ELA, Volume 16, pp. 204-207, August 2007, abstract.

A Note on a Distance Bound Using Eigenvalues of the 
Normalized Laplacian Matrix

Steve Kirkland

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.