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:

- Having zero experience with graph theory, is there a better way to appreciate/interpret what those eigenvalues, heat kernel, … mean?
- 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?
- 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.