These results are fully reproduced by our present calculations, verifying two almost degenerate pairs of end-states, spreading deeper and deeper into the "bulk" region, as the length increases, while their energies tend to almost coincide. This is a prerequisite for QI. (2), developed by the present author and collaborators as the characteristic conductivity of "bare" AGNRs, without any reference to anchors or leads. In eq. ( ) σ depends only on the geometry and the dipole moment D induced by an external field ε. "Geometry" is described in terms of width w and length L, both in Ångstroms, together with the number of zigzag atoms N=W=2Z+1. For more information see section S1 of supporting information (SI). In SI one can also find a larger plot of conductivity versus L for the 13-AGNRs, from which it can be clearly seen that after the CQI region, i.e. for L ≥80Å, the conductivity variation continues practically following the initial L-dependence before CQI, indicating that CQI is something special (to the 13-AGNRs) and a separate feature, superimposed to the "normal" σ(L) variation. This can be verified by connecting the two ends of the CQI region by a smooth (dashed) line. In this case, as we can see in Fig. , we get a continuous σ(L) curve very much similar to the one for 7-, and 9-AGNRs. In addition, in Fig. (a) we can see the aromaticity patterns (i.e. the distribution of aromatic rings, denoted by a red dot in their centers), before (where the aromaticity pattern is not fully developed), and after the "phase transition" at Lc , where the aromaticity pattern is "completed" (within the full "body" of the AGNR). This is indicative of the direct interrelation of CQI with aromaticity, which is well known from other cases. 5-8 For example Markussen et al. emphasized that QI in aromatic molecules is intimately related to the topology of the molecule's π system. In AGNRs the full topology of the π system (the aromaticity pattern) is established just when the end-states first appear. Moreover, the occurrence of end-states, shown in Fig. (b), is directly interrelated with sublattice symmetry and/or imbalance (both related to connectivity and bipartite topology) which in turn is directly correlated with aromaticity, in close analogy to QI in molecular junctions. As we can see in Fig. just before transition (L≈45Å) is around 0.11 eV (comparable to the observed difference by Chen et al. ), whereas just after transition (at the beginning of the CQI region) ΔΕ0,1 becomes 0.045 eV, dropping by more than one order of magnitude, whereas at the end of the CQI region, at a length about 80 Å, becomes practically zero (ΔΕ0,1 ≈ 0.003 eV), as is illustrated in Fig. . We can clearly see in Fig. that the CQI region starts when the HOMO-1 and HOMO end-state energies are practically equal, similarly to the LUMO and LUMO+1 ones, but not exactly equal, so that the difference ΔΕ0,1 would be smaller than 0.05 eV, but larger than 0.005 eV (0.005 eV <ΔΕ0,1< 0.05 eV), as is verified in Fig. ). We can thus conclude that the relevant energy for CQI is ΔΕ0,1 and that CQI occurs for 0.005 eV <ΔΕ0,1< 0.05 eV. This is consistent with the results for the 15-AGNRs shown in Fig. .