Tests that Pass, Tests that Fail

The AMG (alternative molecule generator) is now good enough to run proper tests on, with help from Tobias Kind who has long promised - or threatened, perhaps :) - to test a structure generators. It should lead to software that is of more than theoretical interest.

Currently, there is a download available from github, or it can be built from the project directory if you are familiar with ant and are willing to change the build.properties file to point to a CDK directory. There is an instructions.txt file, with some examples of usages; the -h flag also works as might be expected.

As for passing tests, it currently does better with hydrocarbons - CnH2n + x for x in {-2, 0, 2}. However, it's starting to improve on the more mixed formulae, with oxygen, nitrogen, and so on. The two child-listing methods (filter/symmetric) have different behaviour, annoyingly.

Looking at one of the two pairs of duplicates in the set of C6H4 structures shows why it fails. The method here is the symmetry one, where only the minimal representative of an augmentation under the automorphism group of the parent is chosen. Sadly, this picture shows a case where the method fails:

The parent is highlighted in grey, and the child graphs (A, B) are shown on left and right. The central image shows how both 5a and 5b are adding different sets of bonds. Since the automorphism group of the parent has only the identity permutation (ie: it is trivial) any set of bonds will be equivalent.

I had realised that this could happen, but I foolishly assumed that it was rarer than this. That may be the case for simple graphs, but apparently not for multigraphs like this...

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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

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