Resonance graphs of fullerenes

Niko Tratnik, Petra Žigert Pleteršek


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.


Fullerene, resonance graph, Zhang-Zhang polynomial; cube polynomial; Kekul\'e structure, perfect matching, distributive lattice, median graph

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ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications