Ramanujan graphs

Lubotzky, Alexander; Phillips, R. S.; Sarnak, Peter

doi:10.1007/bf02126799

Cited by 1,230 publications

(1,088 citation statements)

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“…It is a major result discovered by Lubotzky-Phillips-Sarnak [LPS88] (who also coined this term) and independently by Margulis [Mar88] that arbitrarily large dregular Ramanujan graphs exist when d − 1 is prime, and moreover they can be explicitly constructed. Morgenstern [Mor94] extended this to the case when d − 1 is a prime power.…”

Section: Ramanujan Graphsmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

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A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields.But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are "just" one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early '70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even be able to mention. This universality of expanders is becoming more evident as more connections are discovered. It transpires that expansion is a fundamental mathematical concept, well deserving to be thoroughly investigated on its own.In hindsight, one reason that expanders are so ubiquitous is that their very definition can be given in at least three languages: combinatorial/geometric, probabilistic and algebraic. Combinatorially, expanders are graphs which are highly connected; to disconnect a large part of the graph, one has to sever many edges. Equivalently, using the geometric notion of isoperimetry, every s...

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Section: Ramanujan Graphsmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,510

1,355

View full text Add to dashboard Cite

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“…(Autrement dit, s'il n'y a "pas trop" de circuits, les graphes E λ se comportent asymptotiquement comme des "graphes de Ramanujan", au sens de [16], [17].) En effet, d'après ce qui aété au dit au n o 7.1, il existe une mesure ν 0 portée par Ω telle que 1,…”

Section: Jean-pierre Serreunclassified

“…La condition (117) est notamment vérifiée si, pour tout r, on a c r,λ = 0 pour λ assez grand, autrement dit, si le calibre ("girth") de E λ tend vers l'infini avec λ. C'est le cas traité dans [16] et [17].…”

Section: Jean-pierre Serreunclassified

“…On trouve une mesure µ p différente de µ ∞ , cf. n o 2.3; cette mesure intervenait déjà dans [17] et [19],à propos des valeurs propres de certains graphes, et elle a une interprétation simple en termes de mesures de Plancherel, cf. n o 2.3.…”

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“…Le cas où il n'y a "pas trop" de circuits conduità des graphes asymptotiquement du type de Ramanujan (cf. [16], [17]); dans le cas extrême où il y a "très peu" de circuits, on retrouve uné equirépartition suivant la mesure µ q du n o 2.3, cf. [19].…”

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