the complete graph. Lubetzky and Peres proved that the simple random walk exhibits cutoff phenomenon on all Ramanujan graphs. This means that the random walk undergoes a phase transition from being completely unmixed to completely mixed in the total variation norm. This result strongly relies on the graph being Ramanujan, not just an expander—some good expanders are known to not exhibit cutoff. Ramanujan graphs of Pizer have been proposed as the basis for post-quantum elliptic-curve cryptography. Ramanujan graphs can be used to construct expander codes, which are good error correcting codes. See also Expander graph Alon-Boppana bound Expander mixing lemma Spectral graph theory References Further reading External links Survey paper by M. Ram Murty Survey paper by Alexander Lubotzky Survey paper by Hoory, Linial, and Wigderson Graph families Algebraic graph theory Spectral theory Regular graphs Srinivasa Ramanujan