Stanford University CS359G: Graph Partitioning and Expanders
Handout 4
Luca Trevisan
January 13, 2011
Lecture 4
In which we prove the dicult direction of Cheegers inequality.
As in the past lectures, consider an undirected d-regular graph G = (V, E ), cal
Stanford University CS359G: Graph Partitioning and Expanders
Handout M
Luca Trevisan
February 10, 2011
Midterm
To be nished individually. Due on Thursday, February 17, 2011. Submit in class,
or by email to trevisan at stanford dot edu
1. Let G = (V, E ) b
Stanford University CS359G: Graph Partitioning and Expanders
Handout 17
Luca Trevisan
March 1, 2011
Lecture 17
In which we dene and analyze the zig-zag graph product.
1
Replacement Product and Zig-Zag Product
In the previous lecture, we claimed it is poss
Stanford University CS359G: Graph Partitioning and Expanders
Handout 16
Luca Trevisan
February 24, 2011
Lecture 16
In which we give an explicit construction of expander graphs of polylogarithmic degree, state the properties of the zig-zag product of graph
Stanford University CS359G: Graph Partitioning and Expanders
Handout 8
Luca Trevisan
January 27, 2011
Lecture 8
In which we introduce the Leighton-Rao relaxation of sparsest cut.
Let G = (V, E ) be an undirected graph. Unlike past lectures, we will not ne
Stanford University CS359G: Graph Partitioning and Expanders
Handout 11
Luca Trevisan
February 8, 2011
Lecture 11
In which we introduce the Arora-Rao-Vazirani relaxation of sparsest cut, and discuss
why it is solvable in polynomial time.
1
The Arora-Rao-V
Stanford University CS359G: Graph Partitioning and Expanders
Handout 14
Luca Trevisan
February 17, 2011
Lecture 14
In which we begin to discuss the Arora-Rao-Vazirani rounding procedure.
Recall that, in a graph G = (V, E ) with adjacency matrix A, then AR
Stanford University CS359G: Graph Partitioning and Expanders
Handout 9
Luca Trevisan
February 1, 2011
Lecture 9
In which we prove that every metric can be embedded into L1 with logarithmic distortion.
Today we prove the following theorem.
Theorem 1 (Bourg
Stanford University CS359G: Graph Partitioning and Expanders
Handout 9
Luca Trevisan
February 3, 2011
Lecture 10
In which we prove that there are n-point metric spaces that cannot be embedded into
L1 with distortion o(log n), and we see further applicatio
Stanford University CS359G: Graph Partitioning and Expanders
Handout 7
Luca Trevisan
January 25, 2011
Lecture 7
In which we analyze a nearly-linear time algorithm for nding an approximate eigenvector for the second eigenvalue of a graph adjacency matrix,
Stanford University CS359G: Graph Partitioning and Expanders
Handout 1
Luca Trevisan
January 6, 2011
Lecture 2
In which we review linear algebra and introduce spectral graph theory.
1
Eigenvalues and Eigenvectors
Spectral graph theory studies how the eige
Stanford University CS359G: Graph Partitioning and Expanders
Handout 5
Luca Trevisan
January 18, 2011
Lecture 5
In which we introduce the theory of characters of nite abelian groups, which we
will use to compute eigenvalues and eigenvectors of graphs such
Stanford University CS359G: Graph Partitioning and Expanders
Handout 6
Luca Trevisan
January 20, 2011
Lecture 6
In which we talk about the spectrum of Cayley graphs of abelian groups, we compute
the eigenvalues and eigenvectors of the cycle and of the hyp
Stanford University CS359G: Graph Partitioning and Expanders
Handout 1
Luca Trevisan
January 11, 2011
Lecture 3
In which we prove the easy case of Cheegers inequality.
1
Expansion and The Second Eigenvalue
1
Let G = (V, E ) be an undirected d-regular grap
Stanford University CS359G: Graph Partitioning and Expanders
Handout 1
Luca Trevisan
January 4, 2011
Lecture 1
In which we describe what this course is about.
1
Overview
This class is about the following topics:
1. Approximation algorithms for graph parti
Stanford University CS359G: Graph Partitioning and Expanders
Handout 18
Luca Trevisan
March 3, 2011
Lecture 18
In which we prove properties of expander graphs.
1
Quasirandomness of Expander Graphs
1
Recall that if G is a d-regular graph, A is its adjacenc