Equivalence of Zhang-Zhang polynomial and cube polynomial for carbon nanotubes

Petra Žigert Pleteršek
UM FKKT, UM FNM

Niko Tratnik
UM FNM

Martina Berlič
UM FNM

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Minisymposium: CHEMICAL GRAPH THEORY

Content: Benzenoid systems or hexagonal systems are subgraphs of a hexagonal lattice. Open-ended carbon nanotubes alias tubulenes can be seen as an embedding of a benzenoid system to a surface of a cylinder with some perimeter edges being joined. Carbon nanotubes are interesting materials with some unusual properties and have therefore been of great interest for researchers in the last $30$ years. Zhang-Zhang polynomial (also called Clar covering polynomial) of a spherical benzenoid system is a counting polynomial of resonant structures called Clar covers. Cube polynomial is a counting polynomial of induced hypercubes in a graph. In 2013 Zhang et al. established the one-to-one correspondence between Clar covers of a benzenoid system $G$ and hypercubes of its resonance graph $R(G)$. We will extend the equality of these two polynomials to tubulenes.

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