Distance based indices in nanotubical graphs: part 3

Abstract

Nanotubical graphs are obtained by wrapping a hexagonal grid, and then possibly closing the tube with caps. In this paper we derive asymptotics for generalized Wiener index for such graphs. More generally, we show that if $$I^{\lambda }(G)=\sum _{u\ne v}f(u,v)\mathrm{dist}^{\lambda }(u,v)$$ , where $$\lambda \ge -1$$ and f(u, v) is a nonnegative symmetric function which is increasing and which depends only on $$\deg (u)$$ and $$\deg (v)$$ , then the leading term depend only on $$\lambda $$ , f(x, y) when $$\deg (x)=\deg (y)=3$$ , and the circumference of the nanotube. This general form determines the asymptotics also for some other indices as Hararay index, additively weighted Harary index, generalized degree distance, modified generalized degree distance, and possibly others of this kind.