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In this paper, we develop a cascadic multigrid algorithm for fast computationof the Fiedler vector of a graph Laplacian, namely, the eigenvectorcorresponding to the second smallest eigenvalue. This vector has been found tohave applications in fields such as graph partitioning and graph drawing. Thealgorithm is a purely algebraic approach based on a heavy edge coarseningscheme and pointwise smoothing for refinement. To gain theoretical insight, wealso consider the related cascadic multigrid method in the geometric settingfor elliptic eigenvalue problems and show its uniform convergence under certainassumptions. Numerical tests are presented for computing the Fiedler vector ofseveral practical graphs, and numerical results show the efficiency andoptimality of our proposed cascadic multigrid algorithm.

A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians