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 or here 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.