Faulon's Signatures : A Possible Interpretation

Several recent papers by Faulon concern an idea he calls 'signatures'. This post is just a record of what I understood them to be.

Firstly, a signature is a subgraph of a molecular graph. There is a distinction between atomic signatures - which is a tree rooted at a particular atom - and a molecular signature, which is the set of atomic signatures for each atom in a molecule.

A tree is a graph with no cycles, so an atomic signature is not just a subgraph. Like a path, a signature has a length - or rather a height. Here is a picture of signatures of heights 1-4 for a fused ring structure:

The graph G on the left has one of its atoms labelled (a), and each of the trees in the center is a signature rooted at that atom. On the right, is the simple string form of the tree, as a nested list. I should point out that the signatures in these images may not be canonical, as I worked them out by hand (as I have not yet fully implemented the canonization algorithm).

Signatures of the same height may be different for the atoms in a molecule. At a height of zero, it is simply the atoms. A signature of height one is each atom, plus its neighbours. For G, above, there are two distinct height-1 signatures. For greater heights in G, there are more:

These are three subgraphs (SG) of G, rooted at three different atoms (a, b, c). Each one corresponds to a signature tree, which also correspond to different signature strings (not shown). The trees have been given square nodes, instead of circular ones, just to make them look different. From the symmetry of G, it may be clear that the other atoms also have one of these same height-2 signatures.

Finally, there are some odd properties of the trees created from the subgraphs, that become noticable in height-3 signatures of G. As mentioned above, a tree cannot have cycles, so when the paths radiating out from the root atom meet on the same atom, it will appear in the tree twice. Further, when paths cross the same bond - at the same time - both atoms in the bond will appear in both orders across two layers:

The subgraph SG shows the former case, by putting two new atoms corresponding to the duplicate visit to the bridging atom in G. For the subgraph SH of the pentagon H the whole of the last bond visited is duplicated, and the signature tree has a pair of duplicate bonds at the leaves. The tree construction process forbids duplication of bonds except in these two ways.