On the second eigenvalue of random regular graphs

Friedman, Joel; Kahn, Jeff; Szemerédi, Endre

doi:10.1145/73007.73063

Cited by 178 publications

(260 citation statements)

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“…Before we proceed to exemplify this method, let us mention an alternative approach to the same problem due to Kahn and Szemerédi [FKS89]. They show that if A is the adjacency matrix of a random d-regular graph, then A's Rayleigh quotient is likely to be small on all points of an appropriate -net on the unit sphere.…”

Section: Lemma 73 For the Infinite Tree T D The Number Described Abmentioning

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: Lemma 73 For the Infinite Tree T D The Number Described Abmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,501

1,355

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“…A weaker version than Friedman's was obtained independently by Kahn and Szemerédi (cf. [51]). They proved the following results: On the other hand, Alon and Boppana (cf.…”

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confidence: 99%

Eigenvalues in Combinatorial Optimization

Mohar

Poljak

1993

Combinatorial and Graph-Theoretical Problems in Linear Algebra

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“…In particular [3] has lead to further results [10], [11]. The reason for this may be that [3] is based on a rather flexible approach to obtain spectral information about random graphs [12]: Spectral information directly follows from clustering properties known to be typically present in a random graph by (inefficient) counting arguments. We apply this technique here, too.…”

Section: Spectral Partitioning Of Random Graphs With Given Expected Dmentioning

confidence: 99%

“…Note that all entries rtj with the same (•0(i), V'(i)) have the same expected value which makes the analogy possible. In particular we can apply [12]. The vector si is necessary in order to recognize partitions which can be readily recognized just from the row sums of R.…”

Section: Algorithmmentioning

confidence: 99%

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Spectral Partitioning of Random Graphs with Given Expected Degrees

Coja-Oghlan

Goerdt

Lanka

Fourth IFIP International Conference on Theoretical Computer Science- TCS 2006

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Abstract. It is a well established fact, that -in the case of classical random graphs like (variants of) Gn,p or random regular graphsspectral methods yield efficient algorithms for clustering (e. g. colouring or bisection) problems. The theory of large networks emerging recently provides convincing evidence that such networks, albeit looking random in some sense, cannot sensibly be described by classical random graphs. A variety of new types of random graphs have been introduced. One of these types is characterized by the fact that we have a fixed expected degree sequence, that is for each vertex its expected degree is given. Recent theoretical work confirms that spectral methods can be successfully applied to clustering problems for such random graphs, tooprovided that the expected degrees are not too small, in fact > log® n. In this case however the degree of each vertex is concentrated about its expectation. We show how to remove this restriction and apply spectral methods when the expected degrees are bounded below just by a suitable constant. Our results rely on the observation that techniques developed for the classical sparse G",p random graph (that is p = c/n) can be transferred to the present situation, when we consider a suitably normalized adjacency matrix: We divide each entry of the adjacency matrix by the product of the expected degrees of the incident vertices. Given the host of spectral techniques developed for Gn,p this observation should be of independent interest.

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On the second eigenvalue of random regular graphs

Cited by 178 publications

References 10 publications

Expander graphs and their applications

Expander graphs and their applications

Eigenvalues in Combinatorial Optimization

Spectral Partitioning of Random Graphs with Given Expected Degrees

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