Let G be a graph whose eigenvalues are lambda(1), lambda(2),..., lambda(n). The Estrada index of G is equal to Sigma(n)(i=1)e(lambda i) . We point out certain classes of graphs whose characteristic polynomials are closely connected to the Chebyshev polynomials of the second kind. Various relations, in particular approximations, for the Estrada index of these graphs are obtained. (c) 2008 Elsevier B. V. All rights reserved.