Graphene got dramatic attention and lead the two-dimensional material physics after its first successful synthesis in 2004. Its unique electronic properties contain great potential for both scientific and technological applications. RKKY (Ruderman-Kittel-Kasuya Yosida) is an indirect exchange interaction mediated by conduction electrons. In graphene, the interaction strength decay as 1/R³ where R is the distance between the magnetic moments. In the first part of this work, we calculated that applying circular potential on a graphene sheet forms quasi-bound states in the potential region. Via these states, the RKKY interaction is enhanced between magnetic moments on the edge of the potential well. This can be thought of an electronic analog of the Purcell effect. We showed that the interaction strength is even more enhanced if the Fermi level is in resonance with the energies of the quasi-bound states. In the second part, we considered zigzag edged hexagonal nanoflakes. It is known that zigzag edged flakes have zero-energy edge-states. It is also known that the states with closer energies contribute more to RKKY interaction. Thus, we calculated that there is an enhancement between these edge-states. In the third part, we investigated the behavior of RKKY interaction for two dimensional materials with quartic dispersion. An energy dispersion is said to be quartic if it is of the form E = α(k² - kc² )². Here, α and kc are material dependent constants. There are many materials exhibiting the quartic dispersion such as nitrogene, phosphorene, and arsenene. These materials are also sharing two-dimensional hexagonal lattice structure with graphene. What makes quartic dispersion special is that it has van-Hove singularity in its density of states near the band-edge. RKKY interaction is sensitive to the density of states because it depends on the number of electrons contributing spin exchange. Thus, the larger the number of electrons, the stronger the coupling. In this part, we tuned the Fermi level so that it lies on the DOS singularity and then we calculated the interaction strength as a function of R. We found a slowly decaying RKKY interaction for quartic dispersion. If the energy dispersion is pure quartic (i.e. E = ak4), we found the interaction strength depends on 1/(kf R) instead of 1/R which makes the RKKY interaction long range for arbitrarily small Fermi level.