Graphs and Discrete Dirichlet Spaces [Hardcover]

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.

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 Metló/