# Graph Dynamics

CRC Press, Nov 22, 1995 - Mathematics - 248 pages
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.

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### Contents

 INTRODUCTION 1 v Discrete dynamical systems 11 Fixed graphs 18 Nonincreasing parameters and convergence 30 Invariants 36 Connected components 43 Subgraphdefined operators 57 Constructing infinite periodic graphs 65
 Powerlike operators 97 Miscellaneous tools 104 Intersection graph operators 119 Other subgraphdefined operators 132 Powerlike operators 152 Shrinking or expanding operators 171 Composed operators 181 Digraph operators 195

 Admissible graph posets 72 Roots 80 Decision problems 88