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.
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.