Geodesic
Resonance graphs of fullerenes
Ars Mathematica Contemporanea, Tome 11 (2016) no. 2, pp. 425-435.

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A fullerene G is a 3-regular plane graph consisting only of pentagonal and hexagonal faces. The resonance graph R(G) of G reflects the structure of its perfect matchings. The Zhang-Zhang polynomial of a fullerene is a counting polynomial of resonant structures called Clar covers. The cube polynomial is a counting polynomial of induced hypercubes in a graph.In the present paper we show that the resonance graph of every fullerene is bipartite and each connected component has girth 4 or is a path. Also, the equivalence of the Zhang-Zhang polynomial of a fullerene and the cube polynomial of its resonance graph is established. Furthermore, it is shown that every subgraph of the resonance graph isomorphic to a hypercube is an induced subgraph in the resonance graph. For benzenoid systems and tubulenes each connected component of the resonance graph is the covering graph of a distributive lattice; for fullerenes this is not true, as we show with an example.
DOI : 10.26493/1855-3974.1000.8db
Keywords: Fullerene, resonance graph, Zhang-Zhang polynomial, cube polynomial, Kekul\'e structure, perfect matching, distributive lattice, median graph 
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Niko Tratnik; Petra Žigert Pleteršek. Resonance graphs of fullerenes. Ars Mathematica Contemporanea, Tome 11 (2016) no. 2, pp. 425-435. doi : 10.26493/1855-3974.1000.8db. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1000.8db/

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