Combinatorial Matrix Theory

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Combinatorial Matrix Classes

Richard A. Brualdi
Combinatorial Matrix Classes Book PDF

Author : Richard A. Brualdi
Publisher : Cambridge University Press
Page : 26 pages
File Size : 54,5 Mb
Release : 2006-08-10
Category : Mathematics
ISBN : 9780521865654

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Combinatorial Matrix Classes by Richard A. Brualdi Pdf

A natural sequel to the author's previous book Combinatorial Matrix Theory written with H. J. Ryser, this is the first book devoted exclusively to existence questions, constructive algorithms, enumeration questions, and other properties concerning classes of matrices of combinatorial significance. Several classes of matrices are thoroughly developed including the classes of matrices of 0's and 1's with a specified number of 1's in each row and column (equivalently, bipartite graphs with a specified degree sequence), symmetric matrices in such classes (equivalently, graphs with a specified degree sequence), tournament matrices with a specified number of 1's in each row (equivalently, tournaments with a specified score sequence), nonnegative matrices with specified row and column sums, and doubly stochastic matrices. Most of this material is presented for the first time in book format and the chapter on doubly stochastic matrices provides the most complete development of the topic to date.

Combinatorial Matrix Theory

Richard A. Brualdi,Ángeles Carmona,P. van den Driessche,Stephen Kirkland,Dragan Stevanović
Combinatorial Matrix Theory Book PDF

Author : Richard A. Brualdi,Ángeles Carmona,P. van den Driessche,Stephen Kirkland,Dragan Stevanović
Publisher : Birkhäuser
Page : 219 pages
File Size : 49,7 Mb
Release : 2018-03-31
Category : Mathematics
ISBN : 9783319709536

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Combinatorial Matrix Theory by Richard A. Brualdi,Ángeles Carmona,P. van den Driessche,Stephen Kirkland,Dragan Stevanović Pdf

This book contains the notes of the lectures delivered at an Advanced Course on Combinatorial Matrix Theory held at Centre de Recerca Matemàtica (CRM) in Barcelona. These notes correspond to five series of lectures. The first series is dedicated to the study of several matrix classes defined combinatorially, and was delivered by Richard A. Brualdi. The second one, given by Pauline van den Driessche, is concerned with the study of spectral properties of matrices with a given sign pattern. Dragan Stevanović delivered the third one, devoted to describing the spectral radius of a graph as a tool to provide bounds of parameters related with properties of a graph. The fourth lecture was delivered by Stephen Kirkland and is dedicated to the applications of the Group Inverse of the Laplacian matrix. The last one, given by Ángeles Carmona, focuses on boundary value problems on finite networks with special in-depth on the M-matrix inverse problem.

Combinatorics and Random Matrix Theory

Jinho Baik,Percy Deift,Toufic Suidan
Combinatorics and Random Matrix Theory Book PDF

Author : Jinho Baik,Percy Deift,Toufic Suidan
Publisher : American Mathematical Soc.
Page : 461 pages
File Size : 52,6 Mb
Release : 2016-06-22
Category : Combinatorial analysis
ISBN : 9780821848418

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Combinatorics and Random Matrix Theory by Jinho Baik,Percy Deift,Toufic Suidan Pdf

Over the last fifteen years a variety of problems in combinatorics have been solved in terms of random matrix theory. More precisely, the situation is as follows: the problems at hand are probabilistic in nature and, in an appropriate scaling limit, it turns out that certain key quantities associated with these problems behave statistically like the eigenvalues of a (large) random matrix. Said differently, random matrix theory provides a “stochastic special function theory” for a broad and growing class of problems in combinatorics. The goal of this book is to analyze in detail two key examples of this phenomenon, viz., Ulam's problem for increasing subsequences of random permutations and domino tilings of the Aztec diamond. Other examples are also described along the way, but in less detail. Techniques from many different areas in mathematics are needed to analyze these problems. These areas include combinatorics, probability theory, functional analysis, complex analysis, and the theory of integrable systems. The book is self-contained, and along the way we develop enough of the theory we need from each area that a general reader with, say, two or three years experience in graduate school can learn the subject directly from the text.

Spectral Radius of Graphs

Dragan Stevanovic
Spectral Radius of Graphs Book PDF

Author : Dragan Stevanovic
Publisher : Academic Press
Page : 166 pages
File Size : 54,6 Mb
Release : 2014-10-13
Category : Mathematics
ISBN : 9780128020975

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Spectral Radius of Graphs by Dragan Stevanovic Pdf

Spectral Radius of Graphs provides a thorough overview of important results on the spectral radius of adjacency matrix of graphs that have appeared in the literature in the preceding ten years, most of them with proofs, and including some previously unpublished results of the author. The primer begins with a brief classical review, in order to provide the reader with a foundation for the subsequent chapters. Topics covered include spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem. From this introduction, the book delves deeper into the properties of the principal eigenvector; a critical subject as many of the results on the spectral radius of graphs rely on the properties of the principal eigenvector for their proofs. A following chapter surveys spectral radius of special graphs, covering multipartite graphs, non-regular graphs, planar graphs, threshold graphs, and others. Finally, the work explores results on the structure of graphs having extreme spectral radius in classes of graphs defined by fixing the value of a particular, integer-valued graph invariant, such as: the diameter, the radius, the domination number, the matching number, the clique number, the independence number, the chromatic number or the sequence of vertex degrees. Throughout, the text includes the valuable addition of proofs to accompany the majority of presented results. This enables the reader to learn tricks of the trade and easily see if some of the techniques apply to a current research problem, without having to spend time on searching for the original articles. The book also contains a handful of open problems on the topic that might provide initiative for the reader's research. Dedicated coverage to one of the most prominent graph eigenvalues Proofs and open problems included for further study Overview of classical topics such as spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem

A Combinatorial Approach to Matrix Theory and Its Applications

Richard A. Brualdi,Dragos Cvetkovic
A Combinatorial Approach to Matrix Theory and Its Applications Book PDF

Author : Richard A. Brualdi,Dragos Cvetkovic
Publisher : CRC Press
Page : 288 pages
File Size : 54,7 Mb
Release : 2008-08-06
Category : Mathematics
ISBN : 1420082248

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A Combinatorial Approach to Matrix Theory and Its Applications by Richard A. Brualdi,Dragos Cvetkovic Pdf

Unlike most elementary books on matrices, A Combinatorial Approach to Matrix Theory and Its Applications employs combinatorial and graph-theoretical tools to develop basic theorems of matrix theory, shedding new light on the subject by exploring the connections of these tools to matrices. After reviewing the basics of graph theory, elementary counting formulas, fields, and vector spaces, the book explains the algebra of matrices and uses the König digraph to carry out simple matrix operations. It then discusses matrix powers, provides a graph-theoretical definition of the determinant using the Coates digraph of a matrix, and presents a graph-theoretical interpretation of matrix inverses. The authors develop the elementary theory of solutions of systems of linear equations and show how to use the Coates digraph to solve a linear system. They also explore the eigenvalues, eigenvectors, and characteristic polynomial of a matrix; examine the important properties of nonnegative matrices that are part of the Perron–Frobenius theory; and study eigenvalue inclusion regions and sign-nonsingular matrices. The final chapter presents applications to electrical engineering, physics, and chemistry. Using combinatorial and graph-theoretical tools, this book enables a solid understanding of the fundamentals of matrix theory and its application to scientific areas.

Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs

Jason J. Molitierno
Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs Book PDF

Author : Jason J. Molitierno
Publisher : CRC Press
Page : 425 pages
File Size : 49,7 Mb
Release : 2016-04-19
Category : Computers
ISBN : 9781439863398

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Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs by Jason J. Molitierno Pdf

On the surface, matrix theory and graph theory seem like very different branches of mathematics. However, adjacency, Laplacian, and incidence matrices are commonly used to represent graphs, and many properties of matrices can give us useful information about the structure of graphs.Applications of Combinatorial Matrix Theory to Laplacian Matrices o

Matrices in Combinatorics and Graph Theory

Bolian Liu,Hong-Jian Lai
Matrices in Combinatorics and Graph Theory Book PDF

Author : Bolian Liu,Hong-Jian Lai
Publisher : Springer Science & Business Media
Page : 326 pages
File Size : 55,7 Mb
Release : 2000-10-31
Category : Mathematics
ISBN : 0792364694

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Matrices in Combinatorics and Graph Theory by Bolian Liu,Hong-Jian Lai Pdf

Combinatorics and Matrix Theory have a symbiotic, or mutually beneficial, relationship. This relationship is discussed in my paper The symbiotic relationship of combinatorics and matrix theoryl where I attempted to justify this description. One could say that a more detailed justification was given in my book with H. J. Ryser entitled Combinatorial Matrix Theon? where an attempt was made to give a broad picture of the use of combinatorial ideas in matrix theory and the use of matrix theory in proving theorems which, at least on the surface, are combinatorial in nature. In the book by Liu and Lai, this picture is enlarged and expanded to include recent developments and contributions of Chinese mathematicians, many of which have not been readily available to those of us who are unfamiliar with Chinese journals. Necessarily, there is some overlap with the book Combinatorial Matrix Theory. Some of the additional topics include: spectra of graphs, eulerian graph problems, Shannon capacity, generalized inverses of Boolean matrices, matrix rearrangements, and matrix completions. A topic to which many Chinese mathematicians have made substantial contributions is the combinatorial analysis of powers of nonnegative matrices, and a large chapter is devoted to this topic. This book should be a valuable resource for mathematicians working in the area of combinatorial matrix theory. Richard A. Brualdi University of Wisconsin - Madison 1 Linear Alg. Applies., vols. 162-4, 1992, 65-105 2Camhridge University Press, 1991.

Matrices and Matroids for Systems Analysis

Kazuo Murota
Matrices and Matroids for Systems Analysis Book PDF

Author : Kazuo Murota
Publisher : Springer Science & Business Media
Page : 500 pages
File Size : 47,6 Mb
Release : 1999-11-29
Category : Mathematics
ISBN : 3540660240

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Matrices and Matroids for Systems Analysis by Kazuo Murota Pdf

A matroid is an abstract mathematical structure that captures combinatorial properties of matrices. This book offers a unique introduction to matroid theory, emphasizing motivations from matrix theory and applications to systems analysis. This book serves also as a comprehensive presentation of the theory and application of mixed matrices, developed primarily by the present author in the 1990's. A mixed matrix is a convenient mathematical tool for systems analysis, compatible with the physical observation that "fixed constants" and "system parameters" are to be distinguished in the description of engineering systems. This book will be extremely useful to graduate students and researchers in engineering, mathematics and computer science. From the reviews: "...The book has been prepared very carefully, contains a lot of interesting results and is highly recommended for graduate and postgraduate students." András Recski, Mathematical Reviews Clippings 2000m:93006

Graphs and Matrices

Ravindra B. Bapat
Graphs and Matrices Book PDF

Author : Ravindra B. Bapat
Publisher : Springer
Page : 197 pages
File Size : 53,6 Mb
Release : 2014-09-19
Category : Mathematics
ISBN : 9781447165699

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Graphs and Matrices by Ravindra B. Bapat Pdf

This new edition illustrates the power of linear algebra in the study of graphs. The emphasis on matrix techniques is greater than in other texts on algebraic graph theory. Important matrices associated with graphs (for example, incidence, adjacency and Laplacian matrices) are treated in detail. Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph. Such an extensive coverage of the subject area provides a welcome prompt for further exploration. The inclusion of exercises enables practical learning throughout the book. In the new edition, a new chapter is added on the line graph of a tree, while some results in Chapter 6 on Perron-Frobenius theory are reorganized. Whilst this book will be invaluable to students and researchers in graph theory and combinatorial matrix theory, it will also benefit readers in the sciences and engineering.

Matrix Theory and Applications

Charles R. Johnson
Matrix Theory and Applications Book PDF

Author : Charles R. Johnson
Publisher : American Mathematical Soc.
Page : 272 pages
File Size : 55,6 Mb
Release : 1990
Category : Mathematics
ISBN : 9780821801543

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Matrix Theory and Applications by Charles R. Johnson Pdf

This volume contains the lecture notes prepared for the AMS Short Course on Matrix Theory and Applications, held in Phoenix in January, 1989. Matrix theory continues to enjoy a renaissance that has accelerated in the past decade, in part because of stimulation from a variety of applications and considerable interplay with other parts of mathematics. In addition, the great increase in the number and vitality of specialists in the field has dispelled the popular misconception that the subject has been fully researched.

Combinatorial Matrix Theory

Richard A. Brualdi
Combinatorial Matrix Theory Book PDF

Author : Richard A. Brualdi
Publisher : Unknown
Page : 380 pages
File Size : 40,9 Mb
Release : 1991
Category : Combinatorial analysis
ISBN : 1107094003

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Combinatorial Matrix Theory by Richard A. Brualdi Pdf

This is the first book devoted to the exposition of combinatorial matrix theory. It can be used as a graduate course text, but is complete enough for a standard reference work on the basic theory.

Combinatorial Matrix Theory and Generalized Inverses of Matrices

Ravindra B. Bapat,Steve J. Kirkland,K. Manjunatha Prasad,Simo Puntanen
Combinatorial Matrix Theory and Generalized Inverses of Matrices Book PDF

Author : Ravindra B. Bapat,Steve J. Kirkland,K. Manjunatha Prasad,Simo Puntanen
Publisher : Springer Science & Business Media
Page : 283 pages
File Size : 44,5 Mb
Release : 2013-02-11
Category : Mathematics
ISBN : 9788132210535

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Combinatorial Matrix Theory and Generalized Inverses of Matrices by Ravindra B. Bapat,Steve J. Kirkland,K. Manjunatha Prasad,Simo Puntanen Pdf

This book consists of eighteen articles in the area of `Combinatorial Matrix Theory' and `Generalized Inverses of Matrices'. Original research and expository articles presented in this publication are written by leading Mathematicians and Statisticians working in these areas. The articles contained herein are on the following general topics: `matrices in graph theory', `generalized inverses of matrices', `matrix methods in statistics' and `magic squares'. In the area of matrices and graphs, speci_c topics addressed in this volume include energy of graphs, q-analog, immanants of matrices and graph realization of product of adjacency matrices. Topics in the book from `Matrix Methods in Statistics' are, for example, the analysis of BLUE via eigenvalues of covariance matrix, copulas, error orthogonal model, and orthogonal projectors in the linear regression models. Moore-Penrose inverse of perturbed operators, reverse order law in the case of inde_nite inner product space, approximation numbers, condition numbers, idempotent matrices, semiring of nonnegative matrices, regular matrices over incline and partial order of matrices are the topics addressed under the area of theory of generalized inverses. In addition to the above traditional topics and a report on CMTGIM 2012 as an appendix, we have an article on old magic squares from India.

Combinatorial Matrix Theory

Richard A. Brualdi,Herbert John Ryser
Combinatorial Matrix Theory Book PDF

Author : Richard A. Brualdi,Herbert John Ryser
Publisher : Unknown
Page : 380 pages
File Size : 40,9 Mb
Release : 2014-05-14
Category : MATHEMATICS
ISBN : 1107087759

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Combinatorial Matrix Theory by Richard A. Brualdi,Herbert John Ryser Pdf

This book, first published in 1991, is devoted to the exposition of combinatorial matrix theory.

Introduction to Modern Algebra and Matrix Theory

O. Schreier,E. Sperner
Introduction to Modern Algebra and Matrix Theory Book PDF

Author : O. Schreier,E. Sperner
Publisher : Courier Corporation
Page : 402 pages
File Size : 45,9 Mb
Release : 2013-05-13
Category : Mathematics
ISBN : 9780486278650

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Introduction to Modern Algebra and Matrix Theory by O. Schreier,E. Sperner Pdf

This unique text provides students with a basic course in both calculus and analytic geometry. It promotes an intuitive approach to calculus and emphasizes algebraic concepts. Minimal prerequisites. Numerous exercises. 1951 edition.

An Introduction to Random Matrices

Greg W. Anderson,Alice Guionnet,Ofer Zeitouni
An Introduction to Random Matrices Book PDF

Author : Greg W. Anderson,Alice Guionnet,Ofer Zeitouni
Publisher : Cambridge University Press
Page : 507 pages
File Size : 40,9 Mb
Release : 2010
Category : Mathematics
ISBN : 9780521194525

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An Introduction to Random Matrices by Greg W. Anderson,Alice Guionnet,Ofer Zeitouni Pdf

A rigorous introduction to the basic theory of random matrices designed for graduate students with a background in probability theory.