Elementary Number Theory, Group Theory and Ramanujan Graphs

Davidoff, Giuliana P.; Sarnak, Peter; Valette, Alain

doi:10.1017/cbo9780511615825

Cited by 217 publications

(246 citation statements)

References 0 publications

Supporting

Mentioning

240

Contrasting

Unclassified

Order By: Relevance

“…There are several available proofs of this theorem, e.g., [DSV03], [Fri93], [Nil04]. These papers also give the proper credit to the theorem's originator, J.-P. Serre.…”

Section: Theorem 58 (Serre) For Every Integer D and > 0 There Is A mentioning

confidence: 99%

See 1 more Smart Citation

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,502

1,355

View full text Add to dashboard Cite

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...

show abstract

“…There are several available proofs of this theorem, e.g., [DSV03], [Fri93], [Nil04]. These papers also give the proper credit to the theorem's originator, J.-P. Serre.…”

Section: Theorem 58 (Serre) For Every Integer D and > 0 There Is A mentioning

confidence: 99%

“…Here we only state the result and describe the construction. The book by Davidoff, Sarnak, and Valette [DSV03] offers a self-contained description of the beautiful mathematics around it. Lubotzky's book [Lub94] should be consulted as well.…”

Section: Ramanujan Graphsmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,502

1,355

View full text Add to dashboard Cite

show abstract

“…Expanders are highly-connected sparse graphs widely used in computer science, in areas ranging from parallel computation to complexity theory and cryptography; recently they also have found some remarkable applications in pure mathematics; see [5], [10], [15], [20], [21] and references therein. Given an undirected d-regular graph G and a subset X of V , the expansion of X, c(X), is defined to be the ratio |∂(X)|/|X|, where ∂(X) = {y ∈ G : distance(y, X) = 1}.…”

Section: Introductionmentioning

confidence: 99%

“…The idea of obtaining spectral gap results by exploiting high multiplicity together with the upper bound on the number of short closed geodesics is due to Sarnak and Xue [22]; it was subsequently applied in [5] and [7]. In these works the upper bound was achieved by reduction to an appropriate diophantine problem.…”

Section: Introductionmentioning

confidence: 99%

Uniform expansion bounds for Cayley graphs of SL₂(𝔽_p)

2008

View full text Add to dashboard Cite

show abstract

“…It is known [8] that for a connected graph −n ≤ λ i ≤ n where λ i is any eigenvalue and that the second largest eigenvalue λ is closely related to the expansion properties of the graph. Large random graphs and known families with good expansion properties have λ = 2 √ n − 1 [7]. We quote the following bounds on the minimum distances.…”

Section: Bounds On the Minimum Distancementioning

confidence: 99%