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” Mii′. Two molecules with the same aromatic core but different pairs of electrode connection sites (i,i′ and j,j′, respectively) possess different magic integers Mii′ and Mjj′. 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 (Mii′/Mjj′)^2. The MRR is exact for a tight-binding representation of a molecule and a qualitative guide for real molecules.