Introduction to Analysis on Graphs

Cover
Title page
Preface
Chapter 1. The Laplace operator on graphs
	1.1. The notion of a graph
	1.2. Cayley graphs
	1.3. Random walks
	1.4. The Laplace operator
	1.5. The Dirichlet problem
Chapter 2. Spectral properties of the Laplace operator
	2.1. Green’s formula
	2.2. Eigenvalues of the Laplace operator
	2.3. Convergence to equilibrium
	2.4. More about the eigenvalues
	2.5. Convergence to equilibrium for bipartite graphs
	2.6. Eigenvalues of ℤ_{????}
	2.7. Products of graphs
	2.8. Eigenvalues and mixing time in ℤ_{????}ⁿ, ???? odd.
	2.9. Eigenvalues and mixing time in a binary cube
Chapter 3. Geometric bounds for the eigenvalues
	3.1. Cheeger’s inequality
	3.2. Eigenvalues on a path graph
	3.3. Estimating ????₁ via diameter
	3.4. Expansion rate
Chapter 4. Eigenvalues on infinite graphs
	4.1. Dirichlet Laplace operator
	4.2. Cheeger’s inequality
	4.3. Isoperimetric and Faber-Krahn inequalities
	4.4. Estimating ????₁(Ω) via inradius
	4.5. Isoperimetric inequalities on Cayley graphs
	4.6. Solving the Dirichlet problem by iterations
Chapter 5. Estimates of the heat kernel
	5.1. The notion and basic properties of the heat kernel
	5.2. One-dimensional simple random walk
	5.3. Carne-Varopoulos estimate
	5.4. On-diagonal upper estimates of the heat kernel
	5.5. On-diagonal lower bound via the Dirichlet eigenvalues
	5.6. On-diagonal lower bound via volume growth
	5.7. Escape rate of random walk
Chapter 6. The type problem
	6.1. Recurrence and transience
	6.2. Recurrence and transience on Cayley graphs
	6.3. Volume tests for recurrence
	6.4. Isoperimetric tests for transience
Chapter 7. Exercises
Bibliography
Index
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