We show that eigenvalues and eigenfunctions of the Laplace–Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
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Dmitri Burago, Sergei Ivanov, Yaroslav Kurylev, A graph discretization of the Laplace–Beltrami operator. J. Spectr. Theory 4 (2014), no. 4, pp. 675–714DOI 10.4171/JST/83