Graphs and Discrete Dirichlet Spaces 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. 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.