We use the density functional theory B3LYP/6-31G(d) to study the graphene quantum dots with zigzag edges. The result shows that the ground states of different size graphene quantum dots are all ferromagnetic spin-triplet states. The magnetic properties orignate both from the boundary carbon atom occuping protruding position, and from the carbon atom with unpaired electron. On the whole, the energy gap of other structures decreases with the increase of the number of benzene rings as excepted for the structure of 6b, and the system energy gap decreases significantly by the added charges. In addition, using time-dependent density functional theory (TD-DFT), the excited states of the triangular structure composed of six carbon rings which have an energy gap is 3.83 eV are calculated, and the results show that the 17th excited state has the greatest excitation strength, the excitation energy is 3.93 eV, corresponding to a wavelength of 315.8 nm, close to experimental result.