Volume: 71; 2018; 150 pp; Softcover
MSC: Primary 05; 60;
Print ISBN: 978-1-4704-4397-9
Product Code: ULECT/71
List Price: $49.00
AMS Member Price: $39.20
MAA Member Price: $44.10
Electronic ISBN: 978-1-4704-4855-4
Product Code: ULECT/71.E
List Price: $49.00
AMS Member Price: $39.20
MAA Member Price: $44.10
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Supplemental Materials
Introduction to Analysis on Graphs
Share this pageAlexander Grigor’yan
A central object of this book is the discrete
Laplace operator on finite and infinite graphs. The eigenvalues of the
discrete Laplace operator have long been used in graph theory as a
convenient tool for understanding the structure of complex
graphs. They can also be used in order to estimate the rate of
convergence to equilibrium of a random walk (Markov chain) on finite
graphs. For infinite graphs, a study of the heat kernel allows to
solve the type problem—a problem of deciding whether the random walk
is recurrent or transient.
This book starts with elementary properties of the eigenvalues on
finite graphs, continues with their estimates and applications, and
concludes with heat kernel estimates on infinite graphs and their
application to the type problem.
The book is suitable for beginners in the subject and accessible to
undergraduate and graduate students with a background in linear
algebra I and analysis I. It is based on a lecture course taught by
the author and includes a wide variety of exercises. The book will
help the reader to reach a level of understanding sufficient to start
pursuing research in this exciting area.
Readership
Undergraduate and graduate students and researchers interested in random walks on graphs and groups.
Reviews & Endorsements
Anybody who has ever read a mathematical text of the author would agree that his way of presenting complex material is nothing short of marvelous. This new book showcases again the author's unique ability of presenting challenging topics in a clear and accessible manner, and of guiding the reader with ease to a deep understanding of the subject.
-- Matthias Keller, University of Potsdam
Table of Contents
Table of Contents
Introduction to Analysis on Graphs
- Cover Cover11
- Title page iii4
- Preface vii8
- Chapter 1. The Laplace operator on graphs 110
- Chapter 2. Spectral properties of the Laplace operator 2736
- 2.1. Green’s formula 2736
- 2.2. Eigenvalues of the Laplace operator 2837
- 2.3. Convergence to equilibrium 3443
- 2.4. More about the eigenvalues 3948
- 2.5. Convergence to equilibrium for bipartite graphs 4251
- 2.6. Eigenvalues of ℤ_{𝕞} 4352
- 2.7. Products of graphs 4554
- 2.8. Eigenvalues and mixing time in ℤ_{𝕞}ⁿ, 𝕞 odd. 4958
- 2.9. Eigenvalues and mixing time in a binary cube 5160
- Chapter 3. Geometric bounds for the eigenvalues 5362
- Chapter 4. Eigenvalues on infinite graphs 7382
- Chapter 5. Estimates of the heat kernel 8998
- 5.1. The notion and basic properties of the heat kernel 8998
- 5.2. One-dimensional simple random walk 91100
- 5.3. Carne-Varopoulos estimate 96105
- 5.4. On-diagonal upper estimates of the heat kernel 99108
- 5.5. On-diagonal lower bound via the Dirichlet eigenvalues 107116
- 5.6. On-diagonal lower bound via volume growth 112121
- 5.7. Escape rate of random walk 114123
- Chapter 6. The type problem 117126
- Chapter 7. Exercises 131140
- Bibliography 143152
- Index 149158
- Back Cover Back Cover1160