Graphs and Discrete Dirichlet Spaces

The spectral geometry of infinite graphs explores the interplay between the Laplacian, graph geometry, and heat flow using Dirichlet forms. It provides a unified approach to key topics such as self-adjointness, Markov uniqueness, spectral estimates, recurrence, and stochastic completeness, using intrinsic metrics to capture the geometry of graphs. The book is accessible to undergraduates and can serve as a basis for courses on various graph-related topics.

Format: Hardback
Length: 668 pages
Publication date: 23 October 2021
Publisher: Springer Nature Switzerland AG

The spectral geometry of infinite graphs encompasses a rich tapestry of three fundamental themes, intricately intertwined and shaped by their interplay: the spectral theory of the Laplacian, the geometry of the underlying graph, and the captivating realm of heat flow, adorned with probabilistic aspects. Within this comprehensive volume, all three themes converge harmoniously under the guiding lens of Dirichlet forms, offering a powerful and unified framework.

The book delves into a comprehensive exploration of key topics in infinite graphs, including essential self-adjointness, Markov uniqueness, spectral estimates, recurrence, and stochastic completeness. It provides a thorough account of these subjects, offering readers a deep understanding of the underlying principles and applications.

A notable hallmark of this book is its utilization of intrinsic metrics to capture the geometric essence of graphs. These metrics, derived from the study of Dirichlet forms in the context of graphs, provide a structural framework for understanding the interplay between spectral theory, geometry, and probability. While manifolds offer a rich and intricate perspective on these topics, the book's approach in the graph setting is more accessible and inviting, leveraging the discrete nature of the underlying space to unveil the core concepts while preserving the profound insights of the manifold case.

Graphs and Discrete Dirichlet Spaces serves as a comprehensive resource, catering to a wide range of readers, from beginners to advanced scholars. With a modest set of prerequisites, the book can serve as a foundational text for courses spanning various topics, from undergraduate to postgraduate levels. It offers a systematic and coherent approach to the spectral geometry of infinite graphs, fostering a deeper understanding of these fascinating mathematical structures and their applications.

Weight: 1184g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030814588
Edition number: 1st ed. 2021