On the Symmetry of a Zig-Zag and an Armchair Polyhex Carbon Nanotorus
Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan, I. R. Iran * Author to whom correspondence should be addressed: E-mail: Ashrafi@kashanu.ac.ir Received: 17 July 2009; in revised form: 13 September 2009 / Accepted: 14 September 2009 / Published: 8 October 2009 Abstract: A Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [d
On Three Constructions of Nanotori
There are three different approaches for constructing nanotori in the literature: one with three parameters suggested by Altshuler, another with four parameters used mostly in chemistry and physics after the discovery of fullerene molecules, and one with three parameters used in interconnecting networks of computer science known under the name generalized honeycomb tori. Altshuler showed that his method gives all non-isomorphic nanotori, but this was not known for the other two constructions. Here, we show that these three approaches are equivalent and give explicit formulas that convert parameters of one construction into the parameters of the other two constructions. As a consequence, we obtain that the other two approaches also construct all nanotori. The four parameters construction is mainly used in chemistry and physics to describe carbon nanotori molecules. Some properties of the nanotori can be predicted from these four parameters. We characterize when two different quadruples define isomorphic nanotori. Even more, we give an explicit form of all isomorphic nanotori (defined with four parameters). As a consequence, infinitely many 4-tuples correspond to each nanotorus; this is due to redundancy of having an “extra” parameter, which is not a case with the other two constructions. This result significantly narrows the realm of search of the molecule with desired properties. The equivalence of these three constructions can be used for evaluating different graph measures as topological indices, etc.
Construction of the symmetry groups of polymer molecules
A derivation of the line groups, which are the symmetry groups of the stereo-regular polymer molecules, is presented. Every line group is an extension of a one-dimensional translation group by a point group. The point group is either cyclic or a semi-direct product of cyclic groups. The expounded method of derivation of the line groups consists of first extending by the cyclic point groups and then obtaining the rest of the line groups by unification within the Euclidean group. To this purpose, a simple test is given to decide whether two line groups multiply into a third one. The method displays the subgroup structure of the line groups, relevant to the construction of their irreducible representations. All the line groups are derived and tabulated.