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On Canonical Numberings

So, after reading* this (2005) paper : "On Canonical Numbering of Carbon Atoms in Fullerenes : C60 Buckminsterfullerene" (link) I made some pictures to illustrate the difference between it and the numbering scheme used for SMILES (as described here). Er, which is used in the CDK.

Anyway, the point is that the scheme used by Plavšić, Vukičević, and Randić (or PVR as I will refer to them, I hope they don't mind!) numbers the atoms in a way that produces an adjacency matrix with a particular property. If you consider the rows of the matrix to be binary numbers, then the set of numbers is the smallest possible. So, for example:

The structure on the left is cubane, with its adjacency matrix on the right. The column on the far right shows the rows of the matrix in base 10. They are clearly in order. Now what happens for the SMILES? Well:
Here, the rows are neither in order (I'm not sure from their paper whether the ordering is an expected outcome for all structures, nor have I checked...) nor is their sum less than for PVR scheme - 765 vs 753.

Of course, the PVR labelling would be useless for generating SMILES for cubane since there is no way to get a path from it. Indeed, the labels are designed to be maximally unfriendly by pairing the highest with the lowest.

Furthermore their scheme goes on to label bonds and rings:
Which also look quite random; or, as they say :
"... we admit that the final labels ... do not appear »orderly« but one has to recognise that there is no »simple« labelling in [fullerenes] that will appear simple" 
which makes sense. Obviously, not for cubane here, but for C60/C70 and so on it does.
*(probably because the name follows the "On X" paper naming scheme)

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