Resonance graphs of fullerenes
Keywords:Fullerene, resonance graph, Zhang-Zhang polynomial, cube polynomial, Kekul\'e structure, perfect matching, distributive lattice, median graph
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.