  In this paper we improve the spectral convergence rates for graph-based
approximations of Laplace-Beltrami operators constructed from random data. We
utilize regularity of the continuum eigenfunctions and strong pointwise
consistency results to prove that spectral convergence rates are the same as
the pointwise consistency rates for graph Laplacians. In particular, for an
optimal choice of the graph connectivity $\varepsilon$, our results show that
the eigenvalues and eigenvectors of the graph Laplacian converge to those of
the Laplace-Beltrami operator at a rate of $O(n^{-1/(m+4)})$, up to log
factors, where $m$ is the manifold dimension and $n$ is the number of vertices
in the graph. Our approach is general and allows us to analyze a large variety
of graph constructions that include $\varepsilon$-graphs and $k$-NN graphs.
