Applications and Useful Properties
The original motivation for expanders is to build economical robust networks (phone or computer): an expander with bounded valence is precisely an asymptotic robust graph with number of edges growing linearly with size (number of vertices), for all subsets.
Expander graphs have found extensive applications in computer science, in designing algorithms, error correcting codes, extractors, pseudorandom generators, sorting networks (Ajtai, Komlós & Szemerédi (1983)) and robust computer networks. They have also been used in proofs of many important results in computational complexity theory, such as SL=L (Reingold (2008)) and the PCP theorem (Dinur (2007)). In cryptography, expander graphs are used to construct hash functions.
The following are some properties of expander graphs that have proven useful in many areas.
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“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)