Lifts, Discrepancy and Nearly Optimal Spectral Gaps
نویسنده
چکیده
Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map π : H → G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n “new” eigenvalues. We conjecture that every d-regular graph has a 2-lift such that all new eigenvalues are in the range [−2 √ d− 1, 2 √ d− 1] (If true, this is tight , e.g. by the Alon-Boppana bound). Here we show that every graph of maximal degree d has a 2-lift such that all “new” eigenvalues are in the range [−c √
منابع مشابه
Lifts, Discrepancy and Nearly Optimal Spectral Gap*
We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map π :H →G. It is not hard to see that all eigenvalues of G are also eigenvalues of H . In addition, H has n “new” eigenvalues. We conjecture that every...
متن کاملExpansion in Lifts of Graphs
The central goal of this thesis is to better understand, and explicitly construct, expanding towers G1,G2, . . ., which are expander families with the additional constraint that Gn+1 is a lift of Gn . A lift G of H is a graph that locally looks like H , but may be globally di erent; lifts have been proposed as a more structured setting for elementary explicit constructions of expanders, and the...
متن کاملRandomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition
Since being analyzed by Rokhlin, Szlam, and Tygert [1] and popularized by Halko, Martinsson, and Tropp [2], randomized Simultaneous Power Iteration has become the method of choice for approximate singular value decomposition. It is more accurate than simpler sketching algorithms, yet still converges quickly for any matrix, independently of singular value gaps. After Õ(1/ ) iterations, it gives ...
متن کاملParallel Computation of Optimal Parameters for Pseudo Random Number Generation
Two systematic search methods are employed to nd mul-tipliers for linear congruential pseudo-random number generation which are optimal with respect to the discrepancy of pairs of successive pseudo-random numbers. These two methods are compared in terms of their suitability for parallel computation. Experimental results of a MIMD workstationcluster{implementation and an evaluation of the calcul...
متن کاملEquivalent a posteriori error estimates for spectral element solutions of constrained optimal control problem in one dimension
In this paper, we study spectral element approximation for a constrained optimal control problem in one dimension. The equivalent a posteriori error estimators are derived for the control, the state and the adjoint state approximation. Such estimators can be used to construct adaptive spectral elements for the control problems.
متن کامل