By introducing a weight function to the Laplace operator, Bakry and Emery defined the drift Laplacian to study diffusion processes. Our first main result is that, given a Bakry-Emery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $R^n$ and Neumann eigenvalues of domains in $R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to generalize emph{all the results in Riemannian geometry based on gradient estimates to Bakry-Emery manifolds}.
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