## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, February 22, 2011, 12:15 pm

Duration: This information is not available in the database

Location: CAB G51

Speaker: Uli Wagner

## An elementary construction of expanders

Constant degree expanders are graphs that are very sparse but nonetheless highly connected and that imitate many properties of the complete graph. For instance, they have "small" diameter, and a random walk in an expander "mixes quickly", i.e., we can i.e., after "few steps" the distribution of the random vertex we reach will approximate the uniform distribution of the set of vertices.

It is not hard to prove the existence of constant-degree expanders by probabilistic arguments. However, for many applications, one needs to be able to construct them in a way that is "explicit" (in a technical sense to be explained).

Beautiful explicit constructions of expanders were given by Margulis, and later by Gabber-Galil and by Lubotzky-Philips-Sarnak. These constructions are algebraic or number theoretic, and while the graphs are easy to describe, the proofs that they are, in fact, expanders are based on estimating the second eigenvalue of their adjacency matrices and, in some cases, rely on deep mathematical results.

Here, we present a simple iterative construction, due to Alon, Schwartz, and Shapira, for which both the construction and its analysis are rather elementary and combinatorial. The construction uses the notion of "replacement product" or "zig-zag product" due to Reingold, Vadhan, and Wigderson (with the basic idea going back to Gromov).

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