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.
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