  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.
