Diameters, distortion, and eigenvalues

We study the relation between the diameter, the first positive eigenvalue of the discrete p-Laplacian, and the ℓp-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 ℓp-distortion of Pascal graphs, approximating the Sierpinski gasket, is bounded, which allows one to obtain estimates for the convergence to zero of the spectral gap as an application of the main result.