 It is a generalization of the number of Hamiltonian cycles of a digraph as the sum of the products of its Hamiltonian cycles' arc weights (all of which equal unity) for weighted digraphs with arc weights taken from a given commutative ring.In the meantime, for an undirected weighted graph the sum of the products of the edge weights of its Hamiltonian cycles containing any fixed edge ("i","j") can be expressed as the product of the weight of ("i","j") and the Hamiltonian cycle polynomial of the matrix received from its weighted adjacency matrix via removing the "i"-th row and the "j"-th column.In () it was shown that

where formula_4 is the submatrix of formula_5 induced by the rows and columns of formula_5 indexed by formula_7, and formula_8 is the complement of formula_7 in formula_10, while the determinant of the empty submatrix is defined to be 1.