The theoretical and experimental study of the electrical response of biomolecular systems has been pushed forward intensively during the past years due to the potential of such systems in the field of nanoscale electronics for the development of bio-inspired materials or for sensor applications. The sequencing of DNA during the translocation process through natural or artificial nanopores is one of such examples. Hereby the transversal electronic response of individual bases is supposed to be strong enough for discriminating between the four nucleotides in a long DNA sequence when passing through the nanopore. Graphene due to its transport and atom scale thickness comes as a materials of choice, still phenomena related to the influence of the functionalization of the nanopore surface, a reliable electronic coupling between the bases and the electrodes, the influence of the solvent, of the base orientation and of thermal fluctuations need to be thoroughly understood. We investigate the interaction between DNA bases and a graphene nanopore with zigzag edges along the pore. Our results point to the fact that investigating different functionalization strategies of the nanopore edges is fundamental to enhance a purely electric sequencing via graphene nanopores.