Average distance in graphs and eigenvalues

Abstract

Brendan McKay gave the following formula relating the average distance between pairs of vertices in a tree T and the eigenvalues of its Laplacian: (d) over bar (T) = 2/n-1 Sigma(n)(i=2) 1/lambda(i). By modifying Mohar's proof of this result, we prove that for any graph G, its average distance, (d) over bar (G), between pairs of vertices satisfies the following inequality: (d) over bar (G) >= 2/n-1 Sigma(n)(i=2) 1/lambda(i). This solves a conjecture of Graffiti. We also present a generalization of this result to the average of suitably defined distances for k subsets of a graph. (C) 2008