In this work, we develop a cascadic multigrid method for the elliptic eigenvalue problems and show its optimality under certain assumptions.  We also develop an algebraic variant of the cascadic multigrid method for the fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue, and explore the applicability of such an eigensolver to the graph partition and drawing.  Numerical tests for practical graphs are presented to show the efficiency of the proposed cascadic multigrid method. This is a joint work with J. Urschel, J. Xu, and L. Zikatanov at Penn State University.