Experiments using a mechanically controlled break junction and calculations based on density functional theory demonstrate a new magic ratio rule (MRR) that captures the contribution of connectivity to the electrical conductance of graphene-like aromatic molecules. When one electrode is connected to a site i and the other is connected to a site i' of a particular molecule, we assign the molecule a magic integer M-ii'. Two molecules with the same aromatic core but different pairs of electrode connection sites (i,i' and j,j', respectively) possess different magic integers M-ii' and M-jj'. On the basis of connectivity alone, we predict that when the coupling to electrodes is weak and the Fermi energy of the electrodes lies close to the center of the HOMO-LUMO gap, the ratio of their conductances is equal to (M-ii'/M-jj')(2). The MRR is exact for a tight-binding representation of a molecule and a qualitative guide for real molecules.