  We investigate the stability of boron fullerene sets B76, B78 and B82. We
evaluate the ground state energies, nucleus-independent chemical shift (NICS),
the binding energies per atom and the band gap values by means of
first-principles methods. We construct our fullerene design by capping of
pentagons and hexagons of B60 cage in such a way that the total number of atoms
is preserved. In doing so, a new hole density definition is proposed such that
each member of a fullerene group has a different hole density which depends on
the capping process. Our analysis reveal that each boron fullerene set has its
lowest-energy configuration around the same normalized hole density and the
most stable cages are found in the fullerene groups which have relatively large
difference between the maximum and the minimum hole densities. The result is a
new stability measure relating the cage geometry characterized by the hole
density to the relative energy.
