  Experiments using a mechanically-controlled break junction and calculations
based on density functional theory demonstrate a new magic ratio rule
(MRR),which 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'. Based on
connectivity alone, we predict that when the coupling to electrodes is weak and
the Fermi energy of the electrodes lies close to the centre 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.
