In this thesis, we investigate the interaction between two impurity adatoms with high magnetic moment which are located on zigzag graphene nanoribbons that consist of 10516 atoms. The magnetic adatoms communicate with other trough the host electrons such as Ruderman-Kittel-Kasuya-Yoshida (RKKY) interactions. Firstly, in order to numerically calculate the two impurity Anderson model, we use quantum Monte Carlo technique. When the impurity adatoms are located far from edges, the results we obtained are consistent whit the bulk graphene results in the literature. On the other hand, the specific location and orientation of adatoms on the sublattices, significantly affects the spin-spin correlations of the two impurities. However, we observe that while the adatoms approach to the edges, significant differences occur due to the edge effect of zigzag graphene nanoribbon. As a results of this, we found that the magnetic correlations can be also enhanced if the adatoms belong to the same sublattice as the edga atoms, since the states of the adatoms hybridize with edge states. Moreover, we show that chaning chemical potential can crucially affect the magnitude of the correlations of the adatoms, and may lead to aphase transitions from ferromagnetic to antiferromagnetic or vice versa. Besides, we observe that when the width of the zigzag graphene nanoribbons is decreased, the spin-spin correlations are affected.On the other hand, we also calculated spin-spin correlations using mean-field approximation for themean-field Anderson model. We found that results significantly differ from quantum Monte Carlo results. In addition, when the electron-electron interations of he host atoms are taken into account, crucial differences are obtained at the impurity correlations.