i will show how to find uniform finiteness results for certain diophantine equations in terms of the laplace spectrum of an associated graph. the method is to bound the "gonality" of a curve (minimal degree of a map onto a line) by the "stable gonality" of an associated stable reduction graph, and then to bound this stable gonality of the graph (some kind of minimal degree of a map to a tree) in terms of spectral data. the latter bound is a graph theoretical analogue of a famous inequality of li and yau in differential geometry. an example of an application is to bound the degree of the modular parametrisation of elliptic curves over function fields. (joint work with fumiharu kato and janne kool.)