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

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## 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 Kekulé 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.

## Keywords

Carbon nanotube Kekulé structure Resonance graph Distributive lattice Median graph Block graph## Notes

### Acknowledgments

Supported in part by the Ministry of Science of Slovenia under grants \(P1-0297\).

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