Lectures on Spectral Graph Theory

Lectures on Algorithmic Spectral Graph Theory

References on Higher Laplacian Eigenvalues and Combinatorial Graph Properties

This is a collection of references for a series of lectures that I gave in the “boot camp” of the semester on spectral graph theory at the Simons Institute in August, 2014. To keep this document focused, I only refer to results related to the lectures in question, which means that my own work is disproportionally represented. I plan to post a better version of this document in the future, so pl...

Courant-Fischer and Graph Coloring

4.1 Eigenvalues and Optimization I cannot believe that I have managed to teach three lectures on spectral graph theory without giving the characterization of eigenvalues as solutions to optimization problems. It is one of the most useful ways of understanding eigenvalues of symmetric matrices. To begin, let A be a symmetric matrix with eigenvalues α1 ≥ α2 ≥ · · · ≥ αn, and corresponding orthono...

CS 168 : The Modern Algorithmic Toolbox Lectures # 17 and # 18 : Spectral Graph Theory

Most of our lectures thus far have been motivated by concrete problems, and the problems themselves led us to the algorithms and more widely applicable techniques and insights that we then went on to discuss. This section has the opposite structure: we will begin with the very basic observation that graphs can be represented as matrices, and then ask “what happens if we apply the linear algebra...

CS 168 : The Modern Algorithmic Toolbox Lectures # 11 and # 12 : Spectral Graph Theory

Most of our lectures thus far have been motivated by concrete problems, and the problems themselves led us to the algorithms and more widely applicable techniques and insights that we then went on to discuss. This section has the opposite structure: we will begin with the very basic observation that graphs can be represented as matrices, and then ask “what happens if we apply the linear algebra...