Skip to main content

On Canonical Numberings

So, after reading* this (2005) paper : "On Canonical Numbering of Carbon Atoms in Fullerenes : C60 Buckminsterfullerene" (link) I made some pictures to illustrate the difference between it and the numbering scheme used for SMILES (as described here). Er, which is used in the CDK.

Anyway, the point is that the scheme used by Plavšić, Vukičević, and Randić (or PVR as I will refer to them, I hope they don't mind!) numbers the atoms in a way that produces an adjacency matrix with a particular property. If you consider the rows of the matrix to be binary numbers, then the set of numbers is the smallest possible. So, for example:

The structure on the left is cubane, with its adjacency matrix on the right. The column on the far right shows the rows of the matrix in base 10. They are clearly in order. Now what happens for the SMILES? Well:
Here, the rows are neither in order (I'm not sure from their paper whether the ordering is an expected outcome for all structures, nor have I checked...) nor is their sum less than for PVR scheme - 765 vs 753.

Of course, the PVR labelling would be useless for generating SMILES for cubane since there is no way to get a path from it. Indeed, the labels are designed to be maximally unfriendly by pairing the highest with the lowest.

Furthermore their scheme goes on to label bonds and rings:
Which also look quite random; or, as they say :
"... we admit that the final labels ... do not appear »orderly« but one has to recognise that there is no »simple« labelling in [fullerenes] that will appear simple" 
which makes sense. Obviously, not for cubane here, but for C60/C70 and so on it does.
*(probably because the name follows the "On X" paper naming scheme)

Comments

Popular posts from this blog

Generating Dungeons With BSP Trees or Sliceable Rectangles

So, I admit that the original reason for looking at sliceable rectangles was because of this gaming stackoverflow question about generating dungeon maps. The approach described there uses something called a binary split partition tree (BSP Tree) that's usually used in the context of 3D - notably in the rendering engine of the game Doom. Here is a BSP tree, as an example:



In the image, we have a sliced rectangle on the left, with the final rectangles labelled with letters (A-E) and the slices with numbers (1-4). The corresponding tree is on the right, with the slices as internal nodes labelled with 'h' for horizontal and 'v' for vertical. Naturally, only the leaves correspond to rectangles, and each internal node has two children - it's a binary tree.

So what is the connection between such trees and the sliceable dual graphs? Well, the rectangles are related in exactly the expected way:


Here, the same BSP tree is on the left (without some labels), and the slicea…

Common Vertex Matrices of Graphs

There is an interesting set of papers out this year by Milan Randic et al (sorry about the accents - blogger seems to have a problem with accented 'c'...). I've looked at his work before here.

[1] Common vertex matrix: A novel characterization of molecular graphs by counting
[2] On the centrality of vertices of molecular graphs

and one still in publication to do with fullerenes. The central idea here (ho ho) is a graph descriptor a bit like path lengths called 'centrality'. Briefly, it is the count of neighbourhood intersections between pairs of vertices. Roughly this is illustrated here:


For the selected pair of vertices, the common vertices are those at the same distance from each - one at a distance of two and one at a distance of three. The matrix element for this pair will be the sum - 2 - and this is repeated for all pairs in the graph. Naturally, this is symmetric:


At the right of the matrix is the row sum (∑) which can be ordered to provide a graph invarian…

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…