2004, p. 1-27.

Research output: Contribution to journalArticle

Wilson, Stephen E. / Semi-transitive graphs. In: Journal of Graph Theory. 2004 ; Vol. 45, No. 1. pp. 1-27.
@article{2e172f483ac74c08b6a62b8ad69c4b8d,
title = "Semi-transitive graphs",
abstract = "In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi-transitive and semi-transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi-transitive spider-graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi-transitive graph (if it has more than one) is a bi-transitive graph. We show how the alternets can be used to understand the structure of a semi-transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi-transitive graphs of degree 4 begun by Marušič and Praeger. This classification shows that nearly all such graphs are spider-graphs.",
keywords = "Alternet, Graph automorphism, Graph symmetry, Semi-transitive graph",
author = "Wilson, {Stephen E}",
year = "2004",
month = "1",
doi = "10.1002/jgt.10152",
language = "English (US)",
volume = "45",
pages = "1--27",
journal = "Journal of Graph Theory",
issn = "0364-9024",
publisher = "Wiley-Liss Inc.",
number = "1",

}

TY - JOUR

T1 - Semi-transitive graphs

AU - Wilson, Stephen E

PY - 2004/1

Y1 - 2004/1

N2 - In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi-transitive and semi-transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi-transitive spider-graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi-transitive graph (if it has more than one) is a bi-transitive graph. We show how the alternets can be used to understand the structure of a semi-transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi-transitive graphs of degree 4 begun by Marušič and Praeger. This classification shows that nearly all such graphs are spider-graphs.

AB - In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi-transitive and semi-transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi-transitive spider-graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi-transitive graph (if it has more than one) is a bi-transitive graph. We show how the alternets can be used to understand the structure of a semi-transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi-transitive graphs of degree 4 begun by Marušič and Praeger. This classification shows that nearly all such graphs are spider-graphs.

KW - Alternet

KW - Graph automorphism

KW - Graph symmetry

KW - Semi-transitive graph

UR - http://www.scopus.com/inward/record.url?scp=0347133467&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347133467&partnerID=8YFLogxK

U2 - 10.1002/jgt.10152

DO - 10.1002/jgt.10152

M3 - Article

AN - SCOPUS:0347133467

VL - 45

SP - 1

EP - 27

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -