A Beginner's Guide to Graph Theory

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Springer Science & Business Media, Jun 8, 2007 - Mathematics - 260 pages
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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.

Key features:

* Introductory chapters present the main ideas and topics in graph theory—walks, paths and cycles, radius, diameter, eccentricity, cuts and connectivity, trees

* Subsequent chapters examine specialized topics and applications

* Numerous examples and illustrations

* Comprehensive index and bibliography, with suggested literature for more advanced material

New to the second edition:

* New chapters on labeling and communications networks and small-worlds

* Expanded beginner’s material in the early chapters, including more examples, exercises, hints and solutions to key problems

* Many additional changes, improvements, and corrections throughout resulting from classroom use and feedback

Striking a balance between a theoretical and practical approach 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.

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From a review of the first edition:

"Altogether the book gives a comprehensive introduction to graphs, their theory and their application...The 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

 

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Contents

1 Graphs
1
12 Some Definitions
6
13 Degree
13
2 Walks Paths and Cycles
19
22 Radius Diameter and Eccentricity
23
23 Weighted Distance
25
24 Euler Walks
29
25 Hamilton Cycles
34
9 Labeling
123
92 EdgeMagic Total Labeling
126
93 EdgeMagic Labelings of Complete Graphs
132
10 Ramsey Theory
139
102 Ramsey Multiplicity
144
103 Application of SumFree Sets
146
104 Bounds on Classical Ramsey Numbers
149
105 The General Case of Ramseys Theorem
152

26 The Traveling Salesman Problem
39
3 Connectivity
43
32 Blocks
46
33 Connectivity
49
4 Trees
53
42 Spanning Trees
55
43 Minimal Spanning Trees
60
5 Linear Spaces Associated with Graphs
65
52 The Power Set as a Vector Space
66
53 The Vector Spaces Associated with a Graph
68
54 The Cutset Subspace
70
55 Bases and Spanning Trees
72
6 Factorizations
77
62 Tournament Applications of OneFactorizations
83
63 A General Existence Theorem
85
64 Graphs Without OneFactors
89
7 Graph Colorings
93
72 Brooks Theorem
97
73 Counting Vertex Colorings
99
74 EdgeColorings
104
75 Class 2 Graphs
107
8 Planarity
113
82 Eulers Formula
116
83 Maps Graphs and Planarity
119
11 Digraphs
155
112 Orientations and Tournaments
159
113 Directed Euler Walks
163
12 Critical Paths
167
122 Critical Path Analysis
170
123 Critical Paths Under Uncertainty
176
13 Flows in Networks
181
132 Maximal Flows
186
133 The Max Flow Min Cut Theorem
192
134 The Max Flow Min Cut Algorithm
193
135 Supply and Demand Problems
200
14 Computational Considerations
205
142 Data Structures
208
143 Some Graph Algorithms
209
144 Intractability
213
15 Communications Networks and SmallWorlds
217
152 Functions on Graphs
218
153 Classes of Graphs
220
154 SmallWorld Graphs
222
References
225
Hints
231
Answers and Solutions
235
Index
255
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Magic Graphs
W. D. Wallis
Limited preview - 2001
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