CHAPTER 1
Eigenvalues and the Laplacian of a graph
1.1. Introduction
Spectral graph theory has a long history. In the early days, matrix theory and
linear algebra were used to analyze adjacency matrices of graphs. Algebraic methods have proven to be espec
May 19, 2010
Mary Radcliffe
Introduction to the Heat Kernel
1
Introduction
In this lecture, we introduce the heat kernel for a graph, the heat kernel PageRank, and the zeta
function for a graph. We develop some associated theorems to these functions.
2
He
Approximate Page Rank
1
Approximate Page Rank
Let pr,s = pr(, s) denote the page rank of a graph. Then pr,s is a xed point for the equation
p = s + (1 )pZ . We can compute pr,s approximately using the recurrence relation pt+1 =
s + (1 )pt Z . Recall that
May 5, 2010
Yuh-Jie (Eunice) Chen
Local PageRank Properties
1
Introduction
In this lecture, we will talk about some local PageRank properties. That is, given a graph G(V, E ),
starting vector u, and 0 < 1, we will investigate the behaviors of pr,u on subs
Lecture 11: A Local Cheeger Inequality Using Lazy Walks
Transcriber: Franklin Kenter
The goal of this lecture is to prove an inequality involving hS , the Cheeger constant using the
Dirchelet eigenvalues.
Goal Theorem: hS S
2
4
log |S |
t
Denition of : =
Lecture 10: Bounding
Transcriber: Andy Parrish
Goal: Show that (1 /2)2t
i=0 (1
i /2)2t n(1 2 /4)t , where t is any positive integer.
For a positive integer t and a vertex x, let ft,x = x Z t , where x (v ) = 1 if x = v and 0
otherwise, Z = I +P is the
April 21, 2010
Wensong(Tony) Xu
Cheeger inequality
1
Introduction
In this lecture, we will introduce the Cheeger ratio and the Cheeger inequality.
2
Cheeger ratio
The isoperimetric problem is to nd a plane gure which maximizes its area, given a xed length
April 19, 2010
Wensong(Tony) Xu
Eigenvalue, Diameter, and Polynomial
Convergence
1
Introduction
In this lecture, we will show the polynomial convergence of a random walk on a non-bipartite
undirected graph, and its connection with the second largest eigen
April 14, 2010
Yuh-Jie (Eunice) Chen
Dirichlet Eigenvalues
1
Introduction
In this lecture, we will go over the basics of the Dirichlet eigenvalues and prove a matrix-tree
theorem.
2
2.1
Dirichlet eigenvalues
Boundary
The idea of boundary pervades many div
We are interested in studying random walks in graphs by using the matrix
P . Examples of real life phenomena that can be modeled by random walks
include the orderings of a deck of cards or state graphs. A graph is ergodic
if there exists a unique stationa
Lecture 3: Eigenvalues of the Laplacian
Transcriber: Andy Parrish
In this lecture we will consider only graphs G = (V, E) with no isolated vertices and no self-loops.
Recall that A is the adjacency matrix of a graph, and D is the diagonal matrix of degree
April 5, 2010
Franklin Kenter
PageRank, Spectral Graph Theory, and the
Matrix Tree Theorem
Introduction
1
Introduction
In this lecture, we will go over the basics of the PageRank algorithm and how it relates to graph
theory. Then, we will start our study
Lecture 1: Basic Structures and Denition of
PageRank
Transcriber: Mary Radclie
1
A Glimpse of Spectral Graph Theory
Let G = (V, E ) be a graph, where n = |V |. Dene the adjacency matrix of G,
1 uv
denoted by A, to be a matrix indexed by V , with A(u, v )
CHAPTER 3
Diameters and eigenvalues
3.1. The diameter of a graph
In a graph G, the distance between two vertices u and v , denoted by d(u, v ),
is dened to be the length of a shortest path joining u and v in G. (It is possible
to dene the distance by vari
CHAPTER 2
Isoperimetric problems
2.1. History
One of the earliest problems in geometry was the isoperimetric problem, which
was considered by the ancient Greeks. The problem is to nd, among all closed
curves of a given length, the one which encloses the m
May 26, 2010
Janine LoBue
Analytic Methods and the Heat Kernel
1
Introduction
In this lecture, we employ some standard analytic methods from spectral geometry to better understand the heat kernel. In particular, we determine upper and lower bounds for the