  In this paper, we study extremal values for the determinant of the weighted
graph Laplacian under simple nondegeneracy conditions on the weights. We derive
necessary and sufficient conditions for the determinant of the Laplacian to be
bounded away from zero and for the existence of a minimizing set of weights.
These conditions are given both in terms of properties of random spanning trees
and in terms of a type of density on graphs. These results generalize and
extend the work of [7].
