Distance based indices in nanotubical graphs: part 1

Nanotubical structures are obtained by wrapping a hexagonal grid, and then possibly closing the tube with caps. We show that 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. We also show that the asymptotic for Wiener index, Schultz index (also known as degree distance), and Gutman index for (all) nanotubical graphs of type (k, l) on n vertices are n36(k+l)+O(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{n^3}{6(k+l)}+O(n^2)$$\end{document}, 3n32(k+l)+O(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3n^3}{2(k+l)}+O(n^2)$$\end{document}, and n3k+l+O(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{n^3}{k+l}+O(n^2)$$\end{document}, respectively. In all cases, the leading term 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.