代数图基础
出版时间：2013年版
内容简介
《当代科学技术基础理论与前沿问题研究丛书·中国科学技术大学校友文库：代数图基础》以图的代数表示为起点，着重于多面形、曲面、嵌入和地图等对象，用一个统一的理论框架，揭示在更具普遍性的组合乃至代数构形中，可通过局部对称性反映全局性质。特别是通过多项式型的不变量刻画这些构形在不同拓扑、组合和代数变换下的分类。同时，也提供这些分类在算法上的实现和复杂性分析。虽然《当代科学技术基础理论与前沿问题研究丛书·中国科学技术大学校友文库：代数图基础》中的结论多以作者的前期工作为基础发展得到，但仍有一定数量的新结果。例如，关于图在给定亏格曲面上可嵌入性的识别，沿四个不同理论思路的判准就是新近得到的。在亏格为零的特殊情形下，从它们中的一个可一举导出Euler、Whitney、MacLant和Lefschetz在图的平面性方面沿不同理论路线的结果。《当代科学技术基础理论与前沿问题研究丛书·中国科学技术大学校友文库：代数图基础》可供纯粹数学、应用数学、系统科学以及计算机科学等方面的大学生及相关教师使用，还可供相关专业研究生和数学研究人员阅读。
目录
Preface to the USTC Alumni’S Series Preface Chapter 1 Abstract Graphs 1．1 Graphs and Networks 1．2 Surfaces 1．3 Embeddings 1．4 Abstract Representation 1．5 Nores Chapter 2 Abstract Maps 2．1 Ground Sets 2．2 Basic Permutations 2．3 Conjugate Axiom 2．4 nansitive Axiom 2．5 Included Angles 2．6 Notes Chapter 3 Duality 3．1 Dual Maps 3．2 Deletion of an Edge 3．3 Addition of an Edge 3．4 Basic Transformation 3．5 Nores Chapter 4 Orientability 4.1 Orientation 4.2 Basic Equivalence 4.3 Euler Characteristic 4.4 Pattern Examples 4.5 Notes Chapter 5 Orientable Maps 5.1 Butterflies 5.2 Simplified Butterflies 5.3 Reduced Rules 5.4 Orientable Principles 5.5 Orientable Genus 5.6 Notes Chapter 6 Nonorientable Maps 6.1 Barflies 6.2 Simplified Barflies 6.3 Nonorientable Rules 6.4 Nonorientable Principles 6.5 Nonorientable Genus 6.6 Notes Chapter 7 Isomorphisms of Maps 7.1 Commutativity 7.2 Isomorphism Theorem 7.3 Recognition 7.4 Justification 7.5 Pattern Examples 7.6 Notes Chapter 8 Asymmetrization 8.1 Automorphisms 8.2 Upper Bounds of Group Order 8.3 Determination of the Group 8.4 Rootings 8.5 Notes Chapter 9 Asymmetrized Petal Bundles 9.1 Orientable Petal Bundles 9.2 Planar Pedal Bundles 9.3 Nonorientable Pedal Bundles 9.4 The Number of Pedal Bundles 9.5 Notes Chapter 10 Asymmetrized Maps 10.1 Orientable Equation 10.2 Planar Rooted Maps 10.3 Nonorientable Equation 10.4 Gross Equation 10.5 The Number of Rooted Maps 10.6 Notes Chapter 11 Maps Within Symmetry 11.1 Symmetric Relation 11.2 An Application 11.3 Symmetric Principle 11.4 General Examples 11.5 Notes Chapter 12 Genus Polynomials 12.1 Associate Surfaces 12.2 Layer Division of a Surface 12.3 Handle Polynomials 12.4 Crosscap Polynomials 12.5 Notes Chapter 13 Census with Partitions 13.1 Planted Trees 13.2 Hamiltonian Cubic Maps 13.3 Halin Maps 13.4 Biboundary Inner Rooted Maps 13.5 General Maps 13.6 Pan-Flowers 13.7 Notes Chapter 14 Equations with Partitions 14.1 The Meson Functional 14.2 General Maps on the Sphere 14.3 Nonseparable Maps on the Sphere 14.4 Maps Without Cut-Edge on Surfaces 14.5 Eulerian Maps on the Sphere 14.6 Eulerian Maps on Surfaces 14.7 Notes Chapter 15 Upper Maps of a Graph 15.1 Semi-Automorphisms on a Graph 15.2 Automorphisms on a Graph 15.3 Relationships 15.4 Upper Maps with Symmetry 15.5 Via Asymmetrized Upper Maps 15.6 Notes Chapter 16 Genera of Graphs 16.1 A Recursion Theorem 16.2 Maximum Genus 16.3 Minimum Genus 16.4 Average Genus 16.5 Thickness 16.6 Interlacedness 16.7 Notes Chapter 17 Isogemial Graphs 17.1 Basic Concepts 17.2 Two Operations 17.3 Isogemial Theorem 17.4 Nonisomorphic Isogemial Graphs 17.5 Notes Chapter 18 Surface Embeddability 18.1 Via Tree-Travels 18.2 Via Homology 18.3 Via Joint Trees 18.4 Via Configurations 18.5 Notes Appendix 1 Concepts of Polyhedra, Surfaces, Embeddings and Maps Appendix 2 Table of Genus Polynomials for Embeddings and Maps of Small Size Appendix 3 Atlas of Rooted and Unrooted Maps for Small Graphs Bibliography Terminology Author Index
