Diameters, distortion and eigenvalues

TL;DR

This paper investigates the relationships between graph diameter, eigenvalues of the p-Laplacian, and distortion, providing inequalities and applications to specific graph families, including Cayley, Schreier, and Pascal graphs.

Contribution

It establishes a new inequality linking diameter, eigenvalues, and distortion, and applies it to analyze spectral properties of complex graph families.

Findings

Derived bounds for spectral gap decay in Pascal graphs

Established relations between geometric and spectral graph properties

Applied results to approximate fractal structures like the Sierpinski gasket

Abstract

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.