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.
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Supported in part by the Ministry of Science of Slovenia under grants \(P1-0297\).
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Tratnik, N., Žigert Pleteršek, P. Distributive lattice structure on the set of perfect matchings of carbon nanotubes. J Math Chem 54, 1296–1305 (2016). https://doi.org/10.1007/s10910-016-0622-y
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DOI: https://doi.org/10.1007/s10910-016-0622-y