Note on the asymptotic isomer count of large fullerenes

Jerzy Cioslowski

doi:10.1007/s10910-013-0263-3

Abstract

The leading term in the large-\(N\) asymptotics of the isomer count of fullerenes with \(N\) carbon atoms is extracted from the published enumerations for \(N\le 400\) with the help of methods of series analysis. The uncovered simple \(N^9\) 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 \(N^{25/3}\). A conjecture concerning isomer counts of the IPR fullerenes is also formulated.

Highlights

Recent advances in graph-theoretical algorithms have opened new vistas for enumeration of chemical isomers
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
The present result imposes the same asymptotics for the isomer count MI P R(k) of the IPR fullerenes with 2k carbon atoms as 0 < MI P R(k) < M(k) and limk→∞ MI P R(k)/M(k) → 1

Summary

Introduction

Recent advances in graph-theoretical algorithms have opened new vistas for enumeration of chemical isomers. Keywords Fullerenes · Isomer count · Series analysis Significant progress has been achieved in the case of fullerenes CN , of which all structures with N ≤ 400 have 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.

Results

Conclusion