Spectral Geometry Of Graphs

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Spectral Geometry of Graphs

Pavel Kurasov
Spectral Geometry of Graphs Book PDF

Author : Pavel Kurasov
Publisher : Springer Nature
Page : 644 pages
File Size : 50,6 Mb
Release : 2023-12-09
Category : Science
ISBN : 9783662678725

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Spectral Geometry of Graphs by Pavel Kurasov Pdf

This open access book gives a systematic introduction into the spectral theory of differential operators on metric graphs. Main focus is on the fundamental relations between the spectrum and the geometry of the underlying graph. The book has two central themes: the trace formula and inverse problems. The trace formula is relating the spectrum to the set of periodic orbits and is comparable to the celebrated Selberg and Chazarain-Duistermaat-Guillemin-Melrose trace formulas. Unexpectedly this formula allows one to construct non-trivial crystalline measures and Fourier quasicrystals solving one of the long-standing problems in Fourier analysis. The remarkable story of this mathematical odyssey is presented in the first part of the book. To solve the inverse problem for Schrödinger operators on metric graphs the magnetic boundary control method is introduced. Spectral data depending on the magnetic flux allow one to solve the inverse problem in full generality, this means to reconstruct not only the potential on a given graph, but also the underlying graph itself and the vertex conditions. The book provides an excellent example of recent studies where the interplay between different fields like operator theory, algebraic geometry and number theory, leads to unexpected and sound mathematical results. The book is thought as a graduate course book where every chapter is suitable for a separate lecture and includes problems for home studies. Numerous illuminating examples make it easier to understand new concepts and develop the necessary intuition for further studies.

Geometry of the Spectrum

Robert Brooks,Carolyn Gordon,Peter A. Perry
Geometry of the Spectrum Book PDF

Author : Robert Brooks,Carolyn Gordon,Peter A. Perry
Publisher : American Mathematical Soc.
Page : 299 pages
File Size : 45,8 Mb
Release : 1994
Category : Mathematics
ISBN : 9780821851852

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Geometry of the Spectrum by Robert Brooks,Carolyn Gordon,Peter A. Perry Pdf

Spectral geometry runs through much of contemporary mathematics, drawing on and stimulating developments in such diverse areas as Lie algebras, graph theory, group representation theory, and Riemannian geometry. The aim is to relate the spectrum of the Laplace operator or its graph-theoretic analogue, the adjacency matrix, to underlying geometric and topological data. This volume brings together papers presented at the AMS-IMS-SIAM Joint Summer Research Conference on Spectral Geometry, held in July 1993 at the University of Washington in Seattle. With contributions from some of the top experts in the field, this book presents an excellent overview of current developments in spectral geometry.

Analysis and Geometry on Graphs and Manifolds

Matthias Keller,Daniel Lenz,Radoslaw K. Wojciechowski
Analysis and Geometry on Graphs and Manifolds Book PDF

Author : Matthias Keller,Daniel Lenz,Radoslaw K. Wojciechowski
Publisher : Cambridge University Press
Page : 493 pages
File Size : 48,6 Mb
Release : 2020-08-20
Category : Mathematics
ISBN : 9781108713184

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Analysis and Geometry on Graphs and Manifolds by Matthias Keller,Daniel Lenz,Radoslaw K. Wojciechowski Pdf

A contemporary exploration of the interplay between geometry, spectral theory and stochastics which is explored for graphs and manifolds.

Spectral Graph Theory

Fan R. K. Chung
Spectral Graph Theory Book PDF

Author : Fan R. K. Chung
Publisher : American Mathematical Soc.
Page : 228 pages
File Size : 50,5 Mb
Release : 1997
Category : Eigenvalues
ISBN : 9780821803158

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Spectral Graph Theory by Fan R. K. Chung Pdf

This text discusses spectral graph theory.

Spectral Graph Theory

Fan R. K. Chung
Spectral Graph Theory Book PDF

Author : Fan R. K. Chung
Publisher : American Mathematical Soc.
Page : 228 pages
File Size : 48,6 Mb
Release : 2024-05-04
Category : Mathematics
ISBN : 0821889362

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Spectral Graph Theory by Fan R. K. Chung Pdf

Beautifully written and elegantly presented, this book is based on 10 lectures given at the CBMS workshop on spectral graph theory in June 1994 at Fresno State University. Chung's well-written exposition can be likened to a conversation with a good teacher - one who not only gives you the facts, but tells you what is really going on, why it is worth doing, and how it is related to familiar ideas in other areas. The monograph is accessible to the nonexpert who is interested in reading about this evolving area of mathematics.

Graphs and Discrete Dirichlet Spaces

Matthias Keller,Daniel Lenz,Radosław K. Wojciechowski
Graphs and Discrete Dirichlet Spaces Book PDF

Author : Matthias Keller,Daniel Lenz,Radosław K. Wojciechowski
Publisher : Springer Nature
Page : 675 pages
File Size : 41,8 Mb
Release : 2021-10-22
Category : Mathematics
ISBN : 9783030814595

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Graphs and Discrete Dirichlet Spaces by Matthias Keller,Daniel Lenz,Radosław K. Wojciechowski Pdf

The spectral geometry of infinite graphs deals with three major themes and their interplay: the spectral theory of the Laplacian, the geometry of the underlying graph, and the heat flow with its probabilistic aspects. In this book, all three themes are brought together coherently under the perspective of Dirichlet forms, providing a powerful and unified approach. The book gives a complete account of key topics of infinite graphs, such as essential self-adjointness, Markov uniqueness, spectral estimates, recurrence, and stochastic completeness. A major feature of the book is the use of intrinsic metrics to capture the geometry of graphs. As for manifolds, Dirichlet forms in the graph setting offer a structural understanding of the interaction between spectral theory, geometry and probability. For graphs, however, the presentation is much more accessible and inviting thanks to the discreteness of the underlying space, laying bare the main concepts while preserving the deep insights of the manifold case. Graphs and Discrete Dirichlet Spaces offers a comprehensive treatment of the spectral geometry of graphs, from the very basics to deep and thorough explorations of advanced topics. With modest prerequisites, the book can serve as a basis for a number of topics courses, starting at the undergraduate level.

Spectral Analysis on Graph-like Spaces

Olaf Post
Spectral Analysis on Graph like Spaces Book PDF

Author : Olaf Post
Publisher : Springer
Page : 431 pages
File Size : 47,7 Mb
Release : 2012-01-05
Category : Mathematics
ISBN : 9783642238406

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Spectral Analysis on Graph-like Spaces by Olaf Post Pdf

Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.

Graphs and Discrete Dirichlet Spaces

Matthias Keller,Daniel Lenz,Radosław K. Wojciechowski
Graphs and Discrete Dirichlet Spaces Book PDF

Author : Matthias Keller,Daniel Lenz,Radosław K. Wojciechowski
Publisher : Unknown
Page : 0 pages
File Size : 46,8 Mb
Release : 2021
Category : Electronic
ISBN : 3030814602

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Graphs and Discrete Dirichlet Spaces by Matthias Keller,Daniel Lenz,Radosław K. Wojciechowski Pdf

The spectral geometry of infinite graphs deals with three major themes and their interplay: the spectral theory of the Laplacian, the geometry of the underlying graph, and the heat flow with its probabilistic aspects. In this book, all three themes are brought together coherently under the perspective of Dirichlet forms, providing a powerful and unified approach. The book gives a complete account of key topics of infinite graphs, such as essential self-adjointness, Markov uniqueness, spectral estimates, recurrence, and stochastic completeness. A major feature of the book is the use of intrinsic metrics to capture the geometry of graphs. As for manifolds, Dirichlet forms in the graph setting offer a structural understanding of the interaction between spectral theory, geometry and probability. For graphs, however, the presentation is much more accessible and inviting thanks to the discreteness of the underlying space, laying bare the main concepts while preserving the deep insights of the manifold case. Graphs and Discrete Dirichlet Spaces offers a comprehensive treatment of the spectral geometry of graphs, from the very basics to deep and thorough explorations of advanced topics. With modest prerequisites, the book can serve as a basis for a number of topics courses, starting at the undergraduate level.

Introduction to Quantum Graphs

Gregory Berkolaiko,Peter Kuchment
Introduction to Quantum Graphs Book PDF

Author : Gregory Berkolaiko,Peter Kuchment
Publisher : American Mathematical Soc.
Page : 270 pages
File Size : 40,6 Mb
Release : 2013
Category : Mathematics
ISBN : 9780821892114

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Introduction to Quantum Graphs by Gregory Berkolaiko,Peter Kuchment Pdf

A ``quantum graph'' is a graph considered as a one-dimensional complex and equipped with a differential operator (``Hamiltonian''). Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering when one considers propagation of waves of various nature through a quasi-one-dimensional (e.g., ``meso-'' or ``nano-scale'') system that looks like a thin neighborhood of a graph. Works that currently would be classified as discussing quantum graphs have been appearing since at least the 1930s, and since then, quantum graphs techniques have been applied successfully in various areas of mathematical physics, mathematics in general and its applications. One can mention, for instance, dynamical systems theory, control theory, quantum chaos, Anderson localization, microelectronics, photonic crystals, physical chemistry, nano-sciences, superconductivity theory, etc. Quantum graphs present many non-trivial mathematical challenges, which makes them dear to a mathematician's heart. Work on quantum graphs has brought together tools and intuition coming from graph theory, combinatorics, mathematical physics, PDEs, and spectral theory. This book provides a comprehensive introduction to the topic, collecting the main notions and techniques. It also contains a survey of the current state of the quantum graph research and applications.

Spectral Geometry

Alex Barnett
Spectral Geometry Book PDF

Author : Alex Barnett
Publisher : American Mathematical Soc.
Page : 339 pages
File Size : 54,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821853191

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Spectral Geometry by Alex Barnett Pdf

This volume contains the proceedings of the International Conference on Spectral Geometry, held July 19-23, 2010, at Dartmouth College, Dartmouth, New Hampshire. Eigenvalue problems involving the Laplace operator on manifolds have proven to be a consistently fertile area of geometric analysis with deep connections to number theory, physics, and applied mathematics. Key questions include the measures to which eigenfunctions of the Laplacian on a Riemannian manifold condense in the limit of large eigenvalue, and the extent to which the eigenvalues and eigenfunctions of a manifold encode its geometry. In this volume, research and expository articles, including those of the plenary speakers Peter Sarnak and Victor Guillemin, address the flurry of recent progress in such areas as quantum unique ergodicity, isospectrality, semiclassical measures, the geometry of nodal lines of eigenfunctions, methods of numerical computation, and spectra of quantum graphs. This volume also contains mini-courses on spectral theory for hyperbolic surfaces, semiclassical analysis, and orbifold spectral geometry that prepared the participants, especially graduate students and young researchers, for conference lectures.

Image Analysis and Recognition

Aurélio Campilho,Mohamed Kamel
Image Analysis and Recognition Book PDF

Author : Aurélio Campilho,Mohamed Kamel
Publisher : Springer Science & Business Media
Page : 1146 pages
File Size : 53,9 Mb
Release : 2008-06-16
Category : Computers
ISBN : 9783540698111

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Image Analysis and Recognition by Aurélio Campilho,Mohamed Kamel Pdf

Non-linear image processing -- Color photo denoising via hue, saturation and intensity diffusion / Lei He and Chenyang Xu -- Examining the role of scale in the context of the non-local-means filter / Mehran Ebrahimi and Edward R. Vrscay -- Geometrical mutliscale noise resistant method of edge detection / Agnieszka Lisowska -- A simple, general model for the affine self-similarity of images / SImon K. Alexander, Edward R. Vrscay, and Satoshi Tsurumi -- Image and video coding and encryption -- Efficient bit-rate estimation for mode decision of H. 264 / AVC / Shuwei Sun and Shuming Chen -- Introducing a two dimensional measure for watermarking capacity in images / Farzin Yaghmaee and Mansour Jamzad -- Estimating the detectability of small lesions in high resolution MR compressed images / Juan Paz, Marlen Pérez, Iroel Miranda, and Peter Schelkens -- JPEG artifact removal using error distributions of linear coefficient estimates / Mika Inki --

Spectra of Graphs

Dragoš M. Cvetković,Michael Doob,Horst Sachs
Spectra of Graphs Book PDF

Author : Dragoš M. Cvetković,Michael Doob,Horst Sachs
Publisher : Unknown
Page : 374 pages
File Size : 55,8 Mb
Release : 1980
Category : Mathematics
ISBN : UOM:39015040419585

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Spectra of Graphs by Dragoš M. Cvetković,Michael Doob,Horst Sachs Pdf

The theory of graph spectra can, in a way, be considered as an attempt to utilize linear algebra including, in particular, the well-developed theory of matrices for the purposes of graph theory and its applications. to the theory of matrices; on the contrary, it has its own characteristic features and specific ways of reasoning fully justifying it to be treated as a theory in its own right.

Spectra of Graphs

Andries E. Brouwer,Willem H. Haemers
Spectra of Graphs Book PDF

Author : Andries E. Brouwer,Willem H. Haemers
Publisher : Springer Science & Business Media
Page : 250 pages
File Size : 42,6 Mb
Release : 2011-12-17
Category : Mathematics
ISBN : 9781461419396

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Spectra of Graphs by Andries E. Brouwer,Willem H. Haemers Pdf

This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Recent Results in the Theory of Graph Spectra

D.M. Cvetkovic,M. Doob,I. Gutman,A. Torgašev
Recent Results in the Theory of Graph Spectra Book PDF

Author : D.M. Cvetkovic,M. Doob,I. Gutman,A. Torgašev
Publisher : Elsevier
Page : 305 pages
File Size : 55,5 Mb
Release : 1988-01-01
Category : Mathematics
ISBN : 0080867766

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Recent Results in the Theory of Graph Spectra by D.M. Cvetkovic,M. Doob,I. Gutman,A. Torgašev Pdf

The purpose of this volume is to review the results in spectral graph theory which have appeared since 1978. The problem of characterizing graphs with least eigenvalue -2 was one of the original problems of spectral graph theory. The techniques used in the investigation of this problem have continued to be useful in other contexts including forbidden subgraph techniques as well as geometric methods involving root systems. In the meantime, the particular problem giving rise to these methods has been solved almost completely. This is indicated in Chapter 1. The study of various combinatorial objects (including distance regular and distance transitive graphs, association schemes, and block designs) have made use of eigenvalue techniques, usually as a method to show the nonexistence of objects with certain parameters. The basic method is to construct a graph which contains the structure of the combinatorial object and then to use the properties of the eigenvalues of the graph. Methods of this type are given in Chapter 2. Several topics have been included in Chapter 3, including the relationships between the spectrum and automorphism group of a graph, the graph isomorphism and the graph reconstruction problem, spectra of random graphs, and the Shannon capacity problem. Some graph polynomials related to the characteristic polynomial are described in Chapter 4. These include the matching, distance, and permanental polynomials. Applications of the theory of graph spectra to Chemistry and other branches of science are described from a mathematical viewpoint in Chapter 5. The last chapter is devoted to the extension of the theory of graph spectra to infinite graphs.

Quantum Probability and Spectral Analysis of Graphs

Akihito Hora,Nobuaki Obata
Quantum Probability and Spectral Analysis of Graphs Book PDF

Author : Akihito Hora,Nobuaki Obata
Publisher : Springer Science & Business Media
Page : 384 pages
File Size : 54,5 Mb
Release : 2007-07-05
Category : Science
ISBN : 9783540488637

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Quantum Probability and Spectral Analysis of Graphs by Akihito Hora,Nobuaki Obata Pdf

This is the first book to comprehensively cover quantum probabilistic approaches to spectral analysis of graphs, an approach developed by the authors. The book functions as a concise introduction to quantum probability from an algebraic aspect. Here readers will learn several powerful methods and techniques of wide applicability, recently developed under the name of quantum probability. The exercises at the end of each chapter help to deepen understanding.