Distance based indices on nanotubical graphs

Nanotubical graphs are obtained by wrapping a hexagonal grid into a cylinder, and then possibly closing the tube with patches. Here we consider the asymptotic values of Wiener, generalized Wiener, Schultz (also known as degree distance), Gutman, Balaban, Sum-Balaban, and Harary indices for (all) nanotubical graphs of type (k, l) on n vertices. First, we determine the number of vertices at distance d from a particular vertex in an open (k, l) nanotubical graph. Surprisingly, this number does not depend much on the type of the nanotubical structure, but mainely on its circumference. At the same time, the size of a cap of a closed (k, l)-nanotube is bounded by a function that depends only on k and l, and that those extra vertices of the caps do not influence the obtained asymptotical value of the distance based indices considered here. Consequently the asymptotic values are the same for open and closed nanostructures. Finally, we obtained that the leading term of all considered topological indices depends on the circumference of the nanotubical graph, but not on its specific type. Thus, we conclude that these distance based topological indices seem not to be the most suitable for distinguishing nanotubes with the same circumference and of different type as far as the leading term is concerned.

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