with one vacancy and a pristine graphene are shown in Fig. \[Vac1\]a and Fig. \[Vac2\]a, correspondingly, for comparative study. The band gap of the perfect graphene is closed at the border of the Brillouin zone, yielding two Dirac cones. A single vacancy seems to remove the band crossing degeneracy at $K$-points of the Brillouin zone, and introduces at the same time two extended defect levels corresponding to spin-up and spin-down states at the Fermi level. Assuming that a continuous irradiation of a sample will create isolated vacancies, the magnetization of the sample would monotonically increase. Furthermore, in the absence of spin-spin correlations between localized dangling bonds, the magnetic moments of the vacancies belonging to different sublattice will partially compensate each other and reduce the total magnetization. Nevertheless, the magnetization under this assumption would monotonically increase with vacancy concentration. Our calculations below show that creation of mono vacancies monotonically distributed throughout the structure is not energetically favorable, and there is a tendency to form a cluster or hole of vacancies by coalescing them each other or collapsing around the extended defects like grain boundaries, dislocations or the sample boundary.

A divacancy (DV) is produced by the removal of two nearest-neighboring carbon 