Resonance graphs of fullerenes

english

2016

volume 11, issue 2

cube polynomial   distributive lattice   fullerene   Kekulé structure   median graph   perfect matching   resonance graph   Zhang-Zhang polynomial

Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije
(Obvezni izvod spletne publikacije)

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.

29.05.2020