Graph theory for fused cubic clusters of water dodecamer.

The stable structures of the fused cubic water cluster (H2O)12 are examined using graph theoretical techniques and ab initio calculations. The calculations are obtained by scanning the symmetry of digraph structures of hydrogen-bond network spanning 12 oxygen atom vertexes. Using the Pólya theorem the cycle index expressions for 12 vertexes and 20 edges of a cuboid in point-group symmetry D(4h) are developed. A total of 91 energy-allowed fused cubic structures are obtained, which are classified by 8 point-group symmetries: 1 D(2h), 2 S4, 5 C4, 1 D2, 11 C2, 10 C(i), 1 C(s), and 60 C1. An energy level diagram of the structures reveals 14 bands that correspond to 14 unique two-colored graphs derived from the distributions of four free hydrogens of the cluster.