of all these configurations becomes non-zero irrespective to which sublattice the vacancies belong. This fact show that the Lieb’s rule seems to violate for a graphene mono-layer, and the magnetic moment of the supercell with several vacancies is determined with the number of the dangling bonds but not with the difference of the number of atoms in each sublattices. The band structures of di-vacancy and all other two-vacancy states are presented correspondingly in Fig. \[Vac1\]b and Figs. \[Vac2\]b, c, d, e. It is necessary to pay attention to the fact that two-vacancies (including the DV) generate quasi-localized state (see, the band structures in Fig. \[Vac1\]b and Figs. \[Vac2\]c, e) when they belong to the different sublattices, and instead two-vacancies generate an extended state (see the band structures in Figs. \[Vac2\]b, d) when they belong to the same sublattices. Furthermore, the band structures, corresponding to spin-up and spin-down states, coincide each other only in the presence of the di-vacancy due to absence of a dangling bond. All other two-vacancy structures contain dangling bonds, resulting in a splitting of opposite spin-polarized states in the band structures.

In order to create a tri-vacancy structure we remove three nearest-neighboring carbon atoms belonging to one 