Graphs and Matrices

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.

Author: Ravindra B. Bapat

Publisher: Springer

ISBN: 9781447165699

Category: Mathematics

Page: 193

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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.
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Graphs and Matrices

Whilst this book will be invaluable to students and researchers in graph theory and combinatorial matrix theory who want to be acquainted with matrix theoretic ideas used in graph theory, it will also benefit a wider, cross-disciplinary ...

Author: Ravindra B. Bapat

Publisher: Springer

ISBN: 1848829809

Category: Mathematics

Page: 171

View: 914

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Graphs and Matrices provides a welcome addition to the rapidly expanding selection of literature in this field. As the title suggests, the book’s primary focus is graph theory, with an emphasis on topics relating to linear algebra and matrix theory. Information is presented at a relatively elementary level with the view of leading the student into further research. In the first part of the book matrix preliminaries are discussed and the basic properties of graph-associated matrices highlighted. Further topics include those of graph theory such as regular graphs and algebraic connectivity, Laplacian eigenvalues of threshold graphs, positive definite completion problem and graph-based matrix games. Whilst this book will be invaluable to researchers in graph theory, it may also be of benefit to a wider, cross-disciplinary readership.
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Graphs Matrices and Designs

This book is designed to be of interest to applied mathematicians, computer scientists and communications researchers.

Author: Rees

Publisher: CRC Press

ISBN: 0824787900

Category: Mathematics

Page: 344

View: 233

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Examines partitions and covers of graphs and digraphs, latin squares, pairwise balanced designs with prescribed block sizes, ranks and permanents, extremal graph theory, Hadamard matrices and graph factorizations. This book is designed to be of interest to applied mathematicians, computer scientists and communications researchers.
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The Mutually Beneficial Relationship of Graphs and Matrices

This book is not a comprehensive study of graphs and matrices. The particular content of the lectures was chosen for its accessibility, beauty, and current relevance, and for the possibility of enticing the audience to want to learn more.

Author: Richard A. Brualdi

Publisher: American Mathematical Soc.

ISBN: 9780821853153

Category: Mathematics

Page: 96

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Graphs and matrices enjoy a fascinating and mutually beneficial relationship. This interplay has benefited both graph theory and linear algebra. In one direction, knowledge about one of the graphs that can be associated with a matrix can be used to illuminate matrix properties and to get better information about the matrix. Examples include the use of digraphs to obtain strong results on diagonal dominance and eigenvalue inclusion regions and the use of the Rado-Hall theorem to deduce properties of special classes of matrices. Going the other way, linear algebraic properties of one of the matrices associated with a graph can be used to obtain useful combinatorial information about the graph. The adjacency matrix and the Laplacian matrix are two well-known matrices associated to a graph, and their eigenvalues encode important information about the graph. Another important linear algebraic invariant associated with a graph is the Colin de Verdiere number, which, for instance, characterizes certain topological properties of the graph. This book is not a comprehensive study of graphs and matrices. The particular content of the lectures was chosen for its accessibility, beauty, and current relevance, and for the possibility of enticing the audience to want to learn more.
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Graph Theory

Requiring knowledge of the basic concepts of graph theory and a familiarity with some simple results, the book also includes 100 exercises with solutions to help readers gain experience and 131 diagrams to aid in the understanding of ...

Author: B Andrasfai

Publisher: CRC Press

ISBN: 0852742223

Category: Mathematics

Page: 280

View: 836

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Graph Theory: Flows, Matrices covers a number of topics in graph theory that are important in the major areas of application. It provides graph theoretic tools that can be readily and efficiently applied to problems in operational research, computer science, electrical engineering, and economics. Emphasizing didactic principles, the book derives theorems and proofs from a detailed analysis of the structure of graphs. The easy-to-follow algorithms can be readily converted to computer codes in high-level programming languages. Requiring knowledge of the basic concepts of graph theory and a familiarity with some simple results, the book also includes 100 exercises with solutions to help readers gain experience and 131 diagrams to aid in the understanding of concepts and proofs.
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Matrices And Graphs Theory And Applications To Economics Proceedings Of The Conferences

This volume discusses applications on graphs to the analysis of both causal structure of econometric models and input/output matrices; the relationships between general linear models or covariance and graphical models; the characterization ...

Author: Camiz Sergio

Publisher: World Scientific

ISBN: 9789814546492

Category:

Page: 256

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This volume discusses applications on graphs to the analysis of both causal structure of econometric models and input/output matrices; the relationships between general linear models or covariance and graphical models; the characterization of irreducible matrices through graphs; computational matters of eigenvalues of non-negative and symmetrical matrices; qualitative analysis and the sign theorem; topics on the spectrum distribution for real matrices.
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Algebraic Graph Theory

This highly self-contained book about algebraic graph theory is written with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject.

Author: Ulrich Knauer

Publisher: Walter de Gruyter

ISBN: 9783110255096

Category: Mathematics

Page: 324

View: 932

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This is a highly self-contained book about algebraic graph theory which is written with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject. The focus is on homomorphisms and endomorphisms, matrices and eigenvalues. Graph models are extremely useful for almost all applications and applicators as they play an important role as structuring tools. They allow to model net structures - like roads, computers, telephones - instances of abstract data structures - like lists, stacks, trees - and functional or object oriented programming.
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Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs

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 ...

Author: Jason J. Molitierno

Publisher: CRC Press

ISBN: 9781439863398

Category: Computers

Page: 425

View: 175

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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
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Graphs and Matrices

1 () –3 2 3 0 –1 –2 0 2 () –2 2.60 – 1 Incidence matrix We now consider the
incidence matrix of an undirected graph. Let G be a graph with V(G) = {1, ..., n}
and E(G) = {e1, ..., em). The (vertex-edge) incidence matrix of G, which we denote
by ...

Author: Ravindra B. Bapat

Publisher: Springer Science & Business Media

ISBN: 9781848829817

Category: Mathematics

Page: 171

View: 275

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Graphs and Matrices provides a welcome addition to the rapidly expanding selection of literature in this field. As the title suggests, the book’s primary focus is graph theory, with an emphasis on topics relating to linear algebra and matrix theory. Information is presented at a relatively elementary level with the view of leading the student into further research. In the first part of the book matrix preliminaries are discussed and the basic properties of graph-associated matrices highlighted. Further topics include those of graph theory such as regular graphs and algebraic connectivity, Laplacian eigenvalues of threshold graphs, positive definite completion problem and graph-based matrix games. Whilst this book will be invaluable to researchers in graph theory, it may also be of benefit to a wider, cross-disciplinary readership.
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Combinatorial Matrix Theory and Generalized Inverses of Matrices

This book consists of eighteen articles in the area of `Combinatorial Matrix Theory' and `Generalized Inverses of Matrices'.

Author: Ravindra B. Bapat

Publisher: Springer Science & Business Media

ISBN: 9788132210535

Category: Mathematics

Page: 277

View: 123

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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.
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Graph Theory and Sparse Matrix Computation

This book will be a resource for the researcher or advanced student of either graphs or sparse matrices; it will be useful to mathematicians, numerical analysts and theoretical computer scientists alike.

Author: Alan George

Publisher: Springer Science & Business Media

ISBN: 9781461383697

Category: Mathematics

Page: 245

View: 928

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When reality is modeled by computation, matrices are often the connection between the continuous physical world and the finite algorithmic one. Usually, the more detailed the model, the bigger the matrix, the better the answer, however, efficiency demands that every possible advantage be exploited. The articles in this volume are based on recent research on sparse matrix computations. This volume looks at graph theory as it connects to linear algebra, parallel computing, data structures, geometry, and both numerical and discrete algorithms. The articles are grouped into three general categories: graph models of symmetric matrices and factorizations, graph models of algorithms on nonsymmetric matrices, and parallel sparse matrix algorithms. This book will be a resource for the researcher or advanced student of either graphs or sparse matrices; it will be useful to mathematicians, numerical analysts and theoretical computer scientists alike.
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Probabilistic and Extremal Behavior in Graphs and Matrices

This is asymptotically best possible, as one can easily make any two rows proportional with at most m/2 changes. Moreover, this theorem gives an asymptotic solution to a slightly weakened version of a conjecture made by Van Vu in [Vu08].

Author: Gweneth Ann McKinley

Publisher:

ISBN: OCLC:1191267375

Category:

Page: 82

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This thesis deals with several related questions in probabilistic and extremal graph theory and discrete random matrix theory. First, for any bipartite graph H containing a cycle, we prove an upper bound of [mathematical equation] on the number of labeled H-free graphs on n vertices, given only a fairly natural assumption on the growth rate of. Bounds of the form [mathematical equation] have been proven only for relatively few special graphs H, often with considerable difficulty, and our result unifies all previously known special cases. Next, we give a variety of bounds on the clique numbers of random graphs arising from the theory of graphons. A graphon is a symmetric measurable function [mathematical equation], and each graphon gives rise naturally to a random graph distribution, denoted G(n, W ), that can be viewed as a generalization of the Erdős-Ré́nyi random graph. Recently, Doležal, Hladký, and Máthé gave an asymptotic formula of order log n for the clique number of G(n, W ) when W is bounded away from 0 and 1. We show that if W is allowed to approach 1 at a finite number of points, and displays a moderate rate of growth near these points, then the clique number of G(n, W) will be [theta]([square root n]) almost surely. We also give a family of examples with clique number [theta](n[superscript alpha]) for any [alpha] [element symbol] (0, 1) , and some conditions under which the clique number of G(n, W ) will be [omicron]([square root]n), [lower case omega]([square root]n), or [upper case omega]([superscript alpha]) for [alpha] [element symbol] (0, 1). Finally, for an nxm matrix M of independent Rademacher (±1) random variables, it is well known that if n /- m, then M is of full rank with high probability; we show that this property is resilient to adversarial changes to M. More precisely, if m /- n + n[superscript 1-[epsilon]/6], then even after changing the sign of (1 - [epsilon])m/2 entries, M is still of full rank with high probability. This is asymptotically best possible, as one can easily make any two rows proportional with at most m/2 changes. Moreover, this theorem gives an asymptotic solution to a slightly weakened version of a conjecture made by Van Vu in [Vu08].
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Structural Mechanics

Author: Ali Kaveh

Publisher: Macmillan International Higher Education

ISBN: 9781349877607

Category: Structural analysis (Engineering)

Page: 348

View: 630

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Graphs Matrices and Designs

This book is designed to be of interest to applied mathematicians, computer scientists and communications researchers.

Author: Rees

Publisher: Routledge

ISBN: 9781351444385

Category: Mathematics

Page: 344

View: 830

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Examines partitions and covers of graphs and digraphs, latin squares, pairwise balanced designs with prescribed block sizes, ranks and permanents, extremal graph theory, Hadamard matrices and graph factorizations. This book is designed to be of interest to applied mathematicians, computer scientists and communications researchers.
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Matrices and Graphs in Geometry

This series is devoted to significant topics or themes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration ...

Author: Miroslav Fiedler

Publisher: Cambridge University Press

ISBN: 9780521461931

Category: Mathematics

Page: 197

View: 526

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This series is devoted to significant topics or themes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration of the implications and applications. Books in the Encyclopedia of Mathematics and Its Applications cover their subjects comprehensively. Less important results may be summarized as exercises at the ends of chapters. Each book contains an extensive bibliography. Thus the volumes are encyclopedic references or manageable guides to major subjects.
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Matrices in Combinatorics and Graph Theory

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.

Author: Bolian Liu

Publisher: Springer Science & Business Media

ISBN: 9781475731651

Category: Mathematics

Page: 310

View: 957

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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.
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Graph Theoretical Matrices in Chemistry

This comprehensive volume is an updated, extended version of a former bestseller featuring a series of mathematical chemistry monographs. In this edition, nearly 200 graph-theoretical matrices are included.This sec

Author: Dusanka Janezic

Publisher: CRC Press

ISBN: 9781498701228

Category: Mathematics

Page: 160

View: 364

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Graph-Theoretical Matrices in Chemistry presents a systematic survey of graph-theoretical matrices and highlights their potential uses. This comprehensive volume is an updated, extended version of a former bestseller featuring a series of mathematical chemistry monographs. In this edition, nearly 200 graph-theoretical matrices are included. This second edition is organized like the previous one—after an introduction, graph-theoretical matrices are presented in five chapters: The Adjacency Matrix and Related Matrices, Incidence Matrices, The Distance Matrix and Related Matrices, Special Matrices, and Graphical Matrices. Each of these chapters is followed by a list of references. Among the matrices presented several are novel and some are known only to a few. The properties and potential usefulness of many of the presented graph-theoretical matrices in chemistry have yet to be investigated. Most of the graph-theoretical matrices presented have been used as sources of molecular descriptors usually referred to as topological indices. They are particularly concerned with a special class of graphs that represents chemical structures involving molecules. Due to its multidisciplinary scope, this book will appeal to a broad audience ranging from chemistry and mathematics to pharmacology.
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Matrices and Graphs Stability Problems in Mathematical Ecology

These are often represented by systems of ordinary differential equations or difference equations. Matrices and Graphs covers achievements in the field using concepts from matrix theory and graph theory.

Author: D. Logofet

Publisher: CRC Press

ISBN: 9781351082778

Category: Science

Page: 320

View: 719

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Intuitive ideas of stability in dynamics of a biological population, community, or ecosystem can be formalized in the framework of corresponding mathematical models. These are often represented by systems of ordinary differential equations or difference equations. Matrices and Graphs covers achievements in the field using concepts from matrix theory and graph theory. The book effectively surveys applications of mathematical results pertinent to issues of theoretical and applied ecology. The only mathematical prerequisite for using Matrices and Graphs is a working knowledge of linear algebra and matrices. The book is ideal for biomathematicians, ecologists, and applied mathematicians doing research on dynamic behavior of model populations and communities consisting of multi-component systems. It will also be valuable as a text for a graduate-level topics course in applied math or mathematical ecology.
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