Resonance graphs of fullerenes - Tratnik, Niko (Ars mathematica contemporanea, 2016)
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.
Keywords: cube polynomial; distributive lattice; distributivna mreža; fuleren; fullerene; Kekulé structure; kekuléjska struktura; kockovni polinom; median graph; medianski graf; perfect matching; popolno prirejanje; resonance graph; resonančni graf; Zhang-Zhang polynomial; Zhang-Zhangov polinom