Graphical Enumeration
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Graphical Enumeration deals with the enumeration of various kinds of graphs. Topics covered range from labeled enumeration and George Pólya's theorem to rooted and unrooted trees, graphs and digraphs, and power group enumeration. Superposition, blocks, and asymptotics are also discussed. A number of unsolved enumeration problems are presented. Comprised of 10 chapters, this book begins with an overview of labeled graphs, followed by a description of the basic enumeration theorem of Pólya. The next three chapters count an enormous variety of trees, graphs, and digraphs. The Power Group Enumeration Theorem is then described together with some of its applications, including the enumeration of self-complementary graphs and digraphs and finite automata. Two other chapters focus on the counting of superposition and blocks, while another chapter is devoted to asymptotic numbers that are developed for several different graphical structures. The book concludes with a comprehensive definitive list of unsolved graphical enumeration problems. This monograph will be of interest to both students and practitioners of mathematics.
Table of Contents
Preface
1 Labeled Enumeration
1.1 The Number of Ways to Label a Graph
1.2 Connected Graphs
1.3 Blocks
1.4 Eulerian Graphs
1.5 The Number of k-Colored Graphs
1.6 Acyclic Digraphs
1.7 Trees
1.8 Eulerian Trails in Digraphs
Exercises
2 Pólya's Theorem
2.1 Groups and Graphs
2.2 The Cycle Index of a Permutation Group
2.3 Burnside's Lemma
2.4 Pólya's Theorem
2.5 The Special Figure Series 1 + x
2.6 One-One Functions
Exercises
3 Trees
3.1 Rooted Trees
3.2 Unrooted Trees
3.3 Trees with Specified Properties
3.4 Treelike Graphs
3.5 Two-Trees
Exercises
4 Graphs
4.1 Graphs
4.2 Connected Graphs
4.3 Bicolored Graphs
4.4 Rooted Graphs
4.5 Supergraphs and Colored Graphs
4.6 Boolean Functions
4.7 Eulerian Graphs
Exercises
5 Digraphs
5.1 Digraphs
5.2 Tournaments
5.3 Orientations of a Graph
5.4 Mixed Graphs
Exercises
6 Power Group Enumeration
6.1 Power Group Enumeration Theorem
6.2 Self-Complementary Graphs
6.3 Functions with Weights
6.4 Graphs with Colored Lines
6.5 Finite Automata
6.6 Self-Converse Digraphs
Exercises
7 Superposition
7.1 Redfield's Enumeration Theorem
7.2 Redfield's Decomposition Theorem
7.3 Graphs and Digraphs
7.4 A Generalization of Redfield's Enumeration Theorem
7.5 General Graphs
Exercises
8 Blocks
8.1 A Generalization of Redfield's Lemma
8.2 The Composition Group
8.3 The Composition Theorem
8.4 Connected Graphs
8.5 Cycle Index Sums for Rooted Graphs
8.6 Blocks
8.7 Graphs with Given Blocks
8.8 Acyclic Digraphs
Exercises
9 Asymptotics
9.1 Graphs
9.2 Digraphs
9.3 Graphs with a Given Number of Points and Lines
9.4 Connected Graphs and Blocks
9.5 Trees
Exercises
10 Unsolved Problems
10.1 Labeled Graphs
10.2 Digraphs
10.3 Graphs with Given Structural Properties
10.4 Graphs with Given Parameter
10.5 Subgraphs of a Given Graph
10.6 Supergraphs of a Given Graph
10.7 Graphs and Coloring
10.8 Variations on Graphs
Appendixes
I
II
III
Bibliography
Index
Product details
- No. of pages: 286
- Language: English
- Copyright: © Academic Press 1973
- Published: May 28, 1973
- Imprint: Academic Press
- eBook ISBN: 9781483273785