"The vector-bundle Laplacian on a graph" Rick Kenyon, Brown University The classical matrix-tree theorem states that the determinant of the combinatorial laplacian on a graph counts the number of spanning trees. We generalize this result to the laplacian on a one- or two-dimensional vector bundle on a graph. Its determinant counts so-called "cycle-rooted spanning trees". This result allows us to probe the large-scale structure of spanning trees on planar graphs, as well as introduce a generalization of the uniform spanning tree measure on graphs on Riemann surfaces.