Graphical Enumeration - 1st Edition - ISBN: 9780123242457, 9781483273785

Graphical Enumeration

1st Edition

Authors: Frank Harary Edgar M. Palmer
eBook ISBN: 9781483273785
Imprint: Academic Press
Published Date: 28th May 1973
Page Count: 286
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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


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


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


3 Trees

3.1 Rooted Trees

3.2 Unrooted Trees

3.3 Trees with Specified Properties

3.4 Treelike Graphs

3.5 Two-Trees


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


5 Digraphs

5.1 Digraphs

5.2 Tournaments

5.3 Orientations of a Graph

5.4 Mixed Graphs


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


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


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


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


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








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© Academic Press 1973
Academic Press
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About the Author

Frank Harary

Edgar M. Palmer

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