We have measured the single-molecule conductance of 1,n-alkanedithiol molecular bridges (n = 4, 6, 8, 10, 12) on a graphene substrate using scanning tunneling microscopy (STM)-formed electrical junctions. The conductance values of this homologous series ranged from 2.3 nS (n = 12) to 53 nS (n = 4), with a decay constant n of 0.40 per methylene (-CH2) group. This result is explained by a combination of density functional theory (DFT) and Keldysh-Green function calculations. The obtained decay, which is much lower than the one obtained for symmetric gold junctions, is related to the weak coupling at the molecule-graphene interface and the electronic structure of graphene. As a consequence, we show that using graphene nonsymmetric junctions and appropriate anchoring groups may lead to a much-lower decay constant and more-conductive molecular junctions at longer lengths.