We started today’s lecture with the application of magical graphs for the construction of super-concentrators and for error reduction. Specifically, in the latter example we showed how to reduce the error by a factor of while multiplying the running time by , and using *the same number of random bits*. Constructing super-concentrators with a small multiplicative constant is a question that attracted attention. To the best of my knowledge, the best result is due to Noga Alon and Michael Capalbo. Their paper, Smaller Explicit Superconcentrators will appear in the presentations list.

We then defined , the edge expansion of the -regular graph , and saw some examples. Another definition of expansion is related to the spectral properties of a graph. More specifically, to the spectral gap of a graph, namely, . As we will see next time the spectral gap is related to by Cheeger’s inequality.

Today we have revisited some basic facts from linear algebra and saw that the spectral gap is if and only if the graph is disconnected. Cheeger’s inequality can be viewed as an approximated version of this claim. We’ll prove Cheeger’s inequality next time. For now, you can find the proof in Luca Trevisan’s blog, In Theory: The easy direction and the harder one. There are other proofs.

If you want to learn more about spectral graph theory, Daniel Spielman (the same Spielman from Sipser-Spielman codes that we have seen) gave a course about the subject, and in his site you can find great lecture notes.

Next time we will continue to discuss properties of expanders, and head towards constructing a family of constant degree expanders.

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