Let G be an undirected and loopless finite graph that is not a path. The minimum m such that the iterated line graph Lm(G) is hamiltonian is called the hamiltonian index of G, denoted by h(G). A reduction method to determine the hamiltonian index of a graph G with h(G)≥2 is given here. With it we will establish a sharp lower bound and a sharp upper bound for h(G), respectively, which improves some known results of P.A. Catlin et al. [J. Graph Theory 14 (1990)] and H.-J. Lai [Discrete Mathematics 69 (1988)]. Examples show that h(G) may reach all integers between the lower bound and the upper bound. \u