  For a finite graph, a spectral curve is constructed as the zero set of a
two-variate polynomial with integer coefficients coming from p-adic diffusion
on the graph. It is shown that certain spectral curves can distinguish
non-isomorphic pairs of isospectral graphs, and can even reconstruct the graph.
This allows the graph reconstruction from the spectrum of the associated p-adic
Laplacian operator. As an application to p-adic geometry, it is shown that the
reduction graph of a Mumford curve and the product reduction graph of a p-adic
analytic torus can be recovered from the spectrum of such operators.
