Upper Bounds for the Number of Hamiltonian Cycles

An upper bound for the number of Hamiltonian cycles of symmetric diagraphs is established first in this paper, which is tighter than the famous Minc's bound and the Br$\acute{e}$gman's bound. A transformation on graphs is proposed, so that counting the number of Hamiltonian cycles of an undirected graph can be done by counting the number of Hamiltonian cycles of its corresponding symmetric directed graph. In this way, an upper bound for the number of Hamiltonian cycles of undirected graphs is also obtained.