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 ofthe 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.

Autorentext
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.

Inhalt
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 AgmonAllegrettoPiepenbrink 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 BeurlingDeny Criteria.- Appendix D Semigroups, Resolvents and their Generators.- Appendix E Aspects of Operator Theory.- References.- Index.- Notation Index.