Line Graphs and Double Bonding Systems

After looking at a CDK tool for fixing bond orders for aromatic systems (DeduceBondSystemTool in the smiles package) I wondered if there was a more general approach. That is, the problem is to take a molecular graph with no double bonds and generate all possible double bonded systems.

One possibility might be to first convert the graph (G) into a form known as a line graph (lg(G)) where every vertex in lg(G) is an edge in G. If these vertices are labelled to represent the bond order, then an aromatic system has a particular line graph. For example, here is benzene:

The dashed lines show the construction of the line graph, and the labels '-' and '=' mean single and double. Now obviously, the two resulting graphs are essentially the same, so it would be nice to remove this redundancy. An example of two different bonding systems comes from phenanthrene:

Which is great, but how to generate all non-redundant colorings of the line graphs? Since a line graph is just a graph, it can have a signature, and a signature quotient graph. This can then be colored:

However, in this example it is necessary to 'half-color' some of the vertices of the quotient graph ... which doesn't quite seem to work. The numbers in between the colored quotient graphs show how many line graph vertices are in each of the symmetry classes.

In any case, this is a bit of a toy problem, with only a partial solution, but here is a code repository for a sketch of the code. Note that the algorithm is missing!

Comments

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.

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 ...

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...

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