It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is usually a triangulation of $M$). See [here][1] or [here][2] for details.

What I am interested in is going in the opposite direction. That is: 

> Given a fixed (finite) graph $G$ is there a way to approximate its
> Laplacian $L_{G}$ by the Laplace-Beltrami operator $\mathcal{L}_{M}$
> of some surface $M$?

The motivation for this is that if $G$ is a sufficiently dense grid, then I can take $M=\mathbb{R}^{2}$. Now suppose I add an edge to $G$ - it feels right that there should be a way to modify $M=\mathbb{R}^{2}$ into some new $M^{'}$ by somehow folding the manifold appropriately.


  [1]: https://mathoverflow.net/questions/66892/is-the-laplacian-on-a-manifold-the-limit-of-graph-laplacians
  [2]: http://arxiv.org/abs/1301.2222