Graphical Enumeration

1st Edition - May 28, 1973
Authors: Frank Harary, Edgar M. Palmer
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 7 3 7 8 - 5

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,… Read more

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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.

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