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E-raamat: Beginner's Guide to Graph Theory

  • Formaat: PDF+DRM
  • Ilmumisaeg: 05-May-2010
  • Kirjastus: Birkhauser Boston Inc
  • Keel: eng
  • ISBN-13: 9780817645809
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  • Formaat: PDF+DRM
  • Ilmumisaeg: 05-May-2010
  • Kirjastus: Birkhauser Boston Inc
  • Keel: eng
  • ISBN-13: 9780817645809
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Graph theory continues to be one of the fastest growing areas of modern mathematics because of its wide applicability in such diverse disciplines as computer science, engineering, chemistry, management science, social science, and resource planning. Graphs arise as mathematical models in these fields, and the theory of graphs provides a spectrum of methods of proof. This concisely written textbook is intended for an introductory course in graph theory for undergraduate mathematics majors or advanced undergraduate and graduate students from the many fields that benefit from graph-theoretic applications.

This second edition includes new chapters on labeling and communications networks and small-worlds, as well as expanded beginner's material in the early chapters, including more examples, exercises, hints and solutions to key problems. Many additional changes, improvements, and corrections resulting from classroom use and feedback have been added throughout. With a distinctly applied flavor, this gentle introduction to graph theory consists of carefully chosen topics to develop graph-theoretic reasoning for a mixed audience. Familiarity with the basic concepts of set theory, along with some background in matrices and algebra, and a little mathematical maturity are the only prerequisites.

Arvustused

From the reviews:

"Altogether the book gives a comprehensive introduction to graphs, their theory and their applicationThe use of the text is optimized when the exercises are solved. The obtained skills improve understanding of graph theory as well It is very useful that the solutions of these exercises are collected in an appendix." (Simulation News Europe)

From the reviews of the second edition:

"This book is a gentle introduction to graph theory, presenting the main ideas and topics . It is accessible to everyone . This introductory book is addressed to a mixed audience undergraduate mathematics majors, computer scientists, engineers . this book is ideal as well for self-reading. The style is always concise and the essential techniques are well highlighted . It is highly recommended to any student, or working scientist, wishing to explore for the first time this fascinating area of mathematics." (Fabio Mainardi, The Mathematical Association of America, August, 2009)

This book is intended as an introductory course in Graph Theory, one of the fastest growing disciplines of modern Mathematics. The book is nicely written, the presentation is comprehensible but at the same time mathematically precise. The text is supplemented with many figures, with historical notes to many topics and with many examples. Summarizing, this is a nice book, useful not only as an introductory reading for beginners in Graph Theory, but also for those who teach introductory courses in Graph Theory. (Zdenk Ryjáek, Mathematica Bohemica, Issue 2, 2010)

Preface to the Second Edition vii
Preface to the First Edition ix
List of Figures
xvii
Graphs
1(18)
Sets, Binary Relations and Graphs
1(5)
Some Definitions
6(7)
Degree
13(6)
Walks, Paths and Cycles
19(24)
Basic Ideas
19(4)
Radius, Diameter and Eccentricity
23(2)
Weighted Distance
25(4)
Euler Walks
29(5)
Hamilton Cycles
34(5)
The Traveling Salesman Problem
39(4)
Connectivity
43(10)
Cutpoints and Bridges
43(3)
Blocks
46(3)
Connectivity
49(4)
Trees
53(12)
Characterizations of Trees
53(2)
Spanning Trees
55(5)
Minimal Spanning Trees
60(5)
Linear Spaces Associated with Graphs
65(12)
Finite Fields and Vector Spaces
65(1)
The Power Set as a Vector Space
66(2)
The Vector Spaces Associated with a Graph
68(2)
The Cutset Subspace
70(2)
Bases and Spanning Trees
72(5)
Factorizations
77(16)
Definitions; One-Factorizations
77(6)
Tournament Applications of One-Factorizations
83(2)
A General Existence Theorem
85(4)
Graphs Without One-Factors
89(4)
Graph Colorings
93(20)
Vertex Colorings
93(4)
Brooks' Theorem
97(2)
Counting Vertex Colorings
99(5)
Edge-Colorings
104(3)
Class 2 Graphs
107(6)
Planarity
113(10)
Representations and Crossings
113(3)
Euler's Formula
116(3)
Maps, Graphs and Planarity
119(4)
Labeling
123(16)
Introduction; Graceful Labelings
123(3)
Edge-Magic Total Labeling
126(6)
Edge-Magic Labelings of Complete Graphs
132(7)
Ramsey Theory
139(16)
The Graphical Case of Ramsey's Theorem
139(5)
Ramsey Multiplicity
144(2)
Application of Sum-Free Sets
146(3)
Bounds on Classical Ramsey Numbers
149(3)
The General Case of Ramsey's Theorem
152(3)
Digraphs
155(12)
Basic Ideas
155(4)
Orientations and Tournaments
159(4)
Directed Euler Walks
163(4)
Critical Paths
167(14)
Activity Digraphs
167(3)
Critical Path Analysis
170(6)
Critical Paths Under Uncertainty
176(5)
Flows in Networks
181(24)
Transportation Networks and Flows
181(5)
Maximal Flows
186(6)
The Max Flow Min Cut Theorem
192(1)
The Max Flow Min Cut Algorithm
193(7)
Supply and Demand Problems
200(5)
Computational Considerations
205(12)
Computation Time
205(3)
Data Structures
208(1)
Some Graph Algorithms
209(4)
Intractability
213(4)
Communications Networks and Small-Worlds
217(8)
Preliminaries
217(1)
Functions on Graphs
218(2)
Classes of Graphs
220(2)
Small-World Graphs
222(3)
References 225(6)
Hints 231(4)
Answers and Solutions 235(20)
Index 255