A ball-milling treatment can be employed to exfoliate graphite through interactions with commercially available melamine under solid conditions. This procedure allows the fast production of relatively large quantities of material with a low presence of defects. The milling treatment can be modulated in order to achieve graphene flakes with different sizes. Once prepared, the graphene samples can be redispersed in organic solvents, water, or culture media, forming stable dispersions that can be used for multiple purposes. In the present work, we have screened electron-rich benzene derivatives along with triazine derivatives in their respective ability to exfoliate graphite. The results suggest that the formation of a hydrogen-bonding network is important for the formation of multipoint interactions with the surfaces of graphene, which can be used for the exfoliation of graphite and the stabilization of graphene in different solvents. Aminotriazine systems were found to be the best partners in the preparation and stabilization of graphene layers in different solvents, while the equivalent benzene derivatives did not show comparable exfoliation ability. Computational studies have also been performed to rationalize the experimental results. The results provide also the basis for further work in the preparation of noncovalently modified graphene, where derivatives of aminotriazines can be designed to form extensive hydrogen-bond 2D networks on the graphene surface with the aim of manipulating their electronic and chemical properties.