Blowing Carbon Bubbles : Expanding 2D Fullerene Layouts to 3D

The concentric face layout code is working well enough now to handle the larger fullerenes - such as that old favourite, C60. Since coloring the vertices by equivalence class is not always terribly informative, here is a view of the ring equivalence classes :

Where C60 is on the left, and a more colourful C70-D5h is on the right. One difficulty, however, is to understand the symmetries of these structures when they are distorted like this. The further away from the center of the layout, the more stretched the rings become.

So, an obvious next step was to 'blow up' these 2D layouts into 3D. It turns out that is possible, with a combination of inverse stereographic projection and Jmol's minimize command. The first step is necessary since minimizing the 2D coordinates (with a z-coord of zero) just shrinks the diagram down in the plane. Here are before and after shots of these steps:

Clearly the inverse-projection does not give very good 3D positions for the atoms, but they are on the surface of a sphere, and the bonds don't cross. The minimization gives a much better looking version, although there are still dents and a lot of asymmetry.

Next step is to write out the equivalence class colours (vertex/face?)  into a Jmol script, so that they can be visualised in 3D. Oh, and one last thing - it was necessary to scale the flat diagram down so that the radius was closer to a unit circle. If it was contained inside the circle, then the expansion was only a hemisphere. Also, the points were expanded from the centre outwards by a small factor, to improve the bond distances.

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…

Listing Degree Restricted Trees

Although stack overflow is generally just an endless source of questions on the lines of "HALP plz give CODES!? ... NOT homeWORK!! - don't close :(" occasionally you get more interesting ones. For example this one that asks about degree-restricted trees. Also there's some stuff about vertex labelling, but I think I've slightly missed something there.

In any case, lets look at the simpler problem : listing non-isomorphic trees with max degree 3. It's a nice small example of a general approach that I've been thinking about. The idea is to:
Given N vertices, partition 2(N - 1) into N parts of at most 3 -> D = {d0, d1, ... }For each d_i in D, connect the degrees in all possible ways that make trees.Filter out duplicates within each set generated by some d_i. Hmm. Sure would be nice to have maths formatting on blogger....

Anyway, look at this example for partitioning 12 into 7 parts:

At the top are the partitions, in the middle the trees (colored by degree) …

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…