# Combinatorial Skeleton of a Riemannian manifold

In Chung and Yau's paper: "A combinatorial trace formula" (MSN), they proved a combinatorial version of Selberg's trace formula for lattice graphs. I learned also in the setup that it makes sense to define Laplacian, its eigenvalues, and heat kernel of a graph (could be infinite and with loops). It showed a strong analogue to Riemannian geometry!

On both sides, we can extract useful information from the objects' spectra. This was not overwhelmingly surprising to me, however, in the geometric case because I can sort of feel the picture of a laplacian and heat kernel based on my experience.

### Questions

What's curious to me are the following:

1. Having zero experience with graph theory, is there a better way to appreciate/interpret what those eigenvalues, heat kernel, … mean?
2. There must be much more behind it. What are some updated important results? Do people understand this much better than they did 23 years ago?
3. Most interestingly (to me), is there a way to extract a graph $$G$$ from a given Riemannian manifold $$M$$, so that the spectral theory on both sides agree?

Any pointers to related work will be highly appreciated.

• The paper "Discrete Laplace operators: no free lunch" by Wardetzky, Mathur, K{\"a}lberer, and Grinspun (cs.columbia.edu/cg/pdfs/1180993110-laplacian.pdf) discusses several nice properties of the Laplacian (symmetry, positive semi-definiteness, maximum principle, kernel containing the constant functions, and two others) and shows that no linear operator on functions defined on a triangular mesh can have all of these properties. (continued) Oct 31 '19 at 17:18
• As they say in the abstract, this retroactively explains the diversity of discrete Laplacians in the literature. I don't know to what extent this result applies to Chung and Yau's setup. Oct 31 '19 at 17:18