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In [[spectral graph theory]], a '''Ramanujan graph''', named after [[Srinivasa Ramanujan]], is a [[regular graph|regular]] [[graph (mathematics)|graph]] whose [[spectral gap]] is almost as large as possible (see [[extremal graph theory]]). Such graphs are excellent [[expander graph|spectral expanders]].
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Examples of Ramanujan graphs include the [[clique (graph theory)|clique]], the [[biclique]] <math>K_{n,n}</math>, and the [[Petersen graph]].  As [http://www.mast.queensu.ca/~murty/ramanujan.pdf Murty's survey paper] notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, [[number theory]], [[representation theory]], and [[algebraic geometry]]". As an example of this, a regular graph is Ramanujan if and only if its [[Ihara zeta function]] satisfies an analog of the [[Riemann hypothesis]].<ref>Audrey Terras, ''Zeta Functions of Graphs: A Stroll through the Garden'',  volume 128, Cambridge Studies in Advanced Mathematics, [[Cambridge University Press]], (2010).</ref>
 
==Definition==
 
Let <math>G</math> be a connected <math>d</math>-regular graph with <math>n</math> vertices, and let <math>\lambda_0 \geq \lambda_1 \geq \ldots \geq \lambda_{n-1}</math> be the [[eigenvalues]] of the [[adjacency matrix]] of <math>G</math>. Because <math>G</math> is connected and <math>d</math>-regular, its eigenvalues satisfy <math>d = \lambda_0 > \lambda_1 </math> <math> \geq \ldots \geq \lambda_{n-1} \geq -d </math>. Whenever there exists <math>\lambda_i</math> with <math>|\lambda_i| < d</math>, define
 
: <math>\lambda(G) = \max_{|\lambda_i| < d} |\lambda_i|.\,</math>
 
A <math>d</math>-regular graph <math>G</math> is a Ramanujan graph if <math>\lambda(G)</math> is defined and <math>\lambda(G) \leq 2\sqrt{d-1}</math>.
 
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 <math>d</math> and large <math>n</math>, the <math>d</math>-regular, <math>n</math>-vertex Ramanujan graphs minimize <math>\lambda(G)</math>. If <math>G</math> is a <math>d</math>-regular graph with [[Diameter (graph theory)|diameter]] <math>m</math>, a theorem due to [[Noga Alon|Nilli]]<ref>A Nilli, ''On the second eigenualue of a graph'', Discrete Mathematics 91 (1991) pp. 207-210.</ref> states
 
: <math>\lambda_1 \geq 2\sqrt{d-1} - \frac{2\sqrt{d-1} - 1}{\lfloor m/2\rfloor}.</math>
 
Whenever <math>G</math> is <math>d</math>-regular and connected on at least three vertices, <math>|\lambda_1| < d</math>, and therefore <math>\lambda(G) \geq \lambda_1</math>.  Let <math>\mathcal{G}_n^d</math> be the set of all connected <math>d</math>-regular graphs <math>G</math> with at least <math>n</math> vertices.  Because the minimum diameter of graphs in <math>\mathcal{G}_n^d</math> approaches infinity for fixed <math>d</math> and increasing <math>n</math>, Nilli's theorem implies an earlier theorem of [[Noga Alon|Alon]] and Boppana which states
 
: <math>\lim_{n \to \infty} \inf_{G \in \mathcal{G}_n^d} \lambda(G) \geq 2\sqrt{d-1}.</math>
 
==Constructions==
Constructions of Ramanujan graphs are often algebraic. [[Alexander Lubotzky|Lubotzky]], Phillips and [[Peter Sarnak|Sarnak]] show how to construct an infinite family of ''p''&nbsp;+1-regular Ramanujan graphs, whenever ''p''&nbsp;&equiv;&nbsp;1&nbsp;(mod&nbsp;4) is a [[prime number|prime]].  Their proof uses the [[Ramanujan conjecture]], which led to the name of Ramanujan graphs. Morgenstern extended the construction of Lubotzky, Phillips and Sarnak to all prime powers.
 
==References==
{{Reflist}}
* {{cite book | author=Guiliana Davidoff | coauthors=Peter Sarnak, Alain Valette | title=Elementary number theory, group theory and Ramanjuan graphs | publisher=[[Cambridge University Press]] | series=LMS student texts | volume=55 | year=2003 | isbn=0-521-53143-8 | oclc=50253269 }}
* {{cite journal | author=Alexander Lubotzky | coauthors=Ralph Phillips, Peter Sarnak | title=Ramanujan graphs | journal=Combinatorica | volume=8 | year=1988 | pages=261–277 | doi=10.1007/BF02126799 | issue=3 }}
* {{cite journal | author=Moshe Morgenstern | title=Existence and Explicit Constructions of q+1 Regular Ramanujan Graphs for Every Prime Power q | journal=J. Combinatorial Theory, Series B | volume=62 | year=1994 | pages=44–62 | doi=10.1006/jctb.1994.1054 }}
* {{cite journal | author=T. Sunada| title=L-functions in geometry and some applications| journal= Lecture Notes in Math.| volume=1201 | year=1985 | pages=266–284 | doi=10.1007/BFb0075662 | series=Lecture Notes in Mathematics | isbn=978-3-540-16770-9 }}
 
==External links==
* [http://www.mast.queensu.ca/~murty/ramanujan.pdf Survey paper by M. Ram Murty]
 
[[Category:Graph families]]
[[Category:Algebraic graph theory]]
[[Category:Spectral theory]]
[[Category:Regular graphs]]
[[Category:Srinivasa Ramanujan]]

Latest revision as of 02:01, 1 July 2014

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