For a general class of graded-index fiber profiles analytical expressions for propagation constants of guided modes have been evaluated by the evanescent field theory to the order 0(k(-9)) in asymptotic form. The results are compared with the analytical result of the third order perturbation theory. Analytical asymptotic expressions for group delays are also derived. The asymptotic expressions have been applied for a numerical comparison with the WKB theory and the perturbation theory to the third order. Herefrom, we have estimated the accuracy of the WKB calculations of group delays to be approximately 10 psec/km for near parabolic profiles. We show that the evanescent field theory gives a very accurate determination of the propagation constants and group delays of the lower order modes in a near parabolic profile. For profiles which are far from a parabolic shape, we find that the perturbation theory gives wrong results and that the asymptotic error in the evanescent field theory increases. The accuracy of the WKB theory is found to be within the asymptotic error for higher order modes of near-parabolic profiles and all modes of profiles which are far from a parabolic shape.