Formula Debugger

(Please click for bigger - there is quite a bit of detail :)

The left panel shows the search tree of structures; each circle is a structure, the red ones at the tips of some branches are fully connected and saturated. The blue branch has been selected, and is shown in the middle, with the structure at the tip at the top.

This solution is also highlighted (in red - colors are not very consistent) and shown in detail on the right. The right/middle panel is a traditional molecule layout, but with numbers in place of atom symbols. The lower right panel is the spanning tree of the atom highlighted in red in the upper right panel.

Clearly, there are still a lot of unnecessary structures generated since most branches are dead ends. However, this version does at least avoid duplicates - sadly it also misses a few structures :( Clearly refining partitions based on element symbols doesn't totally work...

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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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