Distributive lattice structure on the set of perfect matchings of carbon nanotubes

Abstract

Carbon nanotubes are composed of carbon atoms linked in hexagonal shapes, with each carbon atom covalently bonded to three other carbon atoms. Carbon nanotubes have diameters as small as 1 nm and lengths up to several centimeters. Carbon nanotubes can be open-ended or closed-ended (fullerenes). Open-ended single-walled carbon nanotubes are also called tubulenes. The resonance graph R(T) of a tubulene T reflects interactions between Kekule structures—i.e. perfect matchings of T. With the orientation of edges the resonance digraph \(\overrightarrow{R}(T)\) of a tubulene is obtained. As the main result we show that \(\overrightarrow{R}(T)\) is isomorphic to the Hasse diagram of the direct sum of some distributive lattices. Similar results were proved in [10, 16], but one can not directly apply them to tubulenes. As a consequence of the main result it is proved that every connected component of R(T) is a median graph. Further we show that the block graph of every connected component H of the resonance graph of a tubulene is a path and that H contains at most two vertices of degree one.