Graphs of Groups on Surfaces book cover

Graphs of Groups on Surfaces

Interactions and Models

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.

Hardbound, 378 Pages

Published: April 2001

Imprint: North-holland

ISBN: 978-0-444-50075-5

Reviews

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

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

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