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42
EIGENVALUES AND EXPANDERS
 COMBINATORICA
, 1986
"... 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 mani ..."
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Cited by 408 (19 self)
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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 coNPcomplete.
The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freem ..."
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Cited by 243 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freeman & Co., New York, 1979 (hereinafter referred to as "[G&J]"; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
The Laplacian spectrum of graphs
 Graph Theory, Combinatorics, and Applications
, 1991
"... Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, m ..."
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Cited by 227 (2 self)
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Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidthtype parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,
Lineartime Encodable and Decodable ErrorCorrecting Codes
, 1996
"... We present a new class of asymptotically good, linear errorcorrecting codes. These codes can be both encoded and decoded in linear time. They can also be encoded by logarithmicdepth circuits of linear size and decoded by logarithmic depth circuits of size 0 (n log n). We present both randomized an ..."
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Cited by 149 (5 self)
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We present a new class of asymptotically good, linear errorcorrecting codes. These codes can be both encoded and decoded in linear time. They can also be encoded by logarithmicdepth circuits of linear size and decoded by logarithmic depth circuits of size 0 (n log n). We present both randomized and explicit constructions of these codes.
Lossless condensers, unbalanced expanders, and extractors
 In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing
, 2001
"... Abstract Trevisan showed that many pseudorandom generator constructions give rise to constructionsof explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(log n) additional random bitsthat maps n bi ..."
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Cited by 104 (21 self)
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Abstract Trevisan showed that many pseudorandom generator constructions give rise to constructionsof explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(log n) additional random bitsthat maps n bits strings to poly(log K) bit strings, such that any source with support size Kis mapped almost injectively to the smaller domain. Our construction remains the best lossless condenser to date.By composing our condenser with previous extractors, we obtain new, improved extractors. For small enough minentropies our extractors can output all of the randomness with only O(log n) bits. We also obtain a new disperser that works for every entropy loss, uses an O(log n)bit seed, and has only O(log n) entropy loss. This is the best disperser construction to date,and yields other applications. Finally, our lossless condenser can be viewed as an unbalanced
Expanders that Beat the Eigenvalue Bound: Explicit Construction and Applications
 Combinatorica
, 1993
"... For every n and 0 ! ffi ! 1, we construct graphs on n nodes such that every two sets of size n ffi share an edge, having essentially optimal maximum degree n 1\Gammaffi+o(1) . Using known and new reductions from these graphs, we explicitly construct: 1. A k round sorting algorithm using n 1+1=k ..."
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Cited by 87 (24 self)
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For every n and 0 ! ffi ! 1, we construct graphs on n nodes such that every two sets of size n ffi share an edge, having essentially optimal maximum degree n 1\Gammaffi+o(1) . Using known and new reductions from these graphs, we explicitly construct: 1. A k round sorting algorithm using n 1+1=k+o(1) comparisons. 2. A k round selection algorithm using n 1+1=(2 k \Gamma1)+o(1) comparisons. 3. A depth 2 superconcentrator of size n 1+o(1) . 4. A depth k widesense nonblocking generalized connector of size n 1+1=k+o(1) . All of these results improve on previous constructions by factors of n\Omega\Gamma37 , and are optimal to within factors of n o(1) . These results are based on an improvement to the extractor construction of Nisan & Zuckerman: our algorithm extracts an asymptotically optimal number of random bits from a defective random source using a small additional number of truly random bits. 1
Bounds For Dispersers, Extractors, And DepthTwo Superconcentrators
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2000
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Eigenvalues and Expansion of Regular Graphs
 Journal of the ACM
, 1995
"... The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best known explicit expanders. The spectral method yielded a lower bound of k=4 on the expansion of linear sized subsets of kr ..."
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Cited by 51 (1 self)
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The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best known explicit expanders. The spectral method yielded a lower bound of k=4 on the expansion of linear sized subsets of kregular Ramanujan graphs. We improve the lower bound on the expansion of Ramanujan graphs to approximately k=2. Moreover, we construct a family of kregular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k=2. This shows that k=2 is the best bound one can obtain using the second eigenvalue method. We also show an upper bound of roughly 1 + p k \Gamma 1 on the average degree of linearsized induced subgraphs of Ramanujan graphs. This compares positively with the classical bound 2 p k \Gamma 1. As a byproduct, we obtain improved results on random walks on expanders and construct selection networks (resp. extrovert graphs) of smaller size (resp. degree) th...
Expander Graphs for Digital Stream Authentication and Robust Overlay Networks
 IN PROCEEDINGS OF THE 2002 IEEE SYMPOSIUM ON SECURITY AND PRIVACY
, 2002
"... We use expander graphs to provide efficient new constructions for two security applications: authentication of long digital streams over lossy networks and building scalable, robust overlay networks. Here is a summary of our contributions: (1) To authenticate long digital streams over lossy networks ..."
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Cited by 26 (0 self)
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We use expander graphs to provide efficient new constructions for two security applications: authentication of long digital streams over lossy networks and building scalable, robust overlay networks. Here is a summary of our contributions: (1) To authenticate long digital streams over lossy networks, we provide a construction with a provable lower bound on the ability to authenticate a packet  and that lower bound is independent of the size of the graph. To achieve this, we present an authentication expander graph with constant degree. (Previous work, such as [MS01], used authentication graphs but required graphs with degree linear in the number of vertices.) (2) To build efficient, robust, and scalable overlay networks, we provide a construction using undirected expander graphs with a provable lower bound on the ability of a broadcast message to successfully reach any receiver. This also gives us a new, more efficient solution to the decentralized certificate revocation problem [WLM00].