Ramanujan graph

In spectral graph theory, a Ramanujan graph, named after Srinivasa Ramanujan, is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders.

Examples of Ramanujan graphs include the clique, the biclique $K_{{n,n}}$ , and the Petersen graph. As Murty's survey paper notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". 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.^[1]

Definition

Let $G$ be a connected $d$ -regular graph with $n$ vertices, and let $\lambda _{0}\geq \lambda _{1}\geq \cdots \geq \lambda _{n-1}$ be the eigenvalues of the adjacency matrix of $G$ . Because $G$ is connected and $d$ -regular, its eigenvalues satisfy $d=\lambda _{0}>\lambda _{1}$ $\geq \cdots \geq \lambda _{n-1}\geq -d$ . Whenever there exists $\lambda _{i}$ with $|\lambda _{i}|<d$ , define

\lambda (G)=\max _{{|\lambda _{i}|<d}}|\lambda _{i}|.\,

A $d$ -regular graph $G$ is a Ramanujan graph if $\lambda (G)\leq 2{\sqrt {d-1}}$ .

A Ramanujan graph is characterized as a regular graph whose Ihara zeta function satisfies an analogue of the Riemann Hypothesis.

Extremality of Ramanujan graphs

For a fixed $d$ and large $n$ , the $d$ -regular, $n$ -vertex Ramanujan graphs nearly minimize $\lambda (G)$ . If $G$ is a $d$ -regular graph with diameter $m$ , a theorem due to Noga Alon^[2] states

\lambda _{1}\geq 2{\sqrt {d-1}}-{\frac {2{\sqrt {d-1}}-1}{\lfloor m/2\rfloor }}.

Whenever $G$ is $d$ -regular and connected on at least three vertices, $|\lambda _{1}|<d$ , and therefore $\lambda (G)\geq \lambda _{1}$ . Let ${\mathcal {G}}_{n}^{d}$ be the set of all connected $d$ -regular graphs $G$ with at least $n$ vertices. Because the minimum diameter of graphs in ${\mathcal {G}}_{n}^{d}$ approaches infinity for fixed $d$ and increasing $n$ , this theorem implies an earlier theorem of Alon and Boppana which states

\lim _{{n\to \infty }}\inf _{{G\in {\mathcal {G}}_{n}^{d}}}\lambda (G)\geq 2{\sqrt {d-1}}.

A slightly stronger bound is

\lambda _{1}\geq 2{\sqrt {d-1}}\cdot \left(1-{\frac {c}{m^{2}}}\right),

where $c\approx 2\pi ^{2}$ . The outline of Friedman's proof is the following. Take $k=\left\lfloor {\frac {m}{2}}\right\rfloor -1$ . Let $T_{d,k}$ be the $d$ -regular tree of height $k$ and let $A_{T_{k}}$ be its adjacency matrix. We want to prove that $\lambda (G)\geq \lambda (T_{d,k})=2{\sqrt {d-1}}\cos \theta$ , for some $\theta$ depending only on $m$ . Define a function $g:V(T_{d,k})\rightarrow \mathbb {R}$ by $g(x)=(d-1)^{-\delta /2}\cdot \sin((k+1-\delta )\theta )$ , where $\delta$ is the distance from $x$ to the root of $T_{d,k}$ . Choosing a $\theta$ close to $2\pi /m$ it can be proved that $A_{t,k}g=\lambda (T_{d,k})g$ . Now let $s$ and $t$ be a pair of vertices at distance $m$ and define

f(v)={\begin{cases}c_{1}g(v_{s})&{\text{for }}v{\text{ at distance }}\leq k{\text{ from }}s,\\-c_{2}g(v_{t})&{\text{for }}v{\text{ at distance }}\leq k{\text{ from }}t,\\0&{\text{otherwise,}}\end{cases}}

where $v_s$ is a vertex in $T_{d,k}$ which distance to the root is equal to the distance from $v$ to $s$ and the symmetric for $v_{t}$ . By choosing properly $c_{1},c_{2}$ we get $f\perp 1$ , $(Af)_{v}\geq \lambda (T_{d,k})f_{v}$ for $v$ close to $s$ and $(Af)_{v}\leq \lambda (T_{d,k})f_{v}$ for $v$ close to $t$ . Then by the min-max theorem $\lambda (G)\geq fAf^{T}/\|f\|^{2}\geq \lambda (T_{d,k})$ .

Constructions

Constructions of Ramanujan graphs are often algebraic.

Lubotzky, Phillips and Sarnak^[3] show how to construct an infinite family of $(p+1)$ -regular Ramanujan graphs, whenever $p$ is a prime number and $p\equiv 1{\pmod {4}}$ . Their proof uses the Ramanujan conjecture, which led to the name of Ramanujan graphs. Besides being Ramanujan graphs, their construction satisfies some other properties, for example, their girth is $\Omega (\log _{p}(n))$ where $n$ is the number of nodes.
Morgenstern^[4] extended the construction of Lubotzky, Phillips and Sarnak. His extended construction holds whenever $p$ 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.^[5] Like the graphs of Lubotzky, Phillips, and Sarnak, the degrees of these graphs are always a prime number plus one. These graphs have been proposed as the basis for post-quantum elliptic-curve cryptography.^[6]
Adam Marcus, Daniel Spielman and Nikhil Srivastava^[7] proved the existence of $d$ -regular bipartite Ramanujan graphs for any $d$ . Later^[8] they proved that there exist bipartite Ramanujan graphs of every degree and every number of vertices. Michael B. Cohen^[9] showed how to construct these graphs in polynomial time.

References

↑ Terras, Audrey (2011), Zeta functions of graphs: A stroll through the garden, Cambridge Studies in Advanced Mathematics, 128, Cambridge University Press, ISBN 978-0-521-11367-0, MR 2768284
↑ Nilli, A. (1991), "On the second eigenvalue of a graph", Discrete Mathematics, 91 (2): 207–210, doi:10.1016/0012-365X(91)90112-F, MR 1124768 .
↑ Alexander Lubotzky; Ralph Phillips; Peter Sarnak (1988). "Ramanujan graphs". Combinatorica. 8 (3): 261–277. doi:10.1007/BF02126799.
↑ 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.
↑ 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
↑ 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, 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, Lecture Notes in Computer Science, 10822, Cham: Springer, pp. 329–368, doi:10.1007/978-3-319-78372-7_11, MR 3794837
↑ Adam Marcus; Daniel Spielman; Nikhil Srivastava (2013). Interlacing families I: Bipartite Ramanujan graphs of all degrees. Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium.
↑ Adam Marcus; Daniel Spielman; Nikhil Srivastava (2015). Interlacing families IV: Bipartite Ramanujan graphs of all sizes. Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium.
↑ 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.

Guiliana Davidoff; Peter Sarnak; Alain Valette (2003). Elementary number theory, group theory and Ramanjuan graphs. LMS student texts. 55. Cambridge University Press. ISBN 0-521-53143-8. OCLC 50253269.
T. Sunada (1985). "L-functions in geometry and some applications". Lecture Notes in Math. Lecture Notes in Mathematics. 1201: 266–284. doi:10.1007/BFb0075662. ISBN 978-3-540-16770-9.

External links

Survey paper by M. Ram Murty

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[1] Terras, Audrey (2011), Zeta functions of graphs: A stroll through the garden, Cambridge Studies in Advanced Mathematics, 128, Cambridge University Press, ISBN 978-0-521-11367-0, MR 2768284

[2] Nilli, A. (1991), "On the second eigenvalue of a graph", Discrete Mathematics, 91 (2): 207–210, doi:10.1016/0012-365X(91)90112-F, MR 1124768 .

[lps88-3] Alexander Lubotzky; Ralph Phillips; Peter Sarnak (1988). "Ramanujan graphs". Combinatorica. 8 (3): 261–277. doi:10.1007/BF02126799.

[m94-4] 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.

[5] 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

[6] 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, 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, Lecture Notes in Computer Science, 10822, Cham: Springer, pp. 329–368, doi:10.1007/978-3-319-78372-7_11, MR 3794837

[mss13-7] Adam Marcus; Daniel Spielman; Nikhil Srivastava (2013). Interlacing families I: Bipartite Ramanujan graphs of all degrees. Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium.

[mss15-8] Adam Marcus; Daniel Spielman; Nikhil Srivastava (2015). Interlacing families IV: Bipartite Ramanujan graphs of all sizes. Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium.

[c16-9] 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.