## A Beginner's Guide to Graph TheoryGraph 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. ----- From a review of the first edition:
—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 |

255 | |