Topological characterization of five CT8 fullerene isomers

Experimental and theoretical studies on fullerene cages are the topic of several investigations in recent years [ 1-l 11. Since the chemistry of fullerenes is increasingly becoming important and theoretical studies of these cages are on the increase it is necessary to have several mathematical procedures for the study of topological properties of these cages. Topological, graph theoretical and group theoretical studies on the fullerene cages and polycyclic aromatics in particular have been made in recent years [ 1 l-2 11. Graph theory is one of the powerful tools of discrete mathematics which has in particular made significant impact in the structural characterization of fullerene cages. The carbon soot resulting from resistive heating of graphite contains several byproducts in addition to C,, such as (&, CT8 and Cs4. Some of these fullerenes have been spectroscopically characterized through “C NMR spectra of toluene-extracted and chromatographically separated carbon soot. The CT8 fullerene [ 8111 is quite interesting since a simple “isolated-pentagon rule” which stipulates that most stable fullerenes should have isolated pentagons leads to five equally viable candidates for the C,s cluster. The five isomers with isolated pentagons for C,8 [ 8 ] are labelled as lDsh, 2D3h, 3D3, 4CZy and X&. It is not yet unambiguously established as to which is the most stable of these five candidates. For example, the D3,, structure is the favored one according to the Hiickel theory of Fowler et al. [ 81, while the recent ab initio MNDO, AM 1 and PM3 semi-empirical calculations place the CZV structure as the most stable one [lo]. Since quantum-chemical procedures alone cannot always establish the relative ordering of closely spaced isomers for such large systems, alternative procedures are needed to discriminate the five isomers. In fact, even ordinary spectra are similar for some of the isomers. Among the five isomers of CT8 the two D3,, cages [ 81 are predicted to have identical IR, Raman, 13C NMR and polarized Raman spectra. The two different structures with the CZV symmetries are predicted to have identical IR and Raman spectra although their i3C NMR are predicted to differ. The characteristic polynomials of graphs in general [ 17 ] and fullerene graphs [ 181 in particular have been the topic of several studies. There is a question as to the existence of two non-isomorphic cages with the same characteristic polynomial. While for benzenoid graphs there are non-isomorphic graphs with the same characteristic polynomials, this point has not yet been established for the fullerene cages one way or the other. The present study is aimed at illuminating on these points through the computation of the characteristic polynomials of the five isomers of C,*. It is shown that the first 11 coefficients of all five cages are identical. The twelfth and higher terms in the polynomials are shown to discriminate the cages.