A beginner's guide to graph theory

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 on 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. ——- 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 Title Page\r......Page 3 Copyright Page\r......Page 4 Preface to the Second Edition......Page 6 Preface to the First Edition......Page 7 Outline of Topics......Page 8 Acknowledgments......Page 9 Table of Contents\r......Page 10 List of Figures......Page 13 1.1 Sets, Binary Relations and Graphs......Page 16 1.2 Some Definitions......Page 21 1.3 Degree......Page 28 2.1 Basic Ideas......Page 34 2.2 Radius, Diameter and Eccentricity......Page 38 2.3 Weighted Distance......Page 40 2.4 Euler Walks......Page 44 2.5 Hamilton Cycles......Page 49 2.6 The Traveling Salesman Problem......Page 54 3.1 Cutpoints and Bridges......Page 58 3.2 Blocks......Page 61 3.3 Connectivity......Page 64 4.1 Characterizations of Trees......Page 67 4.2 Spanning Trees......Page 69 4.3 Minimal Spanning Trees......Page 74 5.1 Finite Fields and Vector Spaces......Page 79 5.2 The Power Set as a Vector Space......Page 80 5.3 The Vector Spaces Associated with a Graph......Page 82 5.4 The Cutset Subspace......Page 84 5.5 Bases and Spanning Trees......Page 86 6.1 Definitions; One-Factorizations......Page 91 6.2 Tournament Applications of One-Factorizations......Page 97 6.3 A General Existence Theorem......Page 99 6.4 Graphs Without One-Factors......Page 103 7.1 Vertex Colorings......Page 106 7.2 Brooks' Theorem......Page 110 7.3 Counting Vertex Colorings......Page 112 7.4 Edge-Colorings......Page 117 7.5 Class 2 Graphs......Page 120 8.1 Representations and Crossings......Page 125 8.2 Euler's Formula......Page 128 8.3 Maps, Graphs and Planarity......Page 131 9.1 Introduction; Graceful Labelings......Page 135 9.2 Edge-Magic Total Labeling......Page 138 9.3 Edge-Magic Labelings of Complete Graphs......Page 144 10.1 The Graphical Case of Ramsey's Theorem\r......Page 151 10.2 Ramsey Multiplicity......Page 156 10.3 Application of Sum-Free Sets......Page 158 10.4 Bounds on Classical Ramsey Numbers......Page 161 10.5 The General Case of Ramsey's Theorem......Page 164 11.1 Basic Ideas......Page 166 11.2 Orientations and Tournaments......Page 170 11.3 Directed Euler Walks......Page 174 12.1 Activity Digraphs......Page 177 12.2 Critical Path Analysis......Page 180 12.3 Critical Paths Under Uncertainty......Page 186 13.1 Transportation Networks and Flows\r......Page 190 13.2 Maximal Flows......Page 195 13.3 The Max Flow Min Cut Theorem......Page 201 13.4 The Max Flow Min Cut Algorithm......Page 202 13.5 Supply and Demand Problems......Page 209 14.1 Computation Time......Page 214 14.2 Data Structures......Page 217 14.3 Some Graph Algorithms......Page 218 14.4 Intractability......Page 222 15.1 Preliminaries......Page 225 15.2 Functions on Graphs......Page 226 15.3 Classes of Graphs......Page 228 15.4 Small-World Graphs......Page 230 References......Page 233 Hints......Page 238 Answers and Solutions......Page 241 Index......Page 261 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 on 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. -- 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 Concisely written, gentle introduction to graph theory suitable as a textbook or for self-study Graph-theoretic applications from diverse fields (computer science, engineering, chemistry, management science) 2nd ed. includes new chapters on labeling and communications networks and small worlds, as well as expanded beginner's material Many additional changes, improvements, and corrections resulting from classroom use