Square Grids, Cylinders, Spheres, and Toruses

This is straying from the point; but if any graph can be described by canonized trees made from its subgraphs, then what are the properties of very large (regular) graphs? A grid, for example?

Starting with a square, and fusing squares together results in this situation:

two and three fused squares are similar, at the height-two tree level. With four rings, a new type c appears (b becomes b' with three rings). Beyond this point, any number of rings fused in a row like this has the same structure.

Moving to a second dimension of growth, a square grid (G_nn) has the following structure:

On the left are 'snapshots' of the advancing wave of the tree. Looks a lot like a breadth-first search, I suppose. On the right are the trees for each snapshot, with increasing heights. Although this is not shown, 8 of the 20 leaves of the third tree are duplicates. This should be clear from the third snapshot, which has only 12 (filled) circles on the 'wavefront'.

These trees can even be extended to grids wrapped around three-dimensional objects. Or, to put it another way, grids wrapped up as surfaces:

The diamond shapes with lines radiating out are the expanding trees. To the left of each surface is a very rough sketch of what the spanning tree would be like. The dashed lines indicate cutpoints where the wavefronts meet - these are duplicated on the trees. This is similar to the idea of 'gluing' surfaces in topology.

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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

1,2-dichlorocyclopropane and a spiran

As I am reading a book called "Symmetry in Chemistry" (H. H. Jaffé and M. Orchin) I thought I would try out a couple of examples that they use. One is 1,2-dichlorocylopropane : which is, apparently, dissymmetric because it has a symmetry element (a C2 axis) but is optically active. Incidentally, wedges can look horrible in small structures - this is why: The box around the hydrogen is shaded in grey, to show the effect of overlap. A possible fix might be to shorten the wedge, but sadly this would require working out the bounds of the text when calculating the wedge, which has to be done at render time. Oh well. Another interesting example is this 'spiran', which I can't find on ChEBI or ChemSpider: Image again courtesy of JChempaint . I guess the problem marker (the red line) on the N suggests that it is not a real compound? In any case, some simple code to determine potential chiral centres (using signatures) finds 2 in the cyclopropane structure, and 4 in the

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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