Editing Ramanujan graph

{{Short description|Spectral graph theory concept}}
In the mathematical field of [[spectral graph theory]], a '''Ramanujan graph''' is a [[regular graph]] whose [[spectral gap]] is almost as large as possible (see [[extremal graph theory]]). Such graphs are excellent [[expander graph|spectral expanders]]. As Murty's survey paper<ref>[http://www.mast.queensu.ca/~murty/ramanujan.pdf Survey paper by M. Ram Murty]</ref> notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, [[number theory]], [[representation theory]], and [[algebraic geometry]]". 
These graphs are indirectly named after [[Srinivasa Ramanujan]]; their name comes from the [[Ramanujan–Petersson conjecture]], which was used in a construction of some of these graphs.

==Definition==

Let <math>G</math> be a connected <math>d</math>-regular graph with <math>n</math> vertices, and let <math>\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n</math> be the [[eigenvalues]] of the [[adjacency matrix]] of <math>G</math> (or the [[Spectral graph theory|spectrum]] of <math>G</math>). Because <math>G</math> is connected and <math>d</math>-regular, its eigenvalues satisfy <math>d = \lambda_1 > \lambda_2 </math> <math> \geq \cdots \geq \lambda_n \geq -d </math>.

Define <math>\lambda(G) = \max_{i\neq 1}|\lambda_i| = \max(|\lambda_2|,\ldots, |\lambda_n|)</math>. A connected <math>d</math>-regular graph <math>G</math> is a ''Ramanujan graph'' if <math>\lambda(G) \leq 2\sqrt{d-1}</math>.

Many sources uses an alternative definition <math>\lambda'(G) = \max_{|\lambda_i| < d} |\lambda_i|</math> (whenever there exists <math>\lambda_i</math> with <math>|\lambda_i| < d</math>) to define Ramanujan graphs.<ref name="lps88">{{cite journal|author=Alexander Lubotzky|author2=Ralph Phillips|author3=Peter Sarnak|year=1988|title=Ramanujan graphs|journal=Combinatorica|volume=8|issue=3|pages=261–277|doi=10.1007/BF02126799|s2cid=206812625}}</ref> In other words, we allow <math>-d</math> in addition to the "small" eigenvalues. Since <math>\lambda_n = -d</math> if and only if the graph is [[Bipartite graph|bipartite]], we will refer to the graphs that satisfy this alternative definition but not the first definition ''bipartite Ramanujan graphs''. If <math>G</math> is a Ramanujan graph, then <math>G \times K_2</math> is a bipartite Ramanujan graph, so the existence of Ramanujan graphs is stronger.

As observed by [[Toshikazu Sunada]], a regular graph is Ramanujan if and only if its [[Ihara zeta function]] satisfies an analog of the [[Riemann hypothesis]].<ref>{{citation|last=Terras|first=Audrey|title=Zeta functions of graphs: A stroll through the garden|volume=128|year=2011|series=Cambridge Studies in Advanced Mathematics|publisher=Cambridge University Press|isbn=978-0-521-11367-0|mr=2768284|authorlink=Audrey Terras}}</ref>

== Examples and constructions ==

=== Explicit examples ===

* The [[complete graph]] <math>K_{d+1}</math> has spectrum <math>d, -1, -1, \dots, -1</math>, and thus <math>\lambda(K_{d+1}) = 1</math> and the graph is a Ramanujan graph for every <math>d > 1</math>. The [[complete bipartite graph]] <math>K_{d,d}</math> has spectrum <math>d, 0, 0, \dots, 0, -d</math> and hence is a bipartite Ramanujan graph for every <math>d</math>.
* The [[Petersen graph]] has spectrum <math>3, 1, 1, 1, 1, 1, -2, -2, -2, -2</math>, so it is a 3-regular Ramanujan graph. The [[Regular icosahedron|icosahedral graph]] is a 5-regular Ramanujan graph.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Icosahedral Graph|url=http://mathworld.wolfram.com/IcosahedralGraph.html|access-date=2019-11-29|website=mathworld.wolfram.com|language=en}}</ref>
* A [[Paley graph]] of order <math>q</math> is <math>\frac{q-1}{2}</math>-regular with all other eigenvalues being <math>\frac{-1\pm\sqrt{q}}{2}</math>, making Paley graphs an infinite family of Ramanujan graphs.
* More generally, let <math>f(x)</math> be a degree 2 or 3 polynomial over <math>\mathbb{F}_q</math>. Let <math>S = \{f(x)\, :\, x \in \mathbb{F}_q\}</math> be the image of <math>f(x)</math> as a multiset, and suppose <math>S = -S</math>. Then the [[Cayley graph]] for <math>\mathbb{F}_q</math> with generators from <math>S</math> is a Ramanujan graph.

Mathematicians are often interested in constructing infinite families of <math>d</math>-regular Ramanujan graphs for every fixed <math>d</math>. Such families are useful in applications.

=== Algebraic constructions ===
Several explicit constructions of Ramanujan graphs arise as Cayley graphs and are algebraic in nature. See Winnie Li's survey on Ramanujan's conjecture and other aspects of number theory relevant to these results.<ref>{{Cite journal|last=Li|first=Winnie|title=The Ramanujan conjecture and its applications|url=https://doi.org/10.1098/rsta.2018.0441|journal=Philosophical Transactions of the Royal Society A|year=2020|volume=378-2163|issue=2163|doi=10.1098/rsta.2018.0441| pmc=6939229|pmid=31813366|bibcode=2020RSPTA.37880441W}}</ref>

[[Alexander Lubotzky|Lubotzky]], [[Ralph S. Phillips|Phillips]] and [[Peter Sarnak|Sarnak]]<ref name="lps88" /> and independently [[Grigory Margulis|Margulis]]<ref>{{Cite journal|last=Margulis|first=G. A.|date=1988|title=Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=686&option_lang=eng|journal=Probl. Peredachi Inf.|volume=24-1|pages=51–60}}</ref> showed how to construct an infinite family of <math>(p+1)</math>-regular Ramanujan graphs, whenever <math>p</math> is a [[prime number]] and <math>p\equiv 1 \pmod 4</math>. Both proofs use the [[Ramanujan conjecture]], which led to the name of Ramanujan graphs. Besides being Ramanujan graphs, these constructions satisfies some other properties, for example, their [[girth (graph theory)|girth]] is <math>\Omega(\log_{p}(n))</math> where <math>n</math> is the number of nodes.

Let us sketch the Lubotzky-Phillips-Sarnak construction. Let <math>q \equiv 1 \bmod 4</math> be a prime not equal to <math>p</math>. By [[Jacobi's four-square theorem]], there are <math>p+1</math> solutions to the equation <math>p=a_0^2+a_1^2+a_2^2+a_3^2</math> where <math>a_0 > 0</math> is odd and <math>a_1, a_2, a_3</math> are even. To each such solution associate the <math>\operatorname{PGL}(2,\Z/q\Z)</math> matrix <math display="block">\tilde \alpha  = \begin{pmatrix}a_0 + ia_1 & a_2 + ia_3 \\ -a_2 + ia_3 & a_0 - ia_1\end{pmatrix},\qquad i \text{ a fixed solution to } i^2 = -1 \bmod q.</math>If <math>p
</math> is not a quadratic residue modulo <math>q</math> let <math>X^{p,q}</math> be the Cayley graph of <math>\operatorname{PGL}(2,\Z/q\Z)</math> with these <math>p+1</math> generators, and otherwise, let <math>X^{p,q}</math> be the Cayley graph of <math>\operatorname{PSL}(2,\Z/q\Z)</math> with the same generators. Then <math>X^{p,q}</math> is a <math>(p+1)</math>-regular graph on <math>n=q(q^2-1)</math> or <math>q(q^2-1)/2</math> vertices depending on whether or not  <math>p
</math> is a quadratic residue modulo <math>q</math>. It is proved that <math>X^{p,q}</math> is a Ramanujan graph.

Morgenstern<ref name="m94">{{cite journal|author=Moshe Morgenstern|year=1994|title=Existence and Explicit Constructions of q+1 Regular Ramanujan Graphs for Every Prime Power q|journal=[[Journal of Combinatorial Theory]] | series=Series B|volume=62|pages=44–62|doi=10.1006/jctb.1994.1054|doi-access=free}}</ref> later extended the construction of Lubotzky, Phillips and Sarnak. His extended construction holds whenever <math>p</math> is a [[prime power]].

Arnold Pizer proved that the [[supersingular isogeny graph]]s are Ramanujan, although they tend to have lower girth than the graphs of Lubotzky, Phillips, and Sarnak.<ref>{{citation|last=Pizer|first=Arnold K.|title=Ramanujan graphs and Hecke operators|journal=Bulletin of the American Mathematical Society|volume=23|issue=1|pages=127–137|year=1990|series=New Series|doi=10.1090/S0273-0979-1990-15918-X|mr=1027904|doi-access=free}}</ref> Like the graphs of Lubotzky, Phillips, and Sarnak, the degrees of these graphs are always a prime number plus one.

=== Probabilistic examples ===
[[Adam Marcus (mathematician)|Adam Marcus]], [[Daniel Spielman]] and [[Nikhil Srivastava]]<ref name="mss13">{{cite conference|author=Adam Marcus|author1-link=Adam Marcus (mathematician)|author2=Daniel Spielman|author2-link=Daniel Spielman|author3=Nikhil Srivastava|author3-link=Nikhil Srivastava|year=2013|title=Interlacing families I: Bipartite Ramanujan graphs of all degrees|url=https://annals.math.princeton.edu/wp-content/uploads/Marcus_Spielman_SrivastavaIFI.pdf|conference=Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium}}</ref> proved the existence of infinitely many <math>d</math>-regular ''bipartite'' Ramanujan graphs for any <math>d\geq 3</math>. Later<ref name="mss15">{{cite conference|author=Adam Marcus|author1-link=Adam Marcus (mathematician)|author2=Daniel Spielman|author2-link=Daniel Spielman|author3=Nikhil Srivastava|author3-link=Nikhil Srivastava|year=2015|title=Interlacing families IV: Bipartite Ramanujan graphs of all sizes|url=https://www.cs.yale.edu/homes/spielman/PAPERS/IF4.pdf|conference=Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium}}</ref> they proved that there exist bipartite Ramanujan graphs of every degree and every number of vertices. Michael B. Cohen<ref name="c16">{{cite conference|author=Michael B. Cohen|year=2016|title=Ramanujan Graphs in Polynomial Time|conference=Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium|arxiv=1604.03544|doi=10.1109/FOCS.2016.37}}</ref> showed how to construct these graphs in polynomial time.

The initial work followed an approach of Bilu and [[Nati Linial|Linial]]. They considered an operation called a 2-lift that takes a <math>d</math>-regular graph <math>G</math> with <math>n</math> vertices and a sign on each edge, and produces a new <math>d</math>-regular graph <math>G'</math> on <math>2n</math> vertices. Bilu & Linial conjectured that there always exists a signing so that every new eigenvalue of <math>G'</math> has magnitude at most <math>2\sqrt{d-1}</math>. This conjecture guarantees the existence of Ramanujan graphs with degree <math>d</math> and <math>2^k(d+1)</math> vertices for any <math>k</math>—simply start with the complete graph <math>K_{d+1}</math>, and iteratively take 2-lifts that retain the Ramanujan property.

Using the method of interlacing polynomials, Marcus, Spielman, and Srivastava<ref name="mss13" /> proved Bilu & Linial's conjecture holds when <math>G</math> is already a bipartite Ramanujan graph, which is enough to conclude the existence result. The sequel<ref name="mss15" /> proved the stronger statement that a sum of <math>d</math> random bipartite matchings is Ramanujan with non-vanishing probability. Hall, Puder and Sawin<ref>{{cite journal|last1=Hall|first1=Chris|last2=Puder|first2=Doron|last3=Sawin|first3=William F.|date=2018|title=Ramanujan coverings of graphs|journal=[[Advances in Mathematics]] |volume=323 |pages=367–410 |doi=10.1016/j.aim.2017.10.042 |arxiv=1506.02335}}</ref> extended the original work of Marcus, Spielman and Srivastava to {{mvar|r}}-lifts.

It is still an open problem whether there are infinitely many <math>d</math>-regular (non-bipartite) Ramanujan graphs for any <math>d\geq 3</math>. In particular, the problem is open for <math>d = 7</math>, the smallest case for which <math>d-1</math> is not a prime power and hence not covered by Morgenstern's construction.

== Ramanujan graphs as expander graphs ==
The constant <math>2\sqrt{d-1}</math> in the definition of Ramanujan graphs is asymptotically sharp. More precisely, the [[Alon-Boppana bound]] states that for every <math>d</math> and <math>\epsilon > 0</math>, there exists <math>n</math> such that all <math>d</math>-regular graphs <math>G</math> with at least <math>n</math> vertices satisfy <math>\lambda(G) > 2\sqrt{d-1} - \epsilon</math>. This means that Ramanujan graphs are essentially the best possible [[expander graph]]s.

Due to achieving the tight bound on <math>\lambda (G)</math>, the [[expander mixing lemma]] gives excellent bounds on the uniformity of the distribution of the edges in Ramanujan graphs, and any [[random walk]]s on the graphs has a logarithmic [[Markov chain mixing time|mixing time]] (in terms of the number of vertices): in other words, the random walk converges to the (uniform) [[stationary distribution]] very quickly. Therefore, the diameter of Ramanujan graphs are also bounded logarithmically in terms of the number of vertices.

=== Random graphs ===
Confirming a conjecture of [[Noga Alon|Alon]], Friedman<ref>{{Cite journal|last=Friedman|first=Joel|date=2003|title=Relative expanders or weakly relatively Ramanujan graphs|journal=Duke Mathematical Journal |volume=118|issue=1|pages=19–35|doi=10.1215/S0012-7094-03-11812-8|mr=1978881}}</ref> showed that many families of random graphs are ''weakly-Ramanujan''. This means that for every <math>d</math> and <math>\epsilon > 0</math> and for sufficiently large <math>n</math>, a random <math>d</math>-regular <math>n</math>-vertex graph <math>G</math> satisfies <math>\lambda(G) < 2\sqrt{d-1} + \epsilon</math> with high probability. While this result shows that random graphs are close to being Ramanujan, it cannot be used to prove the existence of Ramanujan graphs. It is conjectured,<ref>{{cite journal
| last1=Miller | first1=Steven J.
| last2=Novikoff | first2=Tim
| last3=Sabelli | first3=Anthony
| date=2008
| title=The Distribution of the Largest Non-trivial Eigenvalues in Families of Random Regular Graphs
| arxiv=math/0611649
| journal=Experimental Mathematics
| volume=17
| issue=2
| pages=231–244
| doi=10.1080/10586458.2008.10129029
| url=https://projecteuclid.org/journals/experimental-mathematics/volume-17/issue-2/The-Distribution-of-the-Largest-Nontrivial-Eigenvalues-in-Families-of/em/1227118974.full}}</ref> though, that random graphs are Ramanujan with substantial probability (roughly 52%). In addition to direct numerical evidence, there is some theoretical support for this conjecture: the spectral gap of a <math>d</math>-regular graph seems to behave according to a  [[Tracy-Widom distribution]] from random matrix theory, which would predict the same asymptotic.

In 2024 a preprint by Jiaoyang Huang, Theo McKenzieand and [[Horng-Tzer Yau]] proved that <math>\lambda(G) \le 2\sqrt{d-1}</math> with the fraction of eigenvalues that hit the Alon-Boppana bound approximately 69% from proving that [[Random matrix#Edge statistics|edge universality]] holds, that is they follow a [[Tracy–Widom distribution|Tracy-Widom distribution]] associated with the [[Gaussian Orthogonal Ensemble]]<ref>{{cite arXiv | eprint=2412.20263 | last1=Huang | first1=Jiaoyang | last2=McKenzie | first2=Theo | last3=Yau | first3=Horng-Tzer | title=Ramanujan Property and Edge Universality of Random Regular Graphs | date=2024 | class=math.PR }}</ref><ref>{{Cite web |last=Sloman |first=Leila |date=2025-04-18 |title=New Proof Settles Decades-Old Bet About Connected Networks |url=https://www.quantamagazine.org/new-proof-settles-decades-old-bet-about-connected-networks-20250418/ |access-date=2025-05-06 |website=Quanta Magazine |language=en}}</ref>

== 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]].

== See also ==

* [[Expander graph]]
*[[Alon–Boppana bound|Alon-Boppana bound]]
* [[Expander mixing lemma]]
* [[Spectral graph theory]]

==References==
{{Reflist}}

==Further reading==
* {{cite book | author1=Giuliana Davidoff|author1-link=Giuliana Davidoff |author2=Peter Sarnak |authorlink2 =Peter Sarnak |author3=Alain Valette  | title=Elementary number theory, group theory and Ramanujan graphs |title-link= Elementary Number Theory, Group Theory and Ramanujan Graphs | publisher=[[Cambridge University Press]] | series=LMS student texts | volume=55 | year=2003 | isbn=0-521-53143-8 | oclc=50253269 }}
* {{cite conference
 | last = Sunada | first = Toshikazu
 | editor1-last = Shiohama | editor1-first = Katsuhiro
 | editor2-last = Sakai | editor2-first = Takashi
 | editor3-last = Sunada | editor3-first = Toshikazu
 | contribution = {{mvar|L}}-functions in geometry and some applications
 | doi = 10.1007/BFb0075662
 | isbn = 978-3-540-16770-9
 | location = Berlin
 | mr = 859591
 | pages = 266–284
 | publisher = Springer
 | series = Lecture Notes in Mathematics
 | title = Curvature and Topology of Riemannian Manifolds: Proceedings of the 17th International Taniguchi Symposium held in Katata, Japan, August 26–31, 1985
 | volume = 1201
 | year = 1986}}

==External links==
* [http://www.mast.queensu.ca/~murty/ramanujan.pdf Survey paper by M. Ram Murty]
*[https://www.ams.org/journals/bull/2012-49-01/S0273-0979-2011-01359-3/ Survey paper by Alexander Lubotzky]
*[https://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01126-8/S0273-0979-06-01126-8.pdf Survey paper by Hoory, Linial, and Wigderson]

[[Category:Graph families]]
[[Category:Algebraic graph theory]]
[[Category:Spectral theory]]
[[Category:Regular graphs]]
[[Category:Srinivasa Ramanujan]]