  To a graph, Hausel and Proudfoot associate two complex manifolds, B and D,
which behave, respectively like moduli of local systems on a Riemann surface,
and moduli of Higgs bundles. For instance, B is a moduli space of microlocal
sheaves, which generalize local systems, and D carries the structure of a
complex integrable system. We show the Euler characteristics of these varieties
count spanning subtrees of the graph, and the point-count over a finite field
for B is a generating polynomial for spanning subgraphs. This polynomial
satisfies a deletion-contraction relation, which we lift to a
deletion-contraction exact triangle for the cohomology of B. There is a
corresponding triangle for D. Finally, we prove B and D are diffeomorphic, that
the diffeomorphism carries the weight filtration on the cohomology of B to the
perverse Leray filtration on the cohomology of D, and that all these structures
are compatible with the deletion-contraction triangles.
