Consider the graph $G$, each edge of which has weight $T$. Also consider some cut of $G$. We call the weight of the cut - product of all edges included in the cut. Generating function of cuts of $G$ is sum of weights of all cuts.
My question: what good properties have this generating function? For example, may be this function is expressed through the determinant of some matrix?
Motivation: generating function of cuts has strong connection with Ising model.