Graph Dynamics

1st Edition

Chapman and Hall/CRC

248 pages

Hardback: 9780582286962
pub: 1995-11-22
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Description

Study of "graph operators" or "graph-valued functions" such as the line graph, the clique graph, the complement, and powers, raises several immediate questions: Which graphs are fixed under the operator? Which graphs appear as images of graphs? What happens if the operator is iterated? Over the last 30 years these questions have been answered and methods developed for particular operators in literally hundreds of papers on the subject. Nowhere, however, could one find a comprehensive treatment-a unification of terminology, questions, and methods.

Graph Dynamics provides that comprehensive treatment. Its purpose is threefold: it serves as an introductory textbook on the topic, offers an encyclopedic survey of the literature, and reports recent research-both new tools and results on concrete operators. Part I explicitly presents graph dynamics general theory, stating general principles illustrated by application to graph operators. Part 2 addresses the operators themselves. It lists all known graph operators grouped together in families and recounts, with complete references, all that is known about the dynamical behavior of these concrete operators.

Graph Dynamics is the book you need if you are looking for information on a particular operator, need a text for advanced students, or want to review collected research results presented with a common terminology. It is clearly an essential resource for anyone working in or studying algebra, combinatorics, or graph theory.

Reviews

"…provides a mine of information concerning graph-valued functions or graph operators…the book as a whole furnishes an excellent background for any researcher in the field of graph theory."

-Optimization, April 1996

"…recommended to all that are seriously interested in graph theory."

-Zentralblatt für Mathematik No. 848

Introduction

Part I: THEORY FOR GENERAL OPERATORS

Discrete Dynamical Systems

Fixed Graphs

Increasing Parameters, Divergence, and Depth

Non-Increasing Parameters and Convergence

Invariants

Connected Components

Subgraph-Defined Operators

Constructing Infinite Periodic Graphs

Roots

Decision Problems

Powerlike Operators

Miscellaneous Tools

PART II: CONCRETE OPERATORS

Intersection Graph Operators

Other Subgraph-Defined Operators

Powerlike Operators

Shrinking or Expanding Operators

Composed Operators

Digraph Operators