Diameters, distortion and eigenvalues
Group Theory
2011-07-13 v3 Functional Analysis
Metric Geometry
Abstract
We study the relation between the diameter, the first positive eigenvalue of the discrete -Laplacian and the -distortion of a finite graph. We prove an inequality relating these three quantities and apply it to families of Cayley and Schreier graphs. We also show that the -distortion of Pascal graphs, approximating the Sierpinski gasket, is bounded, which allows to obtain estimates for the convergence to zero of the spectral gap as an application of the main result.
Cite
@article{arxiv.1005.2560,
title = {Diameters, distortion and eigenvalues},
author = {Rostislav I. Grigorchuk and Piotr W. Nowak},
journal= {arXiv preprint arXiv:1005.2560},
year = {2011}
}
Comments
Final version, to appear in the European Journal of Combinatorics