图论基础

Preface in Chinese

Chapter 1 Basic concepts

1.1 Graph and simple graph

1.2 Graph operations

1.3 Isomorphism

1.4 Incident and adjacent matrix

1.5 The spectrum of graph

1.6 The spectrum of several graphs

1.7 Results from matrix theory

1.8 About the largest zero of characteristic polynomials

1.9 Spectrum radius

Chapter 2 path and cycle

2.1 The path

2.2 The cycle

2.3 The diameter of a graph and its complement graph

Chapter 3 Tree

3.1 Tree

3.2 Spanning tree

3.3 A bound for the tree number of regular graphs

3.4 Cycle space and bound space of a graph

Chapter 4 Connectivity

4.1 Cut edges

4.2 Cut vertex

4.3 Block

4.4 Connectivity

Chapter 5 Euler and Hamilton graphs

5.1 Euler path and circuits

5.2 Hamilton graph

Chapter 6 Matching and matching polynomial

6.1 Matching

6.2 Bipartite graph and perfect matching

6.3 Matching polynomial

6.4 The relation between spectrum and matching polynomial

6.5 Relation between several graphs

6.6 Several matching equivalent and matching unique graphs

6.7 The Hosoya index of several graphs

6.8 Two trees with minimal Hosoya index

6.9 Recent results in matching

Chapter 7 Laplacian and Quasi-Laplacian spectrum

7.1 Sigma function

7.2 The spanning tree and sigma function

7.3 Quasi-Laplacian Spectrum

7.4 Basic lemmas

7.5 Main results

7.6 Three different spectrum of regular graphs

Chapter 8 More theorems form matrix theory

8.1 The irreducible matrix

8.2 Cauchy's interlacing theorem

8.3 The eigenvalues of A(G) and graph structure

Chapter 9 Chromatic polynomial

9.1 Induction

9.2 Two different formula for chromatic polynomial

9.3 Chromatic polynomials for several type of graphs

9.4 Estimate the color number

References

Bibliography

本书除了介绍图论的基本概念和简单结构理论外，主要研究总结了近些年来比较热门的关于图的邻接谱、图的匹配多项式、图的着色、图的色多项式、图的拉普拉斯多项式和图的准拉普拉斯多项式的一些主要成果。为了方便读者进行研究和参考，我们也采编了一些和图论问题相关的线性代数、矩阵论的理论知识。在书的后面给出了基本符号及中英文对照表，以方便读者查阅参考。本书内容详细，证明简洁，并且在每章后面提供了一些新的研究内容与素材并配以一定量的习题，可作为数学与应用数学专业高年级的专业选修课和图论方向的一年级的硕士研究生课程的教材，也可作为广大图论研究工作者的参考用书。

《图论基础》的特点：内容基础、简单易于初学者掌握。主要侧重图基本结构与图的谱理论的相互关系.另外对多项式理论也进行了讨论。内容新颖，尤其是图的匹配多项式方面介绍了近年来的最新研究成果。采用英语语言编写，可以作为双语教材。