The diameter and Laplacian eigenvalues of directed graphs

For undirected graphs it has been known for some time that one can bound the diameter using the eigenvalues. In this note we give a similar result for the diameter of strongly connected directed graphs G, namely D(G) <= [log(2-lambda)/(2)/2 min(x)log(1/phi(x))] + 1 where lambda is the first non-trivial eigenvalue of the Laplacian and phi is the Perron vector of the transition probability matrix of a random walk on G.