Here's the sort of thing I have in mind. Let $M$ be a Riemannian manifold, compact if it helps, and let $\Delta_M$ be the Laplace-Beltrami operator. Choose a sequence of triangulations of $M$ so that the triangles in the $n$th triangulation all have diameter smaller than $1/n$, and let $\Delta_n$ denote the corresponding graph Laplacian - that is, the operator which sends a function $f$ on the vertices of a graph to the function whose value at a vertex is the average of $f$ on all neighboring vertices.
Is there a precise sense in which $\Delta_n$ converges to $\Delta_M$? If not, can we at least calculate spectral invariants for $\Delta_M$ using spectral invariants for $\Delta_n$?
The question is a little strange since the graph Laplacian acts on a finite dimensional space (for a finite graph) while the ordinary Laplacian is an unbounded operator on infinite dimensional Hilbert space. It is also not clear that the two operators capture the same kind of geometric information; for example, the dimension of $M$ can be recovered from $\Delta_M$, but it is not obvious to me how to calculate the dimension of $M$ from the $\Delta_n$'s. Nevertheless, my intuition tells me that there's something out there.
Added: Thanks everyone for all the links / references - I'll need a few more days to chase them down. I'm glad there is so much literature on this question!