Tensor and Cartesian products for nanotori, nanotubes and zig–zag polyhex nanotubes and their applications to 13C NMR spectroscopy

We have developed novel topological techniques to generate distance degree vector sequences (DDS) and distance degree scalar sequences(SS) of tensor and Cartesian products of graphs. We establish that the tensor products of two distance degree regular graphs is distance degree regular. We apply the mathematical techniques thus developed for both tensor and Cartesian products to enumerate 13C NMR signals and intensity patterns through the partitioning of equivalence classes accomplished by the use of distance degree vector sequences. In particular the techniques are applied to several nano materials such as carbon nanotori, monocyclic cylindrical nanotubes and zig–zag polyhex carbon nanotubes. It is shown that DDS sequences satisfactorily partition carbon vertices into equivalence classes for these nanomaterials. We also demonstrate applications of the developed techniques to diagrammatic representations of energy decompositions in molecular interactions.