Graphs and Discrete Dirichlet Spaces

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.

• 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.

“This is an extremely well-written book, with motivations sprinkled throughout. One useful pedagogic device is the slow beginning with a chapter 0 giving the finite case separately, even though this is technically superfluous. The value of the book is enhanced by its many exercises. Undergraduate students and researchers should find it an extremely useful introduction to a beautiful theory.” (Bhaskar Bagchi, zbMATH 1487.05003, 2022)