Graphene clusters consisting of 24 to 150 carbon atoms and hydrogen termination at the zigzag boundary edges have been studied, as well as clusters disordered by vacancy(s). Density Function Theory and Gaussian03 software were used to calculate graphene relative stability, desorption energy, band gap, density of states, surface shape, dipole momentum and electrical polarization of all clusters by applying the hybrid exchange-correlation functional Beke-Lee-Yang-Parr. Furthermore, infrared frequencies were calculated for two of them. Different basis sets, 6-31g**, 6-31g* and 6-31g, depending on the sizes of clusters are considered to compromise the effect of this selection on the calculated results. We found that relative stability and the gap decreases according to the size increase of the graphene cluster. Mulliken charge variation increase with the size. For about 500 carbon atoms, a zero HOMO-LUMO gap amount is predicted. Vacancy generally reduces the stability and having vacancy affects the stability differently according to the location of vacancies. Surface geometry of each cluster depends on the number of vacancies and their locations. The energy gap changes as with the location of vacancies in each cluster. The dipole momentum is dependent on the location of vacancies with respect to one another. 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, but charge distribution results are completely different.