# A Beginner's Guide to Graph Theory

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

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 Copyright