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The Mutually Beneficial Relationship of Graphs and Matrices

2011-07-06 Mathematics
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Release : 2011-07-06
Genre : Mathematics
Kind : eBook
Book Rating : 155/5 ( reviews)

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Book Synopsis The Mutually Beneficial Relationship of Graphs and Matrices by : Richard A. Brualdi

Download or read book The Mutually Beneficial Relationship of Graphs and Matrices written by Richard A. Brualdi. This book was released on 2011-07-06. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Matrices in Combinatorics and Graph Theory

2013-03-09 Mathematics
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Release : 2013-03-09
Genre : Mathematics
Kind : eBook
Book Rating : 655/5 ( reviews)

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Book Synopsis Matrices in Combinatorics and Graph Theory by : Bolian Liu

Download or read book Matrices in Combinatorics and Graph Theory written by Bolian Liu. This book was released on 2013-03-09. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Asymptotics of Random Matrices and Related Models: The Uses of Dyson-Schwinger Equations

2019-04-29 Green's functions
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Release : 2019-04-29
Genre : Green's functions
Kind : eBook
Book Rating : 275/5 ( reviews)

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Book Synopsis Asymptotics of Random Matrices and Related Models: The Uses of Dyson-Schwinger Equations by : Alice Guionnet

Download or read book Asymptotics of Random Matrices and Related Models: The Uses of Dyson-Schwinger Equations written by Alice Guionnet. This book was released on 2019-04-29. Available in PDF, EPUB and Kindle. Book excerpt: Probability theory is based on the notion of independence. The celebrated law of large numbers and the central limit theorem describe the asymptotics of the sum of independent variables. However, there are many models of strongly correlated random variables: for instance, the eigenvalues of random matrices or the tiles in random tilings. Classical tools of probability theory are useless to study such models. These lecture notes describe a general strategy to study the fluctuations of strongly interacting random variables. This strategy is based on the asymptotic analysis of Dyson-Schwinger (or loop) equations: the author will show how these equations are derived, how to obtain the concentration of measure estimates required to study these equations asymptotically, and how to deduce from this analysis the global fluctuations of the model. The author will apply this strategy in different settings: eigenvalues of random matrices, matrix models with one or several cuts, random tilings, and several matrices models.

Matrix Inequalities for Iterative Systems

2017-02-03 Mathematics
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Release : 2017-02-03
Genre : Mathematics
Kind : eBook
Book Rating : 791/5 ( reviews)

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Book Synopsis Matrix Inequalities for Iterative Systems by : Hanjo Taubig

Download or read book Matrix Inequalities for Iterative Systems written by Hanjo Taubig. This book was released on 2017-02-03. Available in PDF, EPUB and Kindle. Book excerpt: The book reviews inequalities for weighted entry sums of matrix powers. Applications range from mathematics and CS to pure sciences. It unifies and generalizes several results for products and powers of sesquilinear forms derived from powers of Hermitian, positive-semidefinite, as well as nonnegative matrices. It shows that some inequalities are valid only in specific cases. How to translate the Hermitian matrix results into results for alternating powers of general rectangular matrices? Inequalities that compare the powers of the row and column sums to the row and column sums of the matrix powers are refined for nonnegative matrices. Lastly, eigenvalue bounds and derive results for iterated kernels are improved.

Advanced Graph Theory and Combinatorics

2016-11-22 Computers
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Release : 2016-11-22
Genre : Computers
Kind : eBook
Book Rating : 643/5 ( reviews)

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Book Synopsis Advanced Graph Theory and Combinatorics by : Michel Rigo

Download or read book Advanced Graph Theory and Combinatorics written by Michel Rigo. This book was released on 2016-11-22. Available in PDF, EPUB and Kindle. Book excerpt: Advanced Graph Theory focuses on some of the main notions arising in graph theory with an emphasis from the very start of the book on the possible applications of the theory and the fruitful links existing with linear algebra. The second part of the book covers basic material related to linear recurrence relations with application to counting and the asymptotic estimate of the rate of growth of a sequence satisfying a recurrence relation.

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