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The spectral convergence of the weighted graph Laplacian is a theoretical foundation of the Laplacian based algorithms such as spectral clustering,and dimensionality reduction using diffusion maps and Laplacian eigenmaps.In this talk,I will present our recent results showing the eigenvalues and eigenvectors of the weighted graph Laplacian converges to the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of the manifold with the Neumann boundary in the limit of infinitely many sample points.We consider the convergence problem from the point of view of solving the Poisson equations on submanifolds.This new perspective also leads to the methods for computing the eigensystem of the Laplace-Beltrami operator with Dirichlet boundaries and for solving the harmonic extension problem from point clouds.I will also present some numerical results.