  We develop theory and methods that use the graph Laplacian to analyze the
geometry of the underlying manifold of point clouds. Our theory provides
theoretical guarantees and explicit bounds on the functional form of the graph
Laplacian, in the case when it acts on functions defined close to singularities
of the underlying manifold. We also propose methods that can be used to
estimate these geometric properties of the point cloud, which are based on the
theoretical guarantees.
