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Graphs and Discrete Dirichlet Spaces

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  • Presents a thorough study of geometric, analytic and probabilistic aspects of infinite graphs, including recent results

  • Provides a very accessible introduction to general Dirichlet form theory by focusing on discrete spaces

  • Relates spectral theory, the heat equation and intrinsic metrics on graphs

Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 358)

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  • ISBN: 978-3-030-81459-5
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  • Tax calculation will be finalised during checkout
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USD 169.99
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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xv
  2. Prelude

    1. Front Matter

      Pages 1-2
    2. Finite Graphs

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 3-94
  3. Foundations and Fundamental Topics

    1. Front Matter

      Pages 95-96
    2. Infinite Graphs – Key Concepts

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 97-140
    3. Infinite Graphs – Toolbox

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 141-184
    4. Markov Uniqueness and Essential Self-Adjointness

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 185-214
    5. Agmon–Allegretto–Piepenbrink and Persson Theorems

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 215-240
    6. Large Time Behavior of the Heat Kernel

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 241-254
    7. Recurrence

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 255-305
    8. Stochastic Completeness

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 307-360
  4. Classes of Graphs

    1. Front Matter

      Pages 361-362
    2. Uniformly Positive Measure

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 363-377
    3. Weak Spherical Symmetry

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 379-416
    4. Sparseness and Isoperimetric Inequalities

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 417-439
  5. Geometry and Intrinsic Metrics

    1. Front Matter

      Pages 441-442
    2. Intrinsic Metrics: Definition and Basic Facts

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 443-467
    3. Harmonic Functions and Caccioppoli Theory

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 469-505
    4. Spectral Bounds

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 507-524
    5. Volume Growth Criterion for Stochastic Completeness and Uniqueness Class

      • Matthias Keller, Daniel Lenz, Radosław K. Wojciechowski
      Pages 525-550

About this book

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.

Keywords

  • Spectral graph theory
  • Graph Laplacian
  • Dirichlet form
  • Stochastic completeness
  • Intrinsic metrics
  • Heat equation on graphs
  • Markov semigroups
  • Schrödinger operators

Authors and Affiliations

  • Institute for Mathematics, University of Potsdam, Potsdam, Germany

    Matthias Keller

  • Institute for Mathematics, Friedrich Schiller University Jena, Jena, Germany

    Daniel Lenz

  • Department of Mathematics and Computer Science, York College of the City University of New YorkJamaica;, Department of Mathematics, Graduate Center of the City University of New York, New York, USA

    Radosław K. Wojciechowski

About the authors

Matthias Keller studied in Chemnitz and obtained his PhD in Jena. He held positions in Princeton, Jerusalem and Haifa before becoming a professor at the University of Potsdam.

Daniel Lenz obtained his PhD in Frankfurt am Main. After prolonged stays in Jerusalem, Chemnitz and Houston, he is now a professor at the Friedrich Schiller University in Jena.

Radoslaw Wojciechowski got his PhD at the Graduate Center of the City University of New York following his undergraduate studies at Indiana University Bloomington. After a postdoc period in Lisbon he is now a professor at York College and the Graduate Center in New York City.

Bibliographic Information

Buying options

eBook
USD 129.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-81459-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD 169.99
Price excludes VAT (USA)