Ramanujan graph

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 spectral expanders. As Murty's survey paper[1] 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 ${\displaystyle G}$ be a connected ${\displaystyle d}$-regular graph with ${\displaystyle n}$ vertices, and let ${\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}}$ be the eigenvalues of the adjacency matrix of ${\displaystyle G}$ (or the spectrum of ${\displaystyle G}$). Because ${\displaystyle G}$ is connected and ${\displaystyle d}$-regular, its eigenvalues satisfy ${\displaystyle d=\lambda _{1}>\lambda _{2}}$ ${\displaystyle \geq \cdots \geq \lambda _{n}\geq -d}$.

Define ${\displaystyle \lambda (G)=\max _{i\neq 1}|\lambda _{i}|=\max(|\lambda _{2}|,\ldots ,|\lambda _{n}|)}$. A connected ${\displaystyle d}$-regular graph ${\displaystyle G}$ is a Ramanujan graph if ${\displaystyle \lambda (G)\leq 2{\sqrt {d-1}}}$.

Many sources uses an alternative definition ${\displaystyle \lambda '(G)=\max _{|\lambda _{i}|<d}|\lambda _{i}|}$ (whenever there exists ${\displaystyle \lambda _{i}}$ with ${\displaystyle |\lambda _{i}|<d}$) to define Ramanujan graphs.[2] In other words, we allow ${\displaystyle -d}$ in addition to the "small" eigenvalues. Since ${\displaystyle \lambda _{n}=-d}$ if and only if the graph is bipartite, we will refer to the graphs that satisfy this alternative definition but not the first definition bipartite Ramanujan graphs. If ${\displaystyle G}$ is a Ramanujan graph, then ${\displaystyle G\times K_{2}}$ 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.[3]

Examples and constructions

Explicit examples

• The complete graph ${\displaystyle K_{d+1}}$ has spectrum ${\displaystyle d,-1,-1,\dots ,-1}$, and thus ${\displaystyle \lambda (K_{d+1})=1}$ and the graph is a Ramanujan graph for every ${\displaystyle d>1}$. The complete bipartite graph ${\displaystyle K_{d,d}}$ has spectrum ${\displaystyle d,0,0,\dots ,0,-d}$ and hence is a bipartite Ramanujan graph for every ${\displaystyle d}$.
• The Petersen graph has spectrum ${\displaystyle 3,1,1,1,1,1,-2,-2,-2,-2}$, so it is a 3-regular Ramanujan graph. The icosahedral graph is a 5-regular Ramanujan graph.[4]
• A Paley graph of order ${\displaystyle q}$ is ${\displaystyle {\frac {q-1}{2}}}$-regular with all other eigenvalues being ${\displaystyle {\frac {-1\pm {\sqrt {q}}}{2}}}$, making Paley graphs an infinite family of Ramanujan graphs.
• More generally, let ${\displaystyle f(x)}$ be a degree 2 or 3 polynomial over ${\displaystyle \mathbb {F} _{q}}$. Let ${\displaystyle S=\{f(x)\,:\,x\in \mathbb {F} _{q}\}}$ be the image of ${\displaystyle f(x)}$ as a multiset, and suppose ${\displaystyle S=-S}$. Then the Cayley graph for ${\displaystyle \mathbb {F} _{q}}$ with generators from ${\displaystyle S}$ is a Ramanujan graph.

Mathematicians are often interested in constructing infinite families of ${\displaystyle d}$-regular Ramanujan graphs for every fixed ${\displaystyle d}$. 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.[5]

Lubotzky, Phillips and Sarnak[2] and independently Margulis[6] showed how to construct an infinite family of ${\displaystyle (p+1)}$-regular Ramanujan graphs, whenever ${\displaystyle p}$ is a prime number and ${\displaystyle p\equiv 1{\pmod {4}}}$. 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 is ${\displaystyle \Omega (\log _{p}(n))}$ where ${\displaystyle n}$ is the number of nodes.

Let us sketch the Lubotzky-Phillips-Sarnak construction. Let ${\displaystyle q\equiv 1{\bmod {4}}}$ be a prime not equal to ${\displaystyle p}$. By Jacobi's four-square theorem, there are ${\displaystyle p+1}$ solutions to the equation ${\displaystyle p=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$ where ${\displaystyle a_{0}>0}$ is odd and ${\displaystyle a_{1},a_{2},a_{3}}$ are even. To each such solution associate the ${\displaystyle \operatorname {PGL} (2,\mathbb {Z} /q\mathbb {Z} )}$ matrix $${\displaystyle {\tilde {\alpha }}={\begin{pmatrix}a_{0}+ia_{1}&a_{2}+ia_{3}\\-a_{2}+ia_{3}&a_{0}-ia_{1}\end{pmatrix}},\qquad i{\text{ a fixed solution to }}i^{2}=-1{\bmod {q}}.}$$If ${\displaystyle p}$ is not a quadratic residue modulo ${\displaystyle q}$ let ${\displaystyle X^{p,q}}$ be the Cayley graph of ${\displaystyle \operatorname {PGL} (2,\mathbb {Z} /q\mathbb {Z} )}$ with these ${\displaystyle p+1}$ generators, and otherwise, let ${\displaystyle X^{p,q}}$ be the Cayley graph of ${\displaystyle \operatorname {PSL} (2,\mathbb {Z} /q\mathbb {Z} )}$ with the same generators. Then ${\displaystyle X^{p,q}}$ is a ${\displaystyle (p+1)}$-regular graph on ${\displaystyle n=q(q^{2}-1)}$ or ${\displaystyle q(q^{2}-1)/2}$ vertices depending on whether or not ${\displaystyle p}$ is a quadratic residue modulo ${\displaystyle q}$. It is proved that ${\displaystyle X^{p,q}}$ is a Ramanujan graph.

Morgenstern[7] later extended the construction of Lubotzky, Phillips and Sarnak. His extended construction holds whenever ${\displaystyle p}$ is a prime power.

Arnold Pizer proved that the supersingular isogeny graphs are Ramanujan, although they tend to have lower girth than the graphs of Lubotzky, Phillips, and Sarnak.[8] Like the graphs of Lubotzky, Phillips, and Sarnak, the degrees of these graphs are always a prime number plus one.

Probabilistic examples

Adam Marcus, Daniel Spielman and Nikhil Srivastava[9] proved the existence of infinitely many ${\displaystyle d}$-regular bipartite Ramanujan graphs for any ${\displaystyle d\geq 3}$. Later[10] they proved that there exist bipartite Ramanujan graphs of every degree and every number of vertices. Michael B. Cohen[11] showed how to construct these graphs in polynomial time.

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

Using the method of interlacing polynomials, Marcus, Spielman, and Srivastava[9] proved Bilu & Linial's conjecture holds when ${\displaystyle G}$ is already a bipartite Ramanujan graph, which is enough to conclude the existence result. The sequel[10] proved the stronger statement that a sum of ${\displaystyle d}$ random bipartite matchings is Ramanujan with non-vanishing probability. Hall, Puder and Sawin[12] extended the original work of Marcus, Spielman and Srivastava to r-lifts.

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

Ramanujan graphs as expander graphs

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

Due to achieving the tight bound on ${\displaystyle \lambda (G)}$, the expander mixing lemma gives excellent bounds on the uniformity of the distribution of the edges in Ramanujan graphs, and any random walks on the graphs has a logarithmic 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 Alon, Friedman[13] showed that many families of random graphs are weakly-Ramanujan. This means that for every ${\displaystyle d}$ and ${\displaystyle \epsilon >0}$ and for sufficiently large ${\displaystyle n}$, a random ${\displaystyle d}$-regular ${\displaystyle n}$-vertex graph ${\displaystyle G}$ satisfies ${\displaystyle \lambda (G)<2{\sqrt {d-1}}+\epsilon }$ 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,[14] 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 ${\displaystyle d}$-regular graph seems to behave according to a Tracy-Widom distribution from random matrix theory, which would predict the same asymptotic.

Applications of Ramanujan graphs

Expander graphs have many applications to computer science, number theory, and group theory, see e.g Lubotzky's survey on applications to pure and applied math and 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[15] relied on the existence of bipartite Ramanujan graphs of every degree in order to quickly approximate the complete graph.
• Lubetzky and Peres proved that the simple random walk exhibits cutoff phenomenon on all Ramanujan graphs.[16] 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.[17]
• Ramanujan graphs of Pizer have been proposed as the basis for post-quantum elliptic-curve cryptography.[18]
• 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

1. Survey paper by M. Ram Murty
2. Alexander Lubotzky; Ralph Phillips; Peter Sarnak (1988). "Ramanujan graphs". Combinatorica. 8 (3): 261–277. doi:10.1007/BF02126799. S2CID 206812625.
3. Terras, Audrey (2011), Zeta functions of graphs: A stroll through the garden, Cambridge Studies in Advanced Mathematics, vol. 128, Cambridge University Press, ISBN 978-0-521-11367-0, MR 2768284
4. Weisstein, Eric W. "Icosahedral Graph". mathworld.wolfram.com. Retrieved 2019-11-29.
5. Li, Winnie (2020). "The Ramanujan conjecture and its applications". Philosophical Transactions of the Royal Society A. 378–2163 (2163). Bibcode:2020RSPTA.37880441W. doi:10.1098/rsta.2018.0441. PMC 6939229. PMID 31813366.
6. Margulis, G. A. (1988). "Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators". Probl. Peredachi Inf. 24–1: 51–60.
7. Moshe Morgenstern (1994). "Existence and Explicit Constructions of q+1 Regular Ramanujan Graphs for Every Prime Power q". Journal of Combinatorial Theory. Series B. 62: 44–62. doi:10.1006/jctb.1994.1054.
8. Pizer, Arnold K. (1990), "Ramanujan graphs and Hecke operators", Bulletin of the American Mathematical Society, New Series, 23 (1): 127–137, doi:10.1090/S0273-0979-1990-15918-X, MR 1027904
9. Adam Marcus; Daniel Spielman; Nikhil Srivastava (2013). Interlacing families I: Bipartite Ramanujan graphs of all degrees (PDF). Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium.
10. Adam Marcus; Daniel Spielman; Nikhil Srivastava (2015). Interlacing families IV: Bipartite Ramanujan graphs of all sizes (PDF). Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium.
11. Michael B. Cohen (2016). Ramanujan Graphs in Polynomial Time. Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium. arXiv:1604.03544. doi:10.1109/FOCS.2016.37.
12. Hall, Chris; Puder, Doron; Sawin, William F. (2018). "Ramanujan coverings of graphs". Advances in Mathematics. 323: 367–410. arXiv:1506.02335. doi:10.1016/j.aim.2017.10.042.
13. Friedman, Joel (2003). "Relative expanders or weakly relatively Ramanujan graphs". Duke Mathematical Journal. 118 (1): 19–35. doi:10.1215/S0012-7094-03-11812-8. MR 1978881.
14. Miller, Steven J.; Novikoff, Tim; Sabelli, Anthony (2008). "The Distribution of the Largest Non-trivial Eigenvalues in Families of Random Regular Graphs". Experimental Mathematics. 17 (2): 231–244. arXiv:math/0611649. doi:10.1080/10586458.2008.10129029.
15. Lee, Yin Tat; Peng, Richard; Spielman, Daniel A. (2015-08-13). "Sparsified Cholesky Solvers for SDD linear systems". arXiv:1506.08204 [cs.DS].
16. Lubetzky, Eyal; Peres, Yuval (2016-07-01). "Cutoff on all Ramanujan graphs". Geometric and Functional Analysis. 26 (4): 1190–1216. arXiv:1507.04725. doi:10.1007/s00039-016-0382-7. ISSN 1420-8970. S2CID 13803649.
17. Lubetzky, Eyal; Sly, Allan (2011-01-01). "Explicit Expanders with Cutoff Phenomena". Electronic Journal of Probability. 16 (none). arXiv:1003.3515. doi:10.1214/EJP.v16-869. ISSN 1083-6489. S2CID 9121682.
18. Eisenträger, Kirsten; Hallgren, Sean; Lauter, Kristin; Morrison, Travis; Petit, Christophe (2018), "Supersingular isogeny graphs and endomorphism rings: Reductions and solutions" (PDF), in Nielsen, Jesper Buus; Rijmen, Vincent (eds.), 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 (PDF), Lecture Notes in Computer Science, vol. 10822, Cham: Springer, pp. 329–368, doi:10.1007/978-3-319-78372-7_11, ISBN 978-3-319-78371-0, MR 3794837, S2CID 4850644

Further reading

• Giuliana Davidoff; Peter Sarnak; Alain Valette (2003). Elementary number theory, group theory and Ramanujan graphs. LMS student texts. Vol. 55. Cambridge University Press. ISBN 0-521-53143-8. OCLC 50253269.
• Sunada, Toshikazu (1986). "L-functions in geometry and some applications". In Shiohama, Katsuhiro; Sakai, Takashi; Sunada, Toshikazu (eds.). Curvature and Topology of Riemannian Manifolds: Proceedings of the 17th International Taniguchi Symposium held in Katata, Japan, August 26–31, 1985. Lecture Notes in Mathematics. Vol. 1201. Berlin: Springer. pp. 266–284. doi:10.1007/BFb0075662. ISBN 978-3-540-16770-9. MR 0859591.

External links

• Survey paper by M. Ram Murty
• Survey paper by Alexander Lubotzky
• Survey paper by Hoory, Linial, and Wigderson