Expanders obtained from affine transformations

Jimbo, Shuichi; Maruoka, Akira

doi:10.1007/bf02579322

Cited by 72 publications

(34 citation statements)

References 8 publications

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“…Their argument uses classical harmonic analysis. In 1987 Jimbo and Marouka [JM87] improved it further using Fourier analysis on the group Z 2 n . We present here a slight simplification of their proof, due to Boppana.…”

Section: Theorem 82 (Gabber-galil [Gg81])mentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,501

1,355

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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: Theorem 82 (Gabber-galil [Gg81])mentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,501

1,355

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“…In [3] we combine similar methods with results of Kazhdan on group representations to obtain many new examples of strong linear expanders. In [5] we apply these methods together with the Fourier analysis method of [17] and the methods of [20] to obtain better explicit expanders than those previously known which enable us to construct an explicit family of linear s.c.-s of density asymptotic to 122.74, better than the previous known constructions. Finally, we have recently found a way of applying our methods to the problem of designing fault tolerant processor arrays, discussed in [29] .…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues and expanders

1986

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Linear expanders have numerous applications to theoretical computer science. Here we show that a regular bipartite graph is an expander ifandonly if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result, which has an analytic analogue for Riemannian manifolds enables one to generate expanders randomly and check efficiently their expanding properties. It also supplies an efficient algorithm for approximating the expanding properties of a graph. The exact determination of these properties is known to be coNP-complete.

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“…Other constructions appeared in Schmidt (1980), Milman (1984, 1985), Jimbo and Maruoka (1985) and Buck (1986).…”

Section: Improved Constructionsmentioning

confidence: 99%

“…Some of these constructions supplied better superconcentrators, of densities 261.5 (Chung (1978)), 218 (Jimbo and Maruoka (1985), and 122.7 (Alon, Galil and Milman (1987)). …”

Section: Improved Constructionsmentioning

confidence: 99%

Explicit Constructions

2001

Random Graphs

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Expanders obtained from affine transformations

Cited by 72 publications

References 8 publications

Expander graphs and their applications

Expander graphs and their applications

Eigenvalues and expanders

Explicit Constructions

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