Note on the asymptotic isomer count of large fullerenes

The leading term in the large-N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} asymptotics of the isomer count of fullerenes with N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} carbon atoms is extracted from the published enumerations for N≤400\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\le 400$$\end{document} with the help of methods of series analysis. The uncovered simple N9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^9$$\end{document} scaling is distinct from isomer counts of most classes of chemical structures that conform to mixed exponential/power-law asymptotics. The second leading asymptotic term is found to be proportional to N25/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{25/3}$$\end{document}. A conjecture concerning isomer counts of the IPR fullerenes is also formulated.


Introduction
Recent advances in graph-theoretical algorithms have opened new vistas for enumeration of chemical isomers. In particular, significant progress has been achieved in the case of fullerenes C N , of which all structures with N ≤ 400 have now been generated [1,2]. The availability of these data has prompted speculations concerning the behavior of the fullerene isomer counts at the N → ∞ limit, both the N 9 [3] and N 19/2 [4] asymptotics being inferred from crude log-log plots and supported by heuristic arguments. For the reason spelled out in the following, this simple power-law scaling appears unlikely at the first glance.
Complete information about isomer counts for a class of chemical species is encoded in the generating function F(t) given by the for formal series where M(k) is the number of isomers comprising k units (such as atoms, bonds, rings, etc.). Since, in general, M(k + 1) ≥ M(k) for all k > 0, the series (1) possesses a finite radius of convergence. Consequently, F(t) possesses at least one singular point, at which it behaves like (t c − t) ζ c , where 0 < t c ≤ 1. The smallest critical point t c and the corresponding critical exponent ζ c determine the leading term in the large-k asymptotics of M(k), which reads where A is a constant. Typically, t c < 1 (e.g. ca. 0.35518 for alkanes [5], ca. 0.20915 for polyenes [5], and 1 5 for catafusenes [5][6][7]), giving rise to the mixed exponential/powerlaw asymptotics (2). On the other hand, the alleged power-law scaling of the fullerene isomer counts would imply t c = 1.
In order to investigate this matter, in this note we invoke the mathematical formalism of series analysis that is commonly used in lattice statistics [8]. Such a formalism has been previously employed in successful extraction of the asymptotic isomer counts of several classes of chemical structures [5]. Let The smallest positive root z c of Q(x) and the quantity η c = 1 − R(z c )/Q (z c ) yield unbiased estimates for t c and ζ c , respectively [5,8]. In general, the accuracy of these estimates increases with both n and m.

Results and conclusions
Application of the aforedescribed formalism to the isomer counts reported in Ref. [1] produces estimates that clearly converge to t c = 1 (Fig. 1) and ζ c = −10 (Fig. 2). Thus, the leading term proportional to N 9 in the large-N asymptotics of the isomer count of fullerenes with N carbon atoms is now firmly established (although not rigorously proven).
The present result imposes the same asymptotics for the isomer count M I P R (k) of the IPR fullerenes with 2k carbon atoms as 0 < M I P R (k) < M(k) and lim k→∞ M I P R (k)/M(k) → 1. Curiously, inspection of the published data [1] allows one to formulate the following conjecture (see Fig. 3): For all k > 53, M(k − 24) < M I P R (k) < M(k − 25), i.e. for all N > 106, the number of the IPR fullerene isomers with N carbon atoms is bracketed by the total numbers of isomers of the C N −50 and C N −48 fullerenes.
The second leading term in the large-k asymptotics of M(k) is also of interest. As revealed by the plot of M(k)/k 9 versus k −2/3 (Fig. 4), this term scales simply as k 25/3 and is negative. The combination of the N 9 and N 25/3 asymptotics explains the apparent N 19/2 scaling deduced from a crude log-log plot [4].