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

@article{Urschel2014ACM, title={A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians}, author={John C. Urschel and Xiaozhe Hu and Jinchao Xu and Ludmil T. Zikatanov}, journal={arXiv: Numerical Analysis}, year={2014} }

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic…

## 27 Citations

Improvement of the Cascadic Multigrid Algorithm with a Gauss Seidel Smoother to Efficiently Compute the Fiedler Vector of a Graph Laplacian

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In this paper, we detail the improvement of the Cascadic Multigrid algorithm with the addition of the Gauss Seidel algorithm in order to compute the Fiedler vector of a graph Laplacian, which is the…

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A novel strategy to compute a posteriori error estimators which aid multilevel iterative solvers for linear systems of graph Laplacians by constructing a Helmholtz decomposition on the graph based on a spanning tree and the corresponding cycle space is proposed.

Fiedler Vector Approximation via Interacting Random Walks

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This paper develops a framework in which a stochastic process is constructed based on a set of interacting random walks on a graph and shows that a suitably scaled version of the stochastics process converges to the Fiedler vector for a sufficiently large number of walks.

Fiedler Vector Approximation via Interacting RandomWalks

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This paper develops a framework in which a stochastic process is constructed based on a set of interacting random walks on a graph and shows that a suitably scaled version of the stochastics process converges to the Fiedler vector for a sufficiently large number of walks.

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This work investigates the effectiveness of the unsmoothed aggregation AMG (UA-AMG) method and shows it is more appropriate for graph-related problems due to its ability to maintain the structure of graphs in the multilevel hierarchy.

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This work proves results regarding the spectral approximation of the Laplacian of the original graph by the Laplin of the disaggregated graph, as well as constructing an alternate disaggregation operator whose eigenvalues interlace those of theOriginal LaplACian.

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We propose a path cover adaptive algebraic multigrid (PC-$\alpha$AMG) method for solving linear systems of weighted graph Laplacians and can also be applied to discretized second order elliptic…

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