On the Symmetry of a Zig-Zag and an Armchair Polyhex Carbon Nanotorus

A Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [d(ij)], where for i not equal j, d(ij) is the Euclidean distance between the nuclei i and j. In this matrix d(ii) can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. The aim of this paper is to compute the automorphism group of the Euclidean graph of a carbon nanotorus. We prove that this group is a semidirect product of a dihedral group by a group of order 2.