Graphs of Groups on Surfaces: Interactions and Models

Arthur T. White (Eds.)0444500758, 9780444500755, 9780585474052

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.

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Table of contents :
Content:
Foreword
Pages v-vii

Chapter 1 Historical setting Original Research Article
Pages 1-4

Chapter 2 A brief introduction to graph theory Original Research Article
Pages 5-12

Chapter 3 The automorphism group of a graph Original Research Article
Pages 13-17

Chapter 4 The Cayley color graph of a group presentation Original Research Article
Pages 19-32

Chapter 5 An introduction to surface topology Original Research Article
Pages 33-48

Chapter 6 Imbedding problems in graph theory Original Research Article
Pages 49-72

Chapter 7 The genus of a group Original Research Article
Pages 73-88

Chapter 8 Map-coloring problems Original Research Article
Pages 89-106

Chapter 9 Quotient graphs and quotient manifolds: Current graphs and the complete graph theorem Original Research Article
Pages 107-117

Chapter 10 Voltage graphs Original Research Article
Pages 119-141

Chapter 11 Nonorientable graph imbeddings Original Research Article
Pages 143-155

Chapter 12 Block designs Original Research Article
Pages 157-171

Chapter 13 Hypergraph imbeddings Original Research Article
Pages 173-183

Chapter 14 Finite fields on surfaces Original Research Article
Pages 185-197

Chapter 15 Finite geometries on surfaces Original Research Article
Pages 199-234

Chapter 16 Map automorphism groups Original Research Article
Pages 235-265

Chapter 17 Enumerating graph imbeddings Original Research Article
Pages 267-279

Chapter 18 Random topological graph theory Original Research Article
Pages 281-294

Chapter 19 Change ringing Original Research Article
Pages 295-321

References
Pages 323-350

References
Pages 351-352

Index of symbols
Pages 353-355

Index of definitions
Pages 357-363