Depending on what kind of zeta functions you want, the Selberg zeta function allows you to relate lengths of closed geodesics to eigenvalues of the Laplacian. In particular, you can use the Selberg zeta function in combination with a trace formula to prove the prime geodesic theorem for compact Riemann surfaces and get Weyl's law. This also leads to construct isospectral manifolds.

Similarly, for graphs one can look at the analogous Ihara zeta function to relate lengths of "geodesics" to certain spectral quantities. In particular, one can get a characterization of Ramanujan graphs in terms of the Ihara zeta function. There are also numerous variants to count different things in graphs (Bartholdi zeta function, path zeta functions), and I have a conjecture with Christina Durfee that zeta functions are better at distinguishing graphs spectrally than the usual (adjacency matrix or Laplacian) spectra considered.