Applying a combinatorial determinant to count weighted cycle systems in a directed graph

One method for counting weighted cycle systems in a graph entails taking the determinant of the identity matrix minus the adjacency matrix of the graph. The result of this operation is the sum over cycle systems of −1 to the power of the number of disjoint cycles times the weight of the cycle system. We use this fact to reprove that the determinant of a matrix of much smaller order can be computed to calculate the number of cycle systems in a hamburger graph. This article deals with counting cycle systems (also called partial cycle covers), which are collections of vertex-disjoint directed cycles in a directed graph. The following combinatorial fact is useful in the study of cycle systems. Theorem 1. Let G = (V, E) be a weighted, directed graph and let M be its adjacency matrix. Let S be the set of cycle systems of G. If C is a cycle system, let |C| denote the number of cycles in C an let wt(C) be the product of the weights of the edges in C. Then