刘彥佩编著的《代数图基础》是中国科学技术大学校友文库之一。本书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。

刘彥佩编著的《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。虽然本书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。

《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。

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