It becomes difficult to find all of the possible structures of the water cluster when the number of water molecules increases. In this report we construct a systematic way of finding all the possible structures using graph theory. In a water cluster water molecules are held together by hydrogen bonding and taken into the hydrogen bond network. Topological patterns of hydrogen bonds in the water clusters correspond to the directed graphs. We counted up directed graphs which correspond to the topological patterns of hydrogen bonds for (H2O)n,n=3-8. Then, for each topological pattern (n=3-5) we examined whether a structure within the pattern corresponds to a local minimum on the potential energy surface with the level of HF/6-31G*. For (H2O)3, 3 topological patterns were found which correspond to stable structures, and 5 topological patterns for (H2O)4, and 22 for (H2O)5, respectively. We carried out ab intio MD calculations (HF/6-31G*) and examined how the water cluster changes its structure from one stable structure to another.
