  We define two natural families of determinantal random subgraphs of a finite
connected graph, one supported by acyclic spanning subgraphs (spanning forests)
with fixed number of components, the other by connected spanning subgraphs with
fixed number of independent cycles. Each family generalizes the uniform
spanning tree and the generating functions of these probability measures
generalize the classical Kirchhoff and Symanzik polynomials. We emphasize the
matroidal nature of this construction, as well as possible generalisations.
