The ground state of graphene and graphene disordered by vacancies

► The study of graphene islands, culminated in the fact that relative stability and the gap decreases according to the size increase. ► Mulliken charge variation increases with the size. ► Vacancy generally reduces the stability and having vacancy affects the stability differently according to the location of vacancies. ► The carbon-carbon length changes according to each covalence band distance from the boundary and vacancies. ► Two basis sets, 6-31g⁎ and 6-31g**, predict equal amount for energy, gap and surface structure.