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 (S_G) 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 S_Gshows the former case, by putting two new atoms corresponding to the duplicate visit to the bridging atom in G. For the subgraph S_H 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.

Comments

Adamantane, Diamantane, Twistane

After cubane, the thought occurred to look at other regular hydrocarbons. If only there was some sort of classification of chemicals that I could use look up similar structures. Oh wate, there is . Anyway, adamantane is not as regular as cubane, but it is highly symmetrical, looking like three cyclohexanes fused together. The vertices fall into two different types when colored by signature: The carbons with three carbon neighbours (degree-3, in the simple graph) have signature (a) and the degree-2 carbons have signature (b). Atoms of one type are only connected to atoms of another - the graph is bipartite . Adamantane connects together to form diamondoids (or, rather, this class have adamantane as a repeating subunit). One such is diamantane , which is no longer bipartite when colored by signature: It has three classes of vertex in the simple graph (a and b), as the set with degree-3 has been split in two. The tree for signature (c) is not shown. The graph is still bipartite accordin

Király's Method for Generating All Graphs from a Degree Sequence

After posting about the Hakimi-Havel theorem, I received a nice email suggesting various relevant papers. One of these was by Zoltán Király called " Recognizing Graphic Degree Sequences and Generating All Realizations ". I have now implemented a sketch of the main idea of the paper, which seems to work reasonably well, so I thought I would describe it. See the paper for details, of course. One focus of Király's method is to generate graphs efficiently , by which I mean that it has polynomial delay. In turn, an algorithm with 'polynomial delay' takes a polynomial amount of time between outputs (and to produce the first output). So - roughly - it doesn't take 1s to produce the first graph, 10s for the second, 2s for the third, 300s for the fourth, and so on. Central to the method is the tree that is traversed during the search for graphs that satisfy the input degree sequence. It's a little tricky to draw, but looks something like this: At the top

General Graph Layout : Putting the Parts Together

An essential tool for graph generation is surely the ability to draw graphs. There are, of course, many methods for doing so along with many implementations of them. This post describes one more (or perhaps an existing method - I haven't checked). Firstly, lets divide a graph up into two parts; a) the blocks, also known as ' biconnected components ', and b) trees connecting those blocks. This is illustrated in the following set of examples on 6 vertices: Trees are circled in green, and blocks in red; the vertices in the overlap between two circles are articulation points. Since all trees are planar, a graph need only have planar blocks to be planar overall. The layout then just needs to do a tree layout on the tree bits and some other layout on the embedding of the blocks. One slight wrinkle is shown by the last example in the image above. There are three parts - two blocks and a tree - just like the one to its left, but sharing a single articulation point. I had

Some Stuff

Search This Blog