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.

The interplay of geometry, spectral theory and stochastics has a long and fruitful history, and is the driving force behind many developments in modern mathematics. Bringing together contributions from a 2017 conference at the University of Potsdam, this volume focuses on global effects of local properties. Exploring the similarities and differences between the discrete and the continuous settings is of great interest to both researchers and graduate students in geometric analysis. The range of survey articles presented in this volume give an expository overview of various topics, including curvature, the effects of geometry on the spectrum, geometric group theory, and spectral theory of Laplacian and Schroedinger operators. Also included are shorter articles focusing on specific techniques and problems, allowing the reader to get to the heart of several key topics.