© 2015 AIP Publishing LLC. The electronic efficiency and binding energy of contacts formed between graphene electrodes and poly-aromatic hydrocarbon (PAH) anchoring groups have been investigated by the non-equilibrium Green's function formalism combined with density functional theory. Our calculations show that PAH molecules always bind in the interior and at the edge of graphene in the AB stacking manner, and that the binding energy increases following the increase of the number of carbon and hydrogen atoms constituting the PAH molecule. When we move to analyzing the electronic transport properties of molecular junctions with a six-carbon alkyne chain as the central molecule, the electronic efficiency of the graphene-PAH contacts is found to depend on the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the corresponding PAH anchoring group, rather than its size. To be specific, the smaller is the HOMO-LUMO gap of the PAH anchoring group, the higher is the electronic efficiency of the graphene-PAH contact. Although the HOMO-LUMO gap of a PAH molecule depends on its specific configuration, PAH molecules with similar atomic structures show a decreasing trend for their HOMO-LUMO gap as the number of fused benzene rings increases. Therefore, graphene-conjugated molecule-graphene junctions with high-binding and high-conducting graphene-PAH contacts can be realized by choosing appropriate PAH anchor groups with a large area and a small HOMO-LUMO gap.