Introductory Graph Theory

by Chartrand, Gary

• ISBN13: 9780486247755
• ISBN10: 0486247759
• Edition: Unabridged
• Format: Paperback
• Copyright: 1984-12-01
• Publisher: Dover Publications

Summary

Clear, lively style covers all basics of theory and application, including mathematical models, elementary concepts of graph theory, transportation problems, connection problems, party problems, diagraphs and mathematical models, games and puzzles, graphs and social psychology, planar graphs and coloring problems, and graphs and other mathematics.

Author Biography

Six Degrees of Paul Erdos
Contrary to popular belief, mathematicians do quite often have fun. Take, for example, the phenomenon of the Erdos number. Paul Erdos (1913–1996), a prominent and productive Hungarian mathematician who traveled the world collaborating with other mathematicians on his research papers. Ultimately, Erdos published about 1,400 papers, by far the most published by any individual mathematician.

About 1970, a group of Erdos's friends and collaborators created the concept of the "Erdos number" to define the "collaborative distance" between Erdos and other mathematicians. Erdos himself was assigned an Erdos number of 0. A mathematician who collaborated directly with Erdos himself on a paper (there are 511 such individuals) has an Erdos number of 1. A mathematician who collaborated with one of those 511 mathematicians would have an Erdos number of 2, and so on — there are several thousand mathematicians with a 2.

From this humble beginning, the mathematical elaboration of the Erdos number quickly became more and more elaborate, involving mean Erdos numbers, finite Erdos numbers, and others. In all, it is believed that about 200,000 mathematicians have an assigned Erdos number now, and 90 percent of the world's active mathematicians have an Erdos number lower than 8. It's somewhat similar to the well-known Hollywood trivia game, Six Degrees of Kevin Bacon. In fact there are some crossovers: Actress-mathematician Danica McKellar, who appeared in TV's The Wonder Years, has an Erdos number of 4 and a Bacon number of 2.

This is all leading up to the fact that Gary Chartrand, author of Dover's Introductory Graph Theory, has an Erdos number of 1 — and is one of many Dover authors who share this honor.

Table of Contents

Chapter 1 Mathematical Models
1.1 Nonmathematical Models
1.2 Mathematical Models
1.3 Graphs
1.4 Graphs as Mathematical Models
1.5 Directed Graphs as Mathematical Models
1.6 Networks as Mathematical Models
Chapter 2 Elementary Concepts of Graph Theory
2.1 The Degree of a Vertex
2.2 Isomorphic Graphs
2.3 Connected Graphs
2.4 Cut-Vertices and Bridges
Chapter 3 Transportation Problems
3.1 The Königsberg Bridge Problem: An Introduction to Eulerian Graphs
3.2 The Salesman's Problem: An Introduction to Hamiltonian Graphs
Chapter 4 Connection Problems
4.1 The Minimal Connector Problem: An Introduction to Trees
*4.2 Trees and Probability
4.3 PERT and the Critical Path Method
Chapter 5 Party Problems
5.1 The Problem of Eccentric Hosts: An Introduction to Ramsey Numbers
5.2 The Dancing Problem: An Introduction to Matching
Chapter 6 Games and Puzzles
6.1 "The Problem of the Four Multicolored Cubes: A Solution to "Instant Insanity"
6.2 The Knight's Tour
6.3 The Tower of Hanoi
6.4 The Three Cannibals and Three Missionaries Problem
Chapter 7 Digraphs and Mathematical Models
7.1 A Traffic System Problem: An Introduction to Orientable Graphs
7.2 Tournaments
7.3 Paired Comparisons and How to Fix Elections
Chapter 8 Graphs and Social Psychology
8.1 The Problem of Balance
8.2 The Problem of Clustering
8.3 Graphs and Transactional Analysis
Chapter 9 Planar Graphs and Coloring Problems
9.1 The Three Houses and Three Utilities Problem: An Introduction to Planar Graphs
9.2 A Scheduling Problem: An Introduction to Chromatic Numbers
9.3 The Four Color Problem
*Chapter 10 Graphs and Other Mathematics
10.1 Graphs and Matrices
10.2 Graphs and Topology
10.3 Graphs and Groups
"Appendix Sets, Relations, Functions, Proofs"
A.1 Sets and Subsets
A.2 Cartesian Products and Relations
A.3 Equivalence Relations
A.4 Functions
A.5 Theorems and Proofs
A.6 Mathematical Induction
"Answers, Hints, and Solutions to Selected Exercises"
Index