$n\geq 3$. The definition of a Hamiltonian cycle of an undirected graph is a closed undirected path that visits each vertex once and only once. We use the notation $m_1m_2\cdots
m_{n}m_1$ to denote a Hamiltonian cycle in an undirected graph (recall $(m_{1},m_{2},\cdots,m_{n},m_{1})$ denotes a Hamiltonian cycle in digraphs). The degree of a vertex in an undirected graph $G$ is defined as the number of its neighbors. Let $\#$HC and $\#$DHC be the problem of counting the number of Hamiltonian cycles in undirected and directed graphs respectively. Now define a symmetric digraph $G'$ corresponding to an undirected graph $G$ by replacing each edge $(i,j)$ of $G$ with two directed edges $(i,j)$ and $(j,i)$. Let $H_G$ and $H_{G'}$ denote the set of the Hamiltonian cycles in $G$ and $G'$ respectively. Let $\mathcal
{P}(H_{G'})$ denote the power set of $H_{G'}$. We will prove the number of Hamiltonian cycles in an undirected graphs equals half of the number of Hamiltonian cycles in its corresponding symmetric digraphs.\
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**Lemma 10.** ([@DFJ94]) *$\#$HC is $\#$P-Complete, even when restricted to graphs $G$ of minimun degree at least $(1-\varepsilon)n$, for any $\varepsilon>0$*\
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**Lemma 11.** *Let $\mathcal{H}=m_1\cdots m_{n}m_1$ be a Hamiltonian cycle in $H_G$. Then there are at 