Posted by: Gil Cohen | May 21, 2012

## Lecture 9

Today we proved Cheeger’s inequality, based on a proof by Luca Trevisan (here and here). Quite recently two papers were published, that study what can we say when the ${k}$ largest eigenvalues of the Laplacian matrix are close to ${1}$. This generalizes the harder direction in Cheeger’s inequality, which is the question for ${k=2}$. The two papers are Multi-way spectral partitioning and higher-order Cheeger inequalities by James R. Lee, Shayan Oveis Gharan and Luca Trevisan, and Algorithmic Extensions of Cheeger’s Inequality to Higher Eigenvalues and Partitions by Anand Louis, Prasad Raghavendra, Prasad Tetali and Santosh Vempala. Both of these beautiful papers will be in the presentations list.

We started to discuss error reduction using expander graphs. In the next lecture we will follow the proof in Arora-Barak book (Theorem 21.12), and give an explicit construction of expanders. For that we will follow Chapter 9 of the book Expander Graphs and their Applications by Shlomo Hoory, Nathan Linial and Avi Wigderson.