Abstract
An upper bound is given on the minimum distance between i subsets of same 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 ith largest eigenvalue for any integer i. Our bounds are shown to be asymptotically tight for explicit families of graphs having an asymptotically 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.
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