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A vector wave analysis is proposed for analyzing the propagation characteristics of the guided modes of the graded-index fibers. Two pairs of the vector wave equations with respect to the transverse components of the electromagnetic fields are derived and solved by the perturbation method in which we use the scalar wave equation as an unperturbed one. As an example, we calculate the eigenvalues and the eigenfunctions of the vector modes of a graded-index fiber with an even polynomial-profile core, and for the eigenvalues we obtain the analytic solutions. Moreover, the result coincides with that obtained by a more sophisticated method.