The minimax rate for estimating eigenpairs of weighted Laplace-Beltrami operators from n samples on a d-dimensional manifold is n^{-2/(d+4)}, and graph Laplacians achieve this rate up to logarithmic factors.
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Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds
The minimax rate for estimating eigenpairs of weighted Laplace-Beltrami operators from n samples on a d-dimensional manifold is n^{-2/(d+4)}, and graph Laplacians achieve this rate up to logarithmic factors.