98 atoms with up to four vacancies in different configurations. The [*C-C*]{} bond length of a pristine graphene lattice is calculated to be $1.409 ~\AA$, $1.412~\AA$, and $1.412~\AA$ for the supercells correspondingly with 50, 98, and 162 atoms, which is in good consistent with the experimental value of $1.42~\AA$.

  ------------------------- ---------------------- -------------------------
          **Type**           **Formation energy**    **Spin polarization**
                                   **(eV)**         **$(Q_{up}-Q_{down})$**
   \[0.5ex\] Single defect           6.99                    1.13
          DV 5-8-5                   5.76                     0.0
         DV 555-888                  4.52                     0.0
        3 vac. 55-10                 8.84                    1.00
        3 vac. 999-3                17.49                    4.00
        3 vac. 555-11               11.91                    1.54
        4 vac. 555-9                 7.91                     0.0
       4 vac. 5-12-5a               11.27                    2.02
       4 vac. 5-12-5b               11.27                    2.02
        4 vac. 55-12                13.33                    1.99
        4 vac. 12-55                13.33                    1.99
           \[1ex\]                                 
  ------------------------- ---------------------- -------------------------

  : Formation energy and total spin polarization for the supercell of 98 atoms with different vacancies.

\[table2\]

Creation of one vacancy, producing pentagon-nonagon type [*5-9*]{} defect in the supercell, results in the appearance of three dangling $\sigma$- bonds, two of which rebind again with each other due to the Jahn-Teller distortion around vacancy. Jahn-Teller distortion deforms the lattice around the vacancy and breaks the threefold symmetry; the geometrical distortion is relaxed off behind the vacancies. Covalent-bond coupling between two dangling bonds of the second-nearest-neighboring 