Results 1 
7 of
7
Expansion of Product Replacement Graphs
 Combinatorica
, 2001
"... . We establish a connection between the expansion coefficient of the product replacement graph \Gamma k (G) and the minimal expansion coefficient of a Cayley graph of G with k generators. In particular, we show that the product replacement graphs \Gamma k \Gamma PSL(2; p) \Delta form an expander ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
. We establish a connection between the expansion coefficient of the product replacement graph \Gamma k (G) and the minimal expansion coefficient of a Cayley graph of G with k generators. In particular, we show that the product replacement graphs \Gamma k \Gamma PSL(2; p) \Delta form an expander family, under assumption that all Cayley graphs of PSL(2; p), with at most k generators are expanders. This gives a new explanation of the outstanding performance of the product replacement algorithm and supports the speculation that all product replacement graphs are expanders [LP,P3].
Landscapes on Spaces of Trees
, 2001
"... Combinatorial optimization problems defined on sets of phylogenetic trees are an important issue in computational biology, for instance the problem of reconstruction a phylogeny using maximum likelihood or parsimony approaches. The collection of possible phylogenetic trees is arranged as a socalled ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Combinatorial optimization problems defined on sets of phylogenetic trees are an important issue in computational biology, for instance the problem of reconstruction a phylogeny using maximum likelihood or parsimony approaches. The collection of possible phylogenetic trees is arranged as a socalled Robinson graph by means of the nearest neighborhood interchange move. The coherent algebra and spectra of Robinson graphs are discussed in some detail as their knowledge is important for an understanding of the landscape structure. We consider simple model landscapes as well as landscapes arising from the maximum parsimony problem, focusing on two complementary measures of ruggedness: the amplitude spectrum arising from projecting the cost functions onto the eigenspaces of the underlying graph and the topology of local minima and their connecting saddle points.
Spectral Techniques for Expander Codes
 in Proceedings of ACM Symposium on Theory of Computing (STOC
, 1997
"... This paper introduces methods based on generalized Fourier analysis for working with a class of errorcorrecting codes constructed in terms of Cayley graphs. Our work is motivated by the recent results of Sipser and Spielman [15] showing graph expansion to be essential for efficient decoding of certa ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
This paper introduces methods based on generalized Fourier analysis for working with a class of errorcorrecting codes constructed in terms of Cayley graphs. Our work is motivated by the recent results of Sipser and Spielman [15] showing graph expansion to be essential for efficient decoding of certain lowdensity paritycheck codes. They leave open the problem of subquadratic encoding for this class of codes, and it is this problem that we address. We show that when the codes are constructed in terms of Cayley graphs, the symmetry of the graphs can be exploited by using the representation theory of the underlying group to devise a subquadratic encoding algorithm that, in the case where the group is PSL 2 (Z=qZ), requires O(n 4=3 ) operations, where n = O(q 3 ) is the block length. Our results indicate that this new class of codes may combine many of the strengths of two of the most powerful and successful, but previously disparate areas of coding theory: the class of cyclic codes...
Survey of Spectra of Laplacians on Finite Symmetric Spaces
"... this paper. 2. FINITE EUCLIDEAN SYMMETRIC SPACES ..."
Eigenvalue Spacings for Quantized Cat Maps
"... According to one of the basic conjectures in Quantum Chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
According to one of the basic conjectures in Quantum Chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms { \cat maps". Our numerical experiments indicate that for \generic" choices of cat maps, the unfolded consecutive spacings distribution in the irreducible components of the Nth quantization (given by the Ndimensional Weil representation) approaches the GOE/GSE law of Random Matrix Theory. For certain special \arithmetic " transformations, related to the Ramanujan graphs of Lubotzky, Phillips and Sarnak, the experiments indicate that the unfolded consecutive spacings distribution follows Poisson statistics; we provide a sharp estimate in that direction.
Level Spacings For Cayley Graphs
 IMA Volumes in Mathematics and its Applications, this volume
"... . We investigate the eigenvalue spacing distributions for randomly generated 4regular Cayley graphs on SL 2 (Fp ), S 10 , and large cyclic groups by numerically calculating their spectra. We present strong evidence that the distributions are Poisson and hence do not follow the Gaussian orthogonal e ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
. We investigate the eigenvalue spacing distributions for randomly generated 4regular Cayley graphs on SL 2 (Fp ), S 10 , and large cyclic groups by numerically calculating their spectra. We present strong evidence that the distributions are Poisson and hence do not follow the Gaussian orthogonal ensemble. Among the Cayley graphs of SL 2 (Fp ) that we consider are the new expander graphs recently discovered by Y. Shalom. In addition, we use a Markov chain method to generate random 4regular graphs, and observe that the average eigenvalue spacings are closely approximated by the Wigner surmise. Key words. Random matrices, Cayley graphs, expander graphs, spacing distribution, Gaussian ensemble, Wigner surmise. AMS(MOS) subject classifications. Primary 05C25, 20C40, 68R10; Secondary 20B25, 20D06, 20C30. 1. Introduction. One of the most remarkable numerical discoveries of the recent past is Odlyzko's finding that the spacings of the zeros of the Riemann zeta function closely follow the...
Level Spacings for SL(2,p)
, 1997
"... . We investigate the eigenvalue spacing distributions for randomly generated 4regular Cayley graphs on SL 2 (Fp ) by numerically calculating their spectra. We present strong evidence that the distributions are Poisson and hence do not follow the Gaussian orthogonal ensemble. Among the Cayley graphs ..."
Abstract
 Add to MetaCart
. We investigate the eigenvalue spacing distributions for randomly generated 4regular Cayley graphs on SL 2 (Fp ) by numerically calculating their spectra. We present strong evidence that the distributions are Poisson and hence do not follow the Gaussian orthogonal ensemble. Among the Cayley graphs of SL 2 (Fp ) we consider are the new expander graphs recently discovered by Y. Shalom. In addition, we use a Markov chain method to generate random 4regular graphs, and observe that the average eigenvalue spacings are closely approximated by the Wigner surmise. LEVEL SPACINGS FOR SL 2 (F p ) JOHN D. LAFFERTY AND DANIEL N. ROCKMORE 1. Introduction. One of the most remarkable numerical discoveries of the recent past is Odlyzko's finding that the spacings of the zeros of the Riemann zeta function closely follow the Gaussian unitary ensemble of random matrix theory [15]. As a result of this work, attention has turned to the spacing distributions for the spectrum of other natural classes of op...