ISOPERIMETRIC-INEQUALITIES AND EIGENVALUES

An upper bound is given on the minimum distance between i subsets of s ame size of a regular graph in terms of the ith largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the it h largest eigenvalue for any integer i. Our bounds are shown to be asy mptotically tight for explicit families of graphs having an asymptotic ally optimal ith largest eigenvalue, A result by Quenell [Adv. Math., 106 (1994), pp. 122-148] relating the diameter, the second eigenvalue, and the girth of a regular graph is obtained as a by-product.