Graph Theory Modeling, Applications, and Algorithms

ISBN-10: 0131423843
ISBN-13: 9780131423848
Edition: 2007
List price: $117.80
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Description: Once considered an ldquo;unimportantrdquo; branch of topology, graph theory has come into its own through many important contributions to a wide range of fields and is now one of the fastest-growing areas in discrete mathematics and computer  More...

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Book details

List price: $117.80
Copyright year: 2007
Publisher: Prentice Hall PTR
Publication date: 9/22/2006
Binding: Paperback
Pages: 464
Size: 7.00" wide x 9.50" long x 1.00" tall
Weight: 1.782
Language: English

Once considered an ldquo;unimportantrdquo; branch of topology, graph theory has come into its own through many important contributions to a wide range of fields and is now one of the fastest-growing areas in discrete mathematics and computer science. This practical, intuitive book introduces basic concepts, definitions, theorems, and examples from graph theory. Presents a collection of interesting results from mathematics that involve key concepts and proof techniques. Covers design and analysis of computer algorithms for solving problems in graph theory. Discusses applications of graph theory to the sciences. Includes a collection of graph algorithms, written in Java, that are ready for compiling and running. For anyone interested in learning graph theory, discrete structures, or algorithmic design for graph problems.

Preface
Introduction to Graph Theory
Introduction
Why Study Graphs?
Mathematical Preliminaries
The Definition of a Graph
Examples of Common Graphs
Degrees and Regular Graphs
Subgraphs
The Definition of a Directed Graph
Indegrees and Outdegrees in a Digraph
Exercises
Basic Concepts in Graph Theory
Paths and Cycles
Connectivity
Homomorphisms and Isomorphisms of Graphs
More on Isomorphisms on Simple Graphs
Formations and Minors of Graphs
Homomorphisms and Isomorphisms for Digraphs
Digraph Connectivity
Exercises
Trees and Forests
Trees and Some of Their Basic Properties
Characterizations of Trees
Inductive Proofs on Trees
Erdos-Szekeres Theorem on Sequences
Centers in Trees
Rooted Trees
Binary Trees
Levels in Rooted and Binary Trees
Exercises
Spanning Trees
Spanning Trees and Forests
Spanning Trees of the Complete Graph
The Adjacency Matrix of a Graph
The Incidence Matrix of a Graph
The Matrix-Tree Theorem
An Application to Electrical Networks
Minimum Cost Spanning Trees
Exercises
Fundamental Properties of Graphs and Digraphs
Bipartite Graphs
Eulerian Graphs
Hamiltonian Graphs
Hamiltonian Cycles in Weighted Graphs
Eulerian and Hamiltonian Digraphs
Tournament Digraphs
On the Adjacency Matrix of a Digraph
Acyclic Digraphs and Posets
Exercises
Connectivity and Flow
Edge Cuts
Edge Connectivity and Connectivity
Blocks in Separable Graphs
Flows in Networks
The Theorems of Menger
Exercises
Planar Graphs
Embeddings in Surfaces
More on Planar Embeddings
Euler's Formula and Consequences
Characterization of Planar Graphs
Kuratowski and Wagner's Theorem
Plane Duality
Higher Genus
Generalization of Euler's Formula
Crossing Number
Exercises
Graph Coloring
The Chromatic Number of a Graph
Multipartite Graphs
Results for General Graphs
Planar Graphs and Other Surface Graphs
Edge Coloring of a Graph
Tait's Theorem
Exercises
Coloring Enumerations and Chordal Graphs
The Chromatic Polynomial of a Graph
Basic Properties of the Chromatic Polynomial
Interval and Intersection Graphs
Chordal Graphs
Powers of Graphs
Exercises
Independence, Dominance, and Matchings
Independence of Vertices
Domination of Vertices
Matchings in a Graph
Hall's Marriage Theorem
Exercises
Cover Parameters and Matching Polynomials
Covers and Related Parameters
Rook Polynomials and Bipartite Graphs
The Matching Defect Polynomial
Matching Algorithms
Exercises
Graph Counting
Introduction
Basic Counting Results
Generating Functions
Partitions of a Finite Set
The Labeled Counting Lemma
The Exponential Formula
The Number Two and Related Graphs
Two-Regular Graphs
Two-Colorable Graphs
Even Graphs
Exercises
Graph Algorithms
Introduction
Recap of Algorithms Already Presented
Algorithm Efficiency
Breadth-First Search
Depth-First Search
Connected Components
Dijkstra's Shortest Path Algorithm
Java Source Code
Exercises
Appendices
Greek Alphabet
Notation
Top Ten Online References
Bibliography
Index

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