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內容簡介

　　《運籌與管理科學叢書24：A First Course in Graph Theory（圖論基礎教程）》著眼於有嚮圖，將無嚮圖作為特例，在一定的深度和廣度上係統地闡述瞭圖論的基本概念、理論和方法以及基本應用．全書內容共分７章，包括Euler迴與Hamilton圈、樹與圖空間、平麵圖、網絡流與連通度、匹配與獨立集、染色理論、圖與群以及圖在矩陣論、組閤數學、組閤優化、運籌學、綫性規劃、電子學以及通訊和計算機科學等多方麵的應用．每章分為理論和應用兩部分，章末有小結和參考文獻．各章內容之間聯係緊密，許多著名的定理給齣最新最簡單的多種證明。每小節末有大量習題，書末附有記號和名詞索引。

目錄

Contents
Preface
Chapter 1 Basic Concepts of Graphs
1.1 Graph and Graphical Representation.
1.2 Graph Isomorphism.
1.3 Vertex Degrees
1.4 Subgraphs and Operations
1.5 Walks， Paths and Connection

Chapter 2 Advanced Concepts of Graphs
2.1 Distances and Diameters
2.2 Circuits and Cycles
2.3 Eulerian Graphs
2.4 Hamiltonian Graphs
2.5 Matrix Representations of Graphs
2.6 Exponents of Primitive Matrices

Chapter 3 Trees and Graphic Spaces
3.1 Trees and Spanning Trees.
3.2 Vector Spaces of Graphs
3.3 Enumeration of Spanning Trees
3.4 The Minimum Connector Problem.
3.5 The Shortest Path Problem.
3.6 The Electrical Network Equations

Chapter 4 Plane Graphs and Planar Graphs
4.1 Plane Graphs and Euler's Formula
4.2 Kuratowski's Theorem.
4.3 Dual Graphs.
4.4 Regular Polyhedra.
4.5 Layout of Printed Circuits

Chapter 5 Flows and Connectivity.
5.1 Network Flows
5.2 Menger's Theorem.
5.3 Connectivity.
5.4 Design of Transport Schemes
5.5 Design of Optimal Transport Schemes
5.6 The Chinese Postman Problem
5.7 Construction of Squared Rectangles.

Chapter 6 Matchings and Independent Sets
6.1 Matchings
6.2 Independent Sets
6.3 The Personnel Assignment Problem.
6.4 The Optimal Assignment Problem.
6.5 The Travelling Salesman Problem.

Chapter 7 Colorings and Integer Flows
7.1 Vertex-Colorings.
7.2 Edge-Colorings
7.3 Face-Coloring and Four-Color Problem.
7.4 Integer Flows and Cycle Covers

Chapter 8 Graphs and Groups
8.1 Group Representation of Graphs
8.2 Transitive Graphs
8.3 Graphic Representation of Groups
8.4 Design of Interconnection Networks
Bibliography
List of Notations
Index

精彩書摘

　　《運籌與管理科學叢書：圖論基礎教程》：
　　Chapter 1
　　Basic Concepts of Graphs
　　In many real-world situations， it is particularly convenient to describe the specified relationship between pairs of certain given objects by means of a diagram， in which points represent the objects and （directed or undirected） lines represent the relationship between pairs of the objects. For xample， a national traffic map describes a condition of the communication lines among cities in the country， where the points represent cities and the lines represent the highways or the railways oining pairs of cities. Notice that in such diagrams one is mainly interested in whether or not two
　　given points are joined by a line； the manner in which they are joined is immaterial. A mathematical abstraction of situations of this type gives rise to the concept of a graph.
　　In fact， a graph provides the natural structures from which to construct mathematical models that are appropriate to almost all fields of scientific （natural and social） inquiry. The underlying subject f study in these fields is some set of “objects” and one or more “relations” between the objects.
　　In this chapter， we will introduce the concept and the geometric representation of a graph， erminology and natation， basic operations used in the remaining parts of the book. It should， for the beginner specially， be worth noting that most graph theorists use personalized terminology in their books， papers and lectures. Even the meaning of the word “graph” varies with different authors. We will adopt the most standard terminology and notation extensively used by most authors， such as Bondy
　　and Murty1 [42]， with a subject index and a list of notations in the end of the book.
　　1 J. A. Bondy （John Adrian Bondy） is a professor of University of Waterloo and Universit′e Lyon 1， received his Ph.D. from University of Oxford in 1969. U. S. R. Murty （Uppaluri Siva Ramachandra Murty） is a professor of University of Waterloo， received his Ph.D. from Indian Statistical Institute in 1967. Bondy and Murty served as editors-in-chief of Journal of Combinatorial Theory， Series B （1985-2004， see this journal， 2004， 90（1）：1）. They are well known and respected for many contributions to graph theory. Particularly， their joint textbook Graph Theory with Applications [42] is acclaimed by readers. The book’s clear exposition and careful choice of topics made it widely influential， and for many years it was used as the principal reference for graph theory courses around the world. It is this textbook that plays an important role to standardize the terminology and notation of graphs. In 2008， they published the new book Graph Theory [43].
　　1.1 Graph and Graphical Representation
　　Let V be a non-empty set. An ordered pair （x， y） or an unorder pair xy is often used to denote a binary relation between two elements in V ， where （x， y） denotes a unilateral relation from x to y and xy denotes a bilateral relation between x and y. A set of binary relations on V can be denoted as a subset of V × V ， the Cartesian product of V with itself. Mathematically， a graph1 G is a mathematical structure （V，E）， denoted by G = （V，E）， where E V × V
　　Example 1.1.1 D = （V （D），E（D）） is a graph， where
　　V （D） = {x1， x2， x3， x4， x5} and
　　E（D） = {a1， a2， a3， a4， a5， a6， a7， a8}，
　　and for each i = 1， 2， ， 8， ai is a unilateral relation defined by
　　a1 = （x1， x2）， a2 = （x3， x2）， a3 = （x3， x3）， a4 = （x4， x3），
　　a5 = （x4， x2）， a6 = （x5， x2）， a7 = （x2， x5）， a8 = （x3， x5）.
　　Example 1.1.2 H = （V （H），E（H）） is a graph， where
　　V （H） = {y1， y2， y3， y4， y5} and
　　E（H） = {b1， b2， b3， b4， b5， b6， b7， b8}，
　　and for each i = 1， 2， ， 8， bi is a unilateral relation defined by
　　b1 = （y1， y2）， b2 = （y3， y2）， b3 = （y3， y3）， b4 = （y4， y3），
　　b5 = （y4， y2）， b6 = （y5， y2）， b7 = （y2， y5）， b8 = （y3， y5）.
　　Example 1.1.3 G = （V （G），E（G）） is a graph， where
　　V （G） = {z1， z2， z3， z4， z5， z6} and
　　E（G） = {e1， e2， e3， e4， e5， e6， e7， e8， e9}，
　　and for each i = 1， 2， ， 9， ei is a bilateral relation defined by
　　e1 = z1z2， e2 = z1z4， e3 = z1z6， e4 = z2z3， e5 = z3z4，
　　e6 = z3z6， e7 = z2z5， e8 = z4z5， e9 = z5z6.
　　A graph G = （V，E） can be drawn on the plane. Each element in V is indicated by a point. For clarity， such a point is often depicted as a small circle. For an
　　1 The word “graph” was first used in this sense by J. J. Sylvester （James Joseph Sylvester， 1814-1897） in 1878 （Chemistry and algebra. Nature， 1877-8， 17： 284）. Sylvester was an English mathematician， played a leadership role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University and as founder of the American Journal of Mathematics.
　　……

前言/序言

運籌與管理科學叢書24：A First Course in Graph Theory（圖論基礎教程） [A First Course in Graph Theory] 下載 mobi epub pdf txt 電子書

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