Inequalities for Graph Eigenvalues

Inequalities for Graph Eigenvalues
Author: Zoran Stanić
Publisher: Cambridge University Press
Total Pages: 311
Release: 2015-07-23
Genre: Mathematics
ISBN: 1107545978

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This book explores the inequalities for eigenvalues of the six matrices associated with graphs. Includes the main results and selected applications.

Matrix Inequalities for Iterative Systems

Matrix Inequalities for Iterative Systems
Author: Hanjo Taubig
Publisher: CRC Press
Total Pages: 219
Release: 2017-02-03
Genre: Mathematics
ISBN: 1498777791

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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.

Eigenvalues, Multiplicities and Graphs

Eigenvalues, Multiplicities and Graphs
Author: Charles R. Johnson
Publisher: Cambridge University Press
Total Pages: 315
Release: 2018-02-12
Genre: Mathematics
ISBN: 110709545X

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This book investigates the influence of the graph of a symmetric matrix on the multiplicities of its eigenvalues.

Spectral Graph Theory

Spectral Graph Theory
Author: Fan R. K. Chung
Publisher: American Mathematical Soc.
Total Pages: 228
Release: 1997
Genre: Mathematics
ISBN: 0821803158

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This text discusses spectral graph theory.

Laplacian Eigenvectors of Graphs

Laplacian Eigenvectors of Graphs
Author: Türker Biyikoglu
Publisher: Springer
Total Pages: 121
Release: 2007-07-07
Genre: Mathematics
ISBN: 3540735100

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This fascinating volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, and graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology. Eigenvectors of graph Laplacians may seem a surprising topic for a book, but the authors show that there are subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs.

Graphs and Matrices

Graphs and Matrices
Author: Ravindra B. Bapat
Publisher: Springer
Total Pages: 197
Release: 2014-09-19
Genre: Mathematics
ISBN: 1447165691

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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.

Finite Sections of Some Classical Inequalities

Finite Sections of Some Classical Inequalities
Author: Herbert S. Wilf
Publisher: Springer Science & Business Media
Total Pages: 90
Release: 2012-12-06
Genre: Mathematics
ISBN: 364286712X

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Hardy, Littlewood and P6lya's famous monograph on inequalities [17J has served as an introduction to hard analysis for many mathema ticians. Some of its most interesting results center around Hilbert's inequality and generalizations. This family of inequalities determines the best bound of a family of operators on /p. When such inequalities are restricted only to finitely many variables, we can then ask for the rate at which the bounds of the restrictions approach the uniform bound. In the context of Toeplitz forms, such research was initiated over fifty years ago by Szego [37J, and the chain of ideas continues to grow strongly today, with fundamental contributions having been made by Kac, Widom, de Bruijn, and many others. In this monograph I attempt to draw together these lines of research from the point of view of sharpenings of the classical inequalities of [17]. This viewpoint leads to the exclusion of some material which might belong to a broader-based discussion, such as the elegant work of Baxter, Hirschman and others on the strong Szego limit theorem, and the inclusion of other work, such as that of de Bruijn and his students, which is basically nonlinear, and is therefore in some sense disjoint from the earlier investigations. I am grateful to Professor Halmos for inviting me to prepare this volume, and to Professors John and Olga Todd for several helpful comments. Philadelphia, Pa. H.S.W.

Geometry and Spectra of Compact Riemann Surfaces

Geometry and Spectra of Compact Riemann Surfaces
Author: Peter Buser
Publisher: Springer Science & Business Media
Total Pages: 473
Release: 2010-10-29
Genre: Mathematics
ISBN: 0817649921

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This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature –1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.

Machine Learning for Neuroscience

Machine Learning for Neuroscience
Author: Chuck Easttom
Publisher: CRC Press
Total Pages: 296
Release: 2023-07-31
Genre: Computers
ISBN: 1000907147

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This book addresses the growing need for machine learning and data mining in neuroscience. The book offers a basic overview of the neuroscience, machine learning and the required math and programming necessary to develop reliable working models. The material is presented in a easy to follow user-friendly manner and is replete with fully working machine learning code. Machine Learning for Neuroscience: A Systematic Approach, tackles the needs of neuroscience researchers and practitioners that have very little training relevant to machine learning. The first section of the book provides an overview of necessary topics in order to delve into machine learning, including basic linear algebra and Python programming. The second section provides an overview of neuroscience and is directed to the computer science oriented readers. The section covers neuroanatomy and physiology, cellular neuroscience, neurological disorders and computational neuroscience. The third section of the book then delves into how to apply machine learning and data mining to neuroscience and provides coverage of artificial neural networks (ANN), clustering, and anomaly detection. The book contains fully working code examples with downloadable working code. It also contains lab assignments and quizzes, making it appropriate for use as a textbook. The primary audience is neuroscience researchers who need to delve into machine learning, programmers assigned neuroscience related machine learning projects and students studying methods in computational neuroscience.