  Laplacian Eigenvectors of the graph constructed from a data set are used in
many spectral manifold learning algorithms such as diffusion maps and spectral
clustering. Given a graph constructed from a random sample of a $d$-dimensional
compact submanifold $M$ in $\mathbb{R}^D$, we establish the spectral
convergence rate of the graph Laplacian. It implies the consistency of the
spectral clustering algorithm via a standard perturbation argument. A simple
numerical study indicates the necessity of a denoising step before applying
spectral algorithms.
