Graphs of Groups on Surfaces

Graphs of Groups on Surfaces

Interactions and Models

1st Edition - April 27, 2001

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  • Author: A.T. White
  • eBook ISBN: 9780080507583

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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.

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

    6-2. Definition of "Imbedding"

    6-3. The Genus of a Graph

    6-4. The Maximum Genus of a Graph

    6-5. Genus Formulae for Graphs

    6-6. Rotation Schemes

    6-7. Imbedding Graphs on Pseudosurfaces

    6-8. Other Topological Parameters for Graphs

    6-9. Applications

    6-10. Problems


    Chapter 7. THE GENUS OF A GROUP


    7-1. Imbeddings of Cayley Color graphs

    7-2. Genus Formulae for Groups

    7-3. Related Results

    7-4. The Characteristic of a Group

    7-5. Problems


    Chapter 8. MAP-COLORING PROBLEMS


    8-1. Definitions and the Six-Color Theorem

    8-2. The Five-Color Theorem

    8-3. The Four-Color Theorem

    8-4. Other Map-Coloring Problems:
    The Heawood Map-Coloring Theorem

    8-5. A Related Problem

    8-6. A Four-Color Theorem for the Torus

    8-7. A Nine-Color Theorem for the Torus and Klein Bottle

    8-8. k-degenerate Graphs

    8-9. Coloring Graphs on Pseudosurfaces

    8-10. The Cochromatic Number of Surfaces

    8-11. Problems


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


    9-1. The Genus of Kn

    9-2. The Theory of Current Graphs as Applied to Kn

    9-3. A Hint of Things to Come

    9-4. Problems


    Chapter 10. VOLTAGE GRAPHS


    10-1. Covering Spaces

    10-2. Voltage Graphs

    10-3. Examples

    10-4. The Heawood Map-coloring Theorem (again)

    10-5. Strong Tensor Products

    10-6. Covering Graphs and Graphical Products

    10-7. Problems


    Chapter 11. NONORIENTABLE GRAPH IMBEDDINGS


    11-1. General Theory

    11-2. Nonorientable Covering Spaces

    11-3. Nonorientable Voltage Graph Imbeddings

    11-4. Examples

    11-5. The Heawood Map-coloring Theorem, Nonorientable Version

    11-6. Other Results

    11-7. Problems


    Chapter 12. BLOCK DESIGNS


    12-1. Balanced Incomplete Block Designs

    12-2. BIBDs and Graph Imbeddings

    12-3. Examples

    12-4. Strongly Regular Graphs

    12-5. Partially Balanced Incomplete Block Designs

    12-6. PBIBDs and Graph Imbeddings

    12-7. Examples

    12-8. Doubling a PBIBD

    12-9. Problems


    Chapter 13. HYPERGRAPH IMBEDDINGS


    13-1. Hypergraphs

    13-2. Associated Bipartite Graphs

    13-3. Imbedding Theory for Hypergraphs

    13-4. The Genus of a Hypergraph

    13-5. The Heawood Map-Coloring Theorem, for Hypergraphs

    13-6. The Genus of a Block Design

    13-7. An Example

    13-8. Nonorientable Analogs

    13-9. Problems


    Chapter 14. FINITE FIELDS ON SURFACES


    14-1. Graphs Modelling Finite Rings

    14-2. Basic Theorems About Finite Fields

    14-3. The Genus of Fp

    14-4. The Genus of Fpr

    14-5. Further Results

    14-6. Problems


    Chapter 15. FINITE GEOMETRIES ON SURFACES


    15-1. Axiom Systems for Geometries

    15-2. n-Point Geometry

    15-3. The Geometries of Fano, Pappus, and Desargues

    15-4. Block Designs as Models for Geometries

    15-5. Surface Models for Geometries

    15-6. Fano, Pappus, and Desargues Revisited

    15-7. 3-Configurations

    15-8. Finite Projective Planes

    15-9. Finite Affine Planes

    15-10. Ten Models for AG(2,3)

    15-11. Completing the Euclidean Plane

    15-12. Problems


    Chapter 16. MAP AUTOMORPHISM GROUPS


    16-1. Map Automorphisms

    16-2. Symmetrical Maps

    16-3. Cayley Maps

    16-4. Complete Maps

    16-5. Other Symmetrical Maps

    16-6. Self -Complementary Graphs

    16-7. Self-dual Maps

    16-8. Paley Maps

    16-9. Problems


    Chapter 17. ENUMERATING GRAPH IMBEDDINGS


    17-1. Counting Labelled Orientable 2-Cell Imbeddings

    17-2. Counting Unlabelled Orientable 2-Cell Imbeddings

    17-3. The Average Number of Symmetries

    17-4. Problems


    Chapter 18. RANDOM TOPOLOGICAL GRAPH THEORY


    18-1. Model I

    18-2. Model II

    18-3. Model III

    18-4. Model IV

    18-5. Model V

    18-6. Model VI- Random Cayley Maps

    18-7. Problems


    Chapter 19. CHANGE RINGING


    19-1. The Setting

    19-2. A Mathematical Model

    19-3. Minimus

    19-4. Doubles

    19-5. Minor

    19-6. Triples and Fabian Stedman

    19-7. Extents on n Bells

    19-8. Summary

    19-9. Problems


    REFERENCES. BIBLIOGRAPHY. INDEX OF SYMBOLS. INDEX OF DEFINITIONS

Product details

  • No. of pages: 378
  • Language: English
  • Copyright: © North Holland 2001
  • Published: April 27, 2001
  • Imprint: North Holland
  • eBook ISBN: 9780080507583

About the Author

A.T. White

Affiliations and Expertise

Western Michigan University, Kalamazoo, MI 49008, USA

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