# Combinatorics of Nonnegative Matrices

ISBN-10: 082182788X

ISBN-13: 9780821827888

Pub. Date: 08/13/2002

Publisher: American Mathematical Society

In this book, the authors focus on the relation of matrices with nonnegative elements to various mathematical structures studied in combinatorics. In addition to applications in graph theory, Markov chains, tournaments, and abstract automata, the authors consider relations between nonnegative matrices and structures such as coverings and minimal coverings of sets

## Overview

In this book, the authors focus on the relation of matrices with nonnegative elements to various mathematical structures studied in combinatorics. In addition to applications in graph theory, Markov chains, tournaments, and abstract automata, the authors consider relations between nonnegative matrices and structures such as coverings and minimal coverings of sets by families of subsets. They also give considerable attention to the study of various properties of matrices and to the classes formed by matrices with a given structure. The authors discuss enumerative problems using both combinatorial and probabilistic methods. They also consider extremal problems related to matrices and problems where nonnegative matrices provide suitable investigative tools. The book contains some classical theorems and a significant number of results not previously published in monograph form, including results obtained by the authors in the last few years. It is appropriate for graduate students, researchers, and engineers interested in combinatorics and its applications.

## Product Details

ISBN-13:
9780821827888
Publisher:
American Mathematical Society
Publication date:
08/13/2002
Series:
Translations of Mathematical Monographs, #213
Pages:
269
Product dimensions:
72.50(w) x 10.00(h) x 7.50(d)

## Related Subjects

 Preface vii List of Notation ix Chapter 1. Matrices and Configurations 1 Introduction 1 1.1. Definitions and examples 2 1.2. Term rank. Arrangement of positive elements 9 1.3. Combinatorial theory of cyclic matrices 27 Chapter 2. Ryser Classes 45 Introduction 45 2.1. A constructive description of Ryser classes 46 2.2. Invariant sets 58 2.3. Estimates of the term rank 69 Chapter 3. Nonnegative Matrices and Extremal Combinatorial Problems 83 Introduction 83 3.1. Forbidden configurations 84 3.2. Covering problem 90 3.3. The van der Waerden-Egorychev-Falikman Theorem 106 Chapter 4. Asymptotic Methods in the Study of Nonnegative Matrices 117 Introduction 117 4.1. Nonnegative matrices and graphs 118 4.2. Asymptotics of the number of primitive (0, 1)-matrices 131 4.3. Asymptotics of the permanent of a random (0, 1)-matrix 135 4.4. Random lattices and Boolean algebras 138 4.5. Coverings of sets and (0, 1)-matrices 143 4.6. Random coverings of sets 151 Chapter 5. Totally Indecomposable, Chainable, and Prime Matrices 159 Introduction 159 5.1. Totally indecomposable and chainable matrices 161 5.2. Rectangular nonnegative matrices 170 5.3. Rectangular nonnegative chainable matrices 184 5.4. Extension of partial diagonals 192 5.5. Prime Boolean matrices 199 5.6. Prime nonnegative matrices 208 Chapter 6. Sequences of Nonnegative Matrices 213 Introduction 213 6.1. Directed graphs of nonnegative matrices 215 6.2. Irreducible and primitive matrices 221 6.3. Tournament matrices 227 6.4. Associated operator 232 6.5. Sequences of powers of a nonnegative matrix 243 6.6. Ergodicity of sequences of nonnegative matrices 250 Bibliography 263 Index 267

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