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Hello, and Welcome to Fun with Expanders
Long in the planning, my online course on graph partitioning algorithms, expanders, and random walks, will start next month. The course page is up for people to sign up. A friend of mine has compared...
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The Cheeger inequality in manifolds
Readers of in theory have heard about Cheeger’s inequality a lot. It is a relation between the edge expansion (or, in graphs that are not regular, the conductance) of a graph and the second smallest...
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applications of the notion of non-expanding sets in undirected graphs
Where you least expect them: a common [definiton] for “population” is a geographical cluster of people who mate more within the cluster than outside of it
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The Riemann hypothesis for graphs
A regular connected graph is Ramanujan if and only if its Ihara zeta function satisfies a Riemann hypothesis. The purpose of this post is to explain all the words in the previous sentence, and to show...
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The spectrum of the infinite tree
The spectral norm of the infinite -regular tree is . We will see what this means and how to prove it. When talking about the expansion of random graphs, abobut the construction of Ramanujan expanders,...
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The Expander Mixing Lemma in Irregular Graphs
Today, after a lecture in the spectral graph theory boot camp at the Simons institute, I was asked what the expander mixing lemma is like in graphs that are not regular. I don’t know if I will have...
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The Alon-Boppana Theorem
Let be a -regular graph, and let be the eigenvalues of the adjacency matrix of counted with multiplicities and sorted in descending order. How good can the spectral expansion of be? 1. Simple Bounds...
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The expansion of the Paley graph
Suppose that we want to construct a very good family of -regular expander graphs. The Alon-Boppana theorem says that the best we can hope for, from the point of view of spectral expansion, is to have...
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References on Laplacian eigenvalues and graph properties
After my lectures in the “boot camp” of the spectral graph theory program at the Simons Institute, I promised I would post some references, because I stated all results without attribution. Here is a a...
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Riemann zeta functions and linear operators
A conjectural approaches to the Riemann hypothesis is to find a correspondence between the zeroes of the zeta function and the eigenvalues of a linear operator. This was first conceived by Hilbert and...
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Harald Helfgott on Growth in Groups
The Bulletin of the AMS is going to publish a 57-page survey on growth in groups, which is already online, and which touches several topics of interest to readers of in theory, including the recent...
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Course on spectral methods and expanders
This semester, starting tomorrow, I am teaching a course on spectral methods and expanders. This is similar to a course I offered twice at Stanford, but this time it will be a 15-week course instead of...
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CS294 Lecture 1: Introduction
In which we describe what this course is about. 1. Overview This is class is about applications of linear algebra to graph theory and to graph algorithms. In the finite-dimensional case, linear...
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