subgraph of G in which indegrees and outdegrees of each vertex are all one. An example of an 1-factor is a spanning union of vertex disjoint directed cycles. Obviously, a Hamiltonian cycle is a special 1-factor with only one cycle. The weight $W(F)$ of an 1-factor $F$ with edge set $\{e\in E\}_{e\in F}$ is defined as $W(F)=\prod_{e\in
F}w_e$. The total weight $W(S)$ of the set $S$ of 1-factors are defined as $W(S)=\sum_{F\in S}W(F)$. Let $W_F(G)$ and $W_H(G)$ denote the total weight of all the 1-factors and Hamiltonian cycles in $G$ respectively. It is easy to see if $w_{ij}=1$ for all $(i,j)\in E$, then $W_F(G)$ and $W_H(G)$ are the number of 1-factors and Hamiltonian cycles in $G$ respectively.

Let $A_G$ be the adjacent matrix associated with $G$ where $A_G(i,j)=w_{ij}$ if $(i,j)\in E$ and $A_G(i,j)=0$ otherwise. For an $n\times n$ matrix $A$, where $n$ is the order of $A$, we use notation $A_{ij}$ to denote the $(n-1)\times (n-1)$ matrix obtained from $A$ by removing row $i$ and column $j$. If there is no confusion, $A^{'}_{ij}$ or $(A_{ij})^{'}$ denotes the $(n-1)\times
(n-1)$ matrix obtained from $A$ by, first permutating row $i$ and row $j$ and then removing row $j$ and column $j$. Next 