Graphs of Groups on Surfaces - 1st Edition - ISBN: 9780444500755, 9780080507583

Graphs of Groups on Surfaces, Volume 188

1st Edition

Interactions and Models

Authors: A.T. White
eBook ISBN: 9780080507583
Hardcover ISBN: 9780444500755
Imprint: North Holland
Published Date: 27th April 2001
Page Count: 378
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Table of Contents

Chapter 1. HISTORICAL SETTING

Chapter 2. A BRIEF INTRODUCTION TO GRAPH THEORY

2-1. Definition of a Graph

2-2. Variations of Graphs
2-3. Additional Definitions

2-4. Operations on Graphs

2-5. Problems

Chapter 3. THE AUTOMORPHISM GROUP OF A GRAPH

3-1. Definitions
3-2. Operations on Permutations Groups
3-3. Computing Automorphism Groups of Graphs

3-4. Graphs with a Given Automorphism Group
3-5. Problems

Chapter 4. THE CAYLEY COLOR GRAPH OF A GROUP PRESENTATION

4-1. Definitions

4-2. Automorphisms

4-3. Properties

4-4. Products

4-5. Cayley Graphs

4-6. Problems

Chapter 5. AN INTRODUCTION TO SURFACE TOPOLOGY

5-1. Definitions

5-2. Surfaces and Other 2-manifolds

5-3. The Characteristic of a Surface

5-4. Three Applications
5-5. Pseudosurfaces

5-6. Problems

Chapter 6. IMBEDDING PROBLEMS IN GRAPH THEORY

6-1. Answers to Some Imbedding Questions<BR> 6-2. Definition of "Imbedding"<BR> 6-3. The Genus of a Graph<BR> 6-4. The Maximum Genus of a Graph<BR> 6-5. Genus Formulae for Graphs<BR> 6-6. Rotation Schemes<BR> 6-7. Imbedding Graphs on Pseudosurfaces<BR> 6-8. Other Topological Parameters for Graphs<BR> 6-9. Applications<BR> 6-10. Problems<BR>

Chapter 7. THE GENUS OF A GROUP

7-1. Imbeddings of Cayley Color graphs<BR> 7-2. Genus Formulae for Groups<BR> 7-3. Related Results<BR> 7-4. The Characteristic of a Group<BR> 7-5. Problems<BR>

Chapter 8. MAP-COLORING PROBLEMS

8-1. Definitions and the Six-Color Theorem<BR> 8-2. The Five-Color Theorem<BR> 8-3. The Four-Color Theorem<BR> 8-4. Other Map-Coloring Problems: The Heawood Map-Coloring Theorem<BR> 8-5. A Related Problem<BR> 8-6. A Four-Color Theorem for the Torus<BR> 8-7. A Nine-Color Theorem for the Torus and Klein Bottle<BR> 8-8. k-degenerate Graphs<BR> 8-9. Coloring Graphs on Pseudosurfaces<BR> 8-10. The Cochromatic Number of Surfaces<BR> 8-11. Problems<BR>

Chapter 9. QUOTIENT GRAPHS AND QUOTIENT MANIFOLDS: CURRENT GRAPHS AND THE COMPLETE GRAPH THEOREM

9-1. The Genus of Kn<BR> 9-2. The Theory of Current Graphs as Applied to Kn<BR> 9-3. A Hint of Things to Come<BR> 9-4. Problems<BR>

Chapter 10. VOLTAGE GRAPHS

10-1. Covering Spaces<BR> 10-2. Voltage Graphs<BR> 10-3. Examples<BR> 10-4. The Heawood Map-coloring Theorem (again)<BR> 10-5. Strong Tensor Products<BR> 10-6. Covering Graphs and Graphical Products<BR> 10-7. Problems<BR>

Chapter 11. NONORIENTABLE GRAPH IMBEDDINGS

11-1. General Theory<BR> 11-2. Nonorientable Covering Spaces<BR> 11-3. Nonorientable Voltage Graph Imbeddings<BR> 11-4. Examples<BR> 11-5. The Heawood Map-coloring Theorem, Nonorientable Version<BR> 11-6. Other Results<BR> 11-7. Problems<BR>

Chapter 12. BLOCK DESIGNS

12-1. Balanced Incomplete Block Designs<BR> 12-2. BIBDs and Graph Imbeddings<BR> 12-3. Examples<BR> 12-4. Strongly Regular Graphs<BR> 12-5. Partially Balanced Incomplete Block Designs<BR> 12-6. PBIBDs and Graph Imbeddings<BR> 12-7. Examples<BR> 12-8. Doubling a PBIBD<BR> 12-9. Problems<BR>

Chapter 13. HYPERGRAPH IMBEDDINGS

13-1. Hypergraphs<BR> 13-2. Associated Bipartite Graphs<BR> 13-3. Imbedding Theory for Hypergraphs<BR> 13-4. The Genus of a Hypergraph<BR> 13-5. The Heawood Map-Coloring Theorem, for Hypergraphs<BR> 13-6. The Genus of a Block Design<BR> 13-7. An Example<BR> 13-8. Nonorientable Analogs<BR> 13-9. Problems<BR>

Chapter 14. FINITE FIELDS ON SURFACES

14-1. Graphs Modelling Finite Rings<BR> 14-2. Basic Theorems About Finite Fields<BR> 14-3. The Genus of Fp<BR> 14-4. The Genus of Fpr<BR> 14-5. Further Results<BR> 14-6. Problems<BR>

Chapter 15. FINITE GEOMETRIES ON SURFACES

15-1. Axiom Systems for Geometries<BR> 15-2. n-Point Geometry<BR> 15-3. The Geometries of Fano, Pappus, and Desargues<BR> 15-4. Block Designs as Models for Geometries<BR> 15-5. Surface Models for Geometries<BR> 15-6. Fano, Pappus, and Desargues Revisited<BR> 15-7. 3-Configurations<BR> 15-8. Finite Projective Planes<BR> 15-9. Finite Affine Planes<BR> 15-10. Ten Models for AG(2,3)<BR> 15-11. Completing the Euclidean Plane<BR> 15-12. Problems<BR>

Chapter 16. MAP AUTOMORPHISM GROUPS

16-1. Map Automorphisms<BR> 16-2. Symmetrical Maps<BR> 16-3. Cayley Maps<BR> 16-4. Complete Maps<BR> 16-5. Other Symmetrical Maps<BR> 16-6. Self -Complementary Graphs<BR> 16-7. Self-dual Maps<BR> 16-8. Paley Maps<BR> 16-9. Problems<BR>

Chapter 17. ENUMERATING GRAPH IMBEDDINGS

17-1. Counting Labelled Orientable 2-Cell Imbeddings<BR> 17-2. Counting Unlabelled Orientable 2-Cell Imbeddings<BR> 17-3. The Average Number of Symmetries<BR> 17-4. Problems<BR>

Chapter 18. RANDOM TOPOLOGICAL GRAPH THEORY

18-1. Model I<BR> 18-2. Model II<BR> 18-3. Model III<BR> 18-4. Model IV<BR> 18-5. Model V<BR> 18-6. Model VI- Random Cayley Maps<BR> 18-7. Problems<BR>

Chapter 19. CHANGE RINGING

19-1. The Setting<BR> 19-2. A Mathematical Model<BR> 19-3. Minimus<BR> 19-4. Doubles<BR> 19-5. Minor<BR> 19-6. Triples and Fabian Stedman<BR> 19-7. Extents on n Bells<BR> 19-8. Summary<BR> 19-9. Problems<BR>

REFERENCES. BIBLIOGRAPHY. INDEX OF SYMBOLS. INDEX OF DEFINITIONS


Description

The book, suitable as both an introductory reference and as a text book in the rapidly growing field of topological graph theory, models both maps (as in map-coloring problems) and groups by means of graph imbeddings on sufaces. Automorphism groups of both graphs and maps are studied. In addition connections are made to other areas of mathematics, such as hypergraphs, block designs, finite geometries, and finite fields. There are chapters on the emerging subfields of enumerative topological graph theory and random topological graph theory, as well as a chapter on the composition of English church-bell music. The latter is facilitated by imbedding the right graph of the right group on an appropriate surface, with suitable symmetries. Throughout the emphasis is on Cayley maps: imbeddings of Cayley graphs for finite groups as (possibly branched) covering projections of surface imbeddings of loop graphs with one vertex. This is not as restrictive as it might sound; many developments in topological graph theory involve such imbeddings.

The approach aims to make all this interconnected material readily accessible to a beginning graduate (or an advanced undergraduate) student, while at the same time providing the research mathematician with a useful reference book in topological graph theory. The focus will be on beautiful connections, both elementary and deep, within mathematics that can best be described by the intuitively pleasing device of imbedding graphs of groups on surfaces.


Details

No. of pages:
378
Language:
English
Copyright:
© North Holland 2001
Published:
Imprint:
North Holland
eBook ISBN:
9780080507583
Hardcover ISBN:
9780444500755

Reviews

@qu:...this is a very well-written and readable book, which I recommend on anyone wanting to learn this particular approach to the subject. @source:Bulletin of the London Mathematical Society


About the Authors

A.T. White Author

Affiliations and Expertise

Western Michigan University, Kalamazoo, MI 49008, USA