# 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

**View all volumes in this series:**North-Holland Mathematics Studies

## 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

Chapter 7. THE GENUS OF A GROUP

Chapter 8. MAP-COLORING PROBLEMS

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

Chapter 10. VOLTAGE GRAPHS

Chapter 11. NONORIENTABLE GRAPH IMBEDDINGS

Chapter 12. BLOCK DESIGNS

Chapter 13. HYPERGRAPH IMBEDDINGS

Chapter 14. FINITE FIELDS ON SURFACES

Chapter 15. FINITE GEOMETRIES ON SURFACES

Chapter 16. MAP AUTOMORPHISM GROUPS

Chapter 17. ENUMERATING GRAPH IMBEDDINGS

Chapter 18. RANDOM TOPOLOGICAL GRAPH THEORY

Chapter 19. CHANGE RINGING

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:
- 27th April 2001

- 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