A note on a distance bound using eigenvalues of the normalized Laplacian matrix

Abstract

Let G be a connected graph, and let X and Y be subsets of its vertex set. A previously
published bound is considered that relates the distance between X and Y to the eigenvalues of the
normalized Laplacian matrix for G, the volumes of X and Y , and the volumes of their complements.
A counterexample is given to the bound, and then a corrected version of the bound is provided.