  The thermodynamic stability of structural isomers of $\mathrm{C}_{24}$,
$\mathrm{C}_{26}$, $\mathrm{C}_{28}$ and $\mathrm{C}_{32}$, including
fullerenes, is studied using density functional and quantum Monte Carlo
methods. The energetic ordering of the different isomers depends sensitively on
the treatment of electron correlation. Fixed-node diffusion quantum Monte Carlo
calculations predict that a $\mathrm{C}_{24}$ isomer is the smallest stable
graphitic fragment and that the smallest stable fullerenes are the
$\mathrm{C}_{26}$ and $\mathrm{C}_{28}$ clusters with $\mathrm{C}_{2v}$ and
$\mathrm{T}_{d}$ symmetry, respectively. These results support proposals that a
$\mathrm{C}_{28}$ solid could be synthesized by cluster deposition.
