R.M. Tanner, Explicit concentrators from generalized N-gons, SIAM J. Alg. Disc. Methods, 5 (1984) 287 - 293.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....this research in part under Grant CCR 8858788, and the Office of Naval Research under Grant N00014 87 K 0467. 1 1 INTRODUCTION 2 The magnitude of j 2 j has received much attention in the literature: for example, it is useful to give some estimate of the expansion properties of G (see [Tan84], Alo86] or the rate of convergence of the Markov process on G (with probabilities 1=d at each edge) to the stable distribution (see [BS87] for more on this and references) The smaller jj the better, usually; intuitively it measures the difference (in L 2 operator norm) between A and the all d=n ....

R.M. Tanner. Explicit concentrators from generalized N-gons. SIAM J. Alg. Disc. Methods, 5:287--293, 1984.

....this lead to an exciting and extensive body of research, developed mainly by mathematicians intrigued by this computer science challenge. Most of this work was guided by the sufficient 1 condition for the expansion of (infinite families of constant degree regular) graphs discovered by Tanner [Tan84] see also [AM85] the second largest eigenvalue of the adjacency matrix should be strictly smaller than the degree. This naturally lead researchers to consider algebraic constructions, where this eigenvalue can be estimated. The celebrated sequence of papers [Mar73, GG81, AM85, AGM87, JM87, ....

....needed to ensure that in a graph of N vertices every two sets of size N=A have an edge between them Random graphs show that degree O(A log A) suffices, but explicit constructions have failed to match this bound. An application of the best known relation between eigenvalues and vertex expansion [Tan84] shows that Ramanujan graphs (e.g. as given by [LPS88, Mar88, Mor94] of degree (A 2 ) suffice. To beat this eigenvalue bound, Wigderson and Zuckerman [WZ99] suggested to build such graphs from extractors and obtained degree A N o(1) which was important for many applications where A ....

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Michael R. Tanner. Explicit concentrators from generalized n-gons. SIAM Journal on Algebraic Discrete Methods, 5(3):287--293, 1984.

....graph G 1 2 d so that df(v) 1 2 d(v) Its quadratic form turns out to have properties similar to the Laplacian operator. In this context, lower bounds on the complexity can be proven in a similar way as in the expander graphs theory using a well known link with the spectral values [12, 2, 1]. In this section let G be an undirected graph with n vertices and let 1 : n be the eigenvalues of the quadratic form G 1 2 d . Lemma 6. If the graph G is connected then 1 = 0; 2 0, and every set U of c vertices satisfies G 1 2 d (U) 2 c (1 Gamma c=n) Proof. 0 is an ....

R. M. Tanner. Explicit Concentrators from Generalized N-gons. In SIAM Journal of Algebraic Discrete Methods, vol. 5, pp. 287--293, 1984.

....to define the (nontrivial) spectral radius of a d regular graph: Definition 2.2. In the context of the theorem above, G) max fjj : 2 spec Q(G) jj 6= dg ; denotes the spectral radius of G. Graphs having a small spectral radius make good magnifiers. This was observed first by Tanner [Ta] in 1984, and by Alon and Milman [A1] A2] AM] Theorem 2.3 (Alon) Let G = V; E) be a d regular graph on n vertices, and its spectral radius. Then G is a i n; d; 2d Gamma2 3d Gamma2 j magnifier. Instantly the question arises, how small can be made. There is an asymptotic ....

R. M. Tanner, Explicit concentrators from generalized N--gons, SIAM J. Alg. Discr. Meth. 5 (1984), 287--294.

....modulo q, there is a (p 1) regular Cayley graph of PSL(2;Z=qZ) with q(q 2 Gamma 1) 2 vertices such that the second largest eigenvalue of the graph is at most 2 p p. One can show that a graph with the eigenvalue separation displayed by these graphs is a good expander by using results from [Tan84, Alo86, Kah92] Other constructions that achieve a similar separation have since appeared [Bie89, Mor95, Mor94] From the fact that these graphs are Cayley graphs, one can show that they have a simple representation: Proposition 3. Each graph described in Theorem 2 can be constructed in time ....

R. M. Tanner. Explicit concentrators from generalized n-gons. SIAM Journal Alg. Disc. Meth., 5(3):287--293, September 1984.

....modulo q, there is a (p 1) regular Cayley graph of PSL(2;Z=qZ) with q(q 2 Gamma 1) 2 vertices such that the second largest eigenvalue of the graph is at most 2 p p. One can show that a graph with the eigenvalue separation displayed by these graphs is a good expander by using results from [Tan84, Alo86, Kah92]. Other constructions that achieve a similar separation have since appeared [Bie89, Mor95, Mor94] From the fact that these graphs are Cayley graphs, one can show that they have a simple representation: Proposition 6. Each graph described in Theorem 5 can be constructed in time polynomial in its ....

R. M. Tanner. Explicit concentrators from generalized n-gons. SIAM Journal Alg. Disc. Meth., 5(3):287--293, September 1984.

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R.M. Tanner, Explicit concentrators from generalized N-gons, SIAM J. Alg. Disc. Methods, 5 (1984) 287 - 293.

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R. Michael Tanner. Explicit Concentrators from Generalized N-gons. SIAM Journal of Algebraic and Discrete Methods, 5(3), September 1984.

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R. M. Tanner, "Explicit concentrators from generalized N-gons," SIAM J. Alg. Disc. Meth. 5 (1984) 287--293.

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