Diameters, distortion and eigenvalues

Details Diameters, distortion and eigenvalues Diameters, distortion and eigenvalues

2010-05-14

arxiv.org/abs/1005.2560v3

Mathematics

Group Theory

Final version, to appear in the European Journal of Combinatorics

Scientific paper

We study the relation between the diameter, the first positive eigenvalue of the discrete $p$-Laplacian and the $\ell_p$-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 $\ell_p$-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.

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