Graphs and discrete Dirichlet spaces / Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski.
Keller, Matthias, author.; Lenz, Daniel, author.; Wojciechowski, Radoslaw K., author.
2021
QA166
Available Online

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Title Graphs and discrete Dirichlet spaces / Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski.
Author Keller, Matthias, author.
ISBN 9783030814595 (electronic bk.)
3030814599 (electronic bk.)
9783030814588 (print)
3030814580
DOI https://doi.org/10.1007/978-3-030-81459-5
Published Cham, Switzerland : Springer, 2021.
Language English
Description 1 online resource (xv, 668 pages) : illustrations
Item Number 10.1007/978-3-030-81459-5 doi
Call Number QA166
Dewey Decimal Classification 511/.5
Summary 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.
Bibliography, etc. Note Includes bibliographical references and indexes.
Access Note Access limited to authorized users.
Digital File Characteristics text file
PDF
Source of Description Online resource; title from PDF title page (SpringerLink, viewed October 26, 2021).
Added Author Lenz, Daniel, author.
Wojciechowski, Radoslaw K., author.
Series Grundlehren der mathematischen Wissenschaften ; v. 358. 2196-9701
Available in Other Form Print version: 9783030814588
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Part 0 Prelude
Chapter 0 Finite Graphs
Part 1 Foundations and Fundamental Topics
Chapter 1 Infinite Graphs
Key Concepts
Chapter 2 Infinite Graphs
Toolbox
Chapter 3 Markov Uniqueness and Essential Self-Adjointness
Chapter 4 Agmon-Allegretto-Piepenbrink and Persson Theorems
Chapter 5 Large Time Behavior of the Heat Kernel
Chapter 6 Recurrence
Chapter 7 Stochastic Completeness
Part 2 Classes of Graphs
Chapter 8 Uniformly Positive Measure
Chapter 9 Weak Spherical Symmetry
Chapter 10 Sparseness and Isoperimetric Inequalities
Part 3 Geometry and Intrinsic Metrics
Chapter 11 Intrinsic Metrics: Definition and Basic Facts
Chapter 12 Harmonic Functions and Caccioppoli Theory
Chapter 13 Spectral Bounds
Chapter 14 Volume Growth Criterion for Stochastic Completeness and Uniqueness Class
Appendix A The Spectral Theorem
Appendix B Closed Forms on Hilbert Spaces
Appendix C Dirichlet Forms and Beurling-Deny Criteria
Appendix D Semigroups, Resolvents and their Generators
Appendix E Aspects of Operator Theory
References
Index
Notation Index.

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