Distances on nanotubical structures

Nanotubical graphs are obtained by wrapping a hexagonal grid into a cylinder, and then possibly closing the tube with patches. In this paper 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. In particular, for $$d\ge 2k$$d≥2k it is $$2\,(k+l)$$2(k+l) for an infinite open nanotube. This result can be used as a tool for precise evaluation of distance based topological indices for nanotubical structures. The presented results imply an interesting conclusion that these indices do not distinguish the type of the nanotubes very well.