Let G be a graph on n vertices, and let lambda(1), lambda(2), ... , lambda(n) be its eigenvalues. The Estrada index of G is a recently introduced molecular structure descriptor, defined as EE = Sigma(n)(i=1) e(lambda i). Using a Monte Carlo approach, and treating the graph eigenvalues as random variables, we deduce approximate expressions for EE, in terms of the number of vertices and number of edges, of very high accuracy. (C) 2007 Elsevier B.V. All rights reserved.