  Vacancy-induced magnetization of a graphene layer is investigated by means of
a first principle DFT method. Calculations of the formation energy and the
magnetization by creating the different number of vacancies in a supercell show
that a clustering with big number of vacancies in the cluster is rather
favorable than that of isolated vacancies, homogeneously distributed in the
layer. The magnetic moment of a cluster with big number of vacancies is shown
to be not proportional with the vacancy concentration, which is in good
agreement with the recent experimental results. Our studies support the idea
that although the vacancies in a graphene create a magnetic moment, they do not
produce a magnetic ordering. It is shown that although the Lieb's rule for the
magnetization in a hexagonal structure violates, two-vacancies, including a
di-vacancy, in the supercell generate quasi-localized state when they belong to
the different sublattices, and instead two-vacancies generate an extended state
when they belong to the same sublattices. Analytical investigation of the
dynamics of carbon atom- and vacancy-concentrations according to the non-linear
continuity equations shows that the vacancies, produced by irradiation at the
middle of a graphene layer, migrate to the edge of the sample resulting in a
specific {\it 'segregation'} of the vacancy concentration and self-healing of
the graphene.
