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Ramanujan graph
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== Applications of Ramanujan graphs == Expander graphs have many [[Expander graph|applications]] to computer science, number theory, and group theory, see e.g [https://www.ams.org/journals/bull/2012-49-01/S0273-0979-2011-01359-3/ Lubotzky's survey] on applications to pure and applied math and [https://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01126-8/S0273-0979-06-01126-8.pdf Hoory, Linial, and Wigderson's survey] which focuses on computer science. Ramanujan graphs are in some sense the best expanders, and so they are especially useful in applications where expanders are needed. Importantly, the Lubotzky, Phillips, and Sarnak graphs can be traversed extremely quickly in practice, so they are practical for applications. Some example applications include * In an application to fast solvers for Laplacian linear systems, Lee, Peng, and Spielman<ref>{{cite arXiv|last1=Lee|first1=Yin Tat|last2=Peng|first2=Richard|last3=Spielman|first3=Daniel A.|date=2015-08-13|title=Sparsified Cholesky Solvers for SDD linear systems|class=cs.DS|eprint=1506.08204}}</ref> relied on the existence of bipartite Ramanujan graphs of every degree in order to quickly approximate the complete graph. *Lubetzky and [[Yuval Peres|Peres]] proved that the simple random walk exhibits [[cutoff phenomenon]] on all Ramanujan graphs.<ref>{{Cite journal|last1=Lubetzky|first1=Eyal|last2=Peres|first2=Yuval|date=2016-07-01|title=Cutoff on all Ramanujan graphs|url=https://doi.org/10.1007/s00039-016-0382-7|journal=Geometric and Functional Analysis|language=en|volume=26|issue=4|pages=1190–1216|doi=10.1007/s00039-016-0382-7|arxiv=1507.04725|s2cid=13803649|issn=1420-8970}}</ref> 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.<ref>{{Cite journal|last1=Lubetzky|first1=Eyal|last2=Sly|first2=Allan|date=2011-01-01|title=Explicit Expanders with Cutoff Phenomena|journal=Electronic Journal of Probability|volume=16|issue=none|doi=10.1214/EJP.v16-869|s2cid=9121682|issn=1083-6489|doi-access=free|arxiv=1003.3515}}</ref> * Ramanujan graphs of Pizer have been proposed as the basis for [[post-quantum cryptography|post-quantum]] [[elliptic-curve cryptography]].<ref>{{citation|last1=Eisenträger|first1=Kirsten|title=Advances in Cryptology – EUROCRYPT 2018: 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tel Aviv, Israel, April 29 - May 3, 2018, Proceedings, Part III|url=http://pure-oai.bham.ac.uk/ws/files/47705132/Supersingular.pdf|volume=10822|pages=329–368|year=2018|editor1-last=Nielsen|editor1-first=Jesper Buus|series=Lecture Notes in Computer Science|contribution=Supersingular isogeny graphs and endomorphism rings: Reductions and solutions|contribution-url=https://eprint.iacr.org/2018/371.pdf|location=Cham|publisher=Springer|doi=10.1007/978-3-319-78372-7_11|mr=3794837|last2=Hallgren|first2=Sean|last3=Lauter|first3=Kristin|last4=Morrison|first4=Travis|last5=Petit|first5=Christophe|isbn=978-3-319-78371-0 |s2cid=4850644 |author1-link=Kirsten Eisenträger|author3-link=Kristin Lauter|editor2-last=Rijmen|editor2-first=Vincent|editor2-link=Vincent Rijmen}}</ref> * Ramanujan graphs can be used to construct [[expander code]]s, which are good [[Error correction code|error correcting codes]].
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