Signatures with user-defined edge colors

A bug in the CDK implementation of my signature library turned out to be due to the fact that the bond colors were hard coded to just recognise the labels {"-", "=", "#" }. The relevant code section even had an XXX above it!

Poor show, but it's finally fixed now. So that means I can handle user-defined edge colors/labels - consider the complete graph (K5) below:

So the red/blue colors here are simply those of a chessboard imposed on top of the adjacency matrix - shown here on the right. You might expect there to be at least two vertex signature classes here : {0, 2, 4} and {1, 3} where the first class has vertices with two blue and two red edges, and the second has three blue and two red.

Indeed, here's what happens for K4 to K7:

Clearly even-numbered complete graphs have just one vertex class, while odd-numbered ones have two (at least?). There is a similar situation for complete bipartite graphs:

Although I haven't explored any more of these. Next is to put this fixed version into the CDK...

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chalky

I wanted to show something that hints at the things that the new architecture can afford us: This is using a Java2D graphics Paint object to make it look like chalk...kindof. It's a very simplistic way of doing it by making a small image with a random number of white, gray, lightgray, and black pixels. edit: it doesn't look so good at small scales some tweaking of stroke widths and so on is essential.

The Gale-Ryser Theorem

This is a small aside. While reading a paper by Grüner, Laue, and Meringer on generation by homomorphism they mentioned the Gale-Ryser (GR) theorem. As it turns out, this is a nice small theorem closely related to the better known Erdős-Gallai (EG). So, GR says that given two partitions of an integer ( p and q) there exists a (0, 1) matrix A iff p* dominates q such that the row sum vector r(A) = p and the column sum vector c(A) = q . As with most mathematics, that's quite terse and full of terminology like 'dominates' : but it's relatively simple. Here is an example: The partitions p and q are at the top left, they both sum to 10. Next, p is transposed to get p* = [5, 4, 1] and this is compared to q at the bottom left. Since the sum at each point in the sequence is greater (or equal) for p* than q , the former dominates. One possible matrix is at the top left with the row sum vector to the right, and th...

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 ...

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