  Together with the second generation REBO reactive potential, replica-exchange
molecular dynamics simulations coupled with systematic quenching were used to
generate a broad set of isomers for neutral C$_n$ clusters with $n=24$, 42, and
60. All the minima were sorted in energy and analyzed using order parameters to
monitor the evolution of their structural and chemical properties. The
structural diversity measured by the fluctuations in these various indicators
is found to increase significantly with energy, the number of carbon rings,
especially 6-membered, exhibiting a monotonic decrease in favor of
low-coordinated chains and branched structures. A systematic statistical
analysis between the various parameters indicates that energetic stability is
mainly driven by the amount of sp$^2$ hybridization, more than any geometrical
parameter. The astrophysical relevance of these results is discussed in the
light of the recent detection of C$_{60}$ and C$_{60}^+$ fullerenes in the
interstellar medium.
