### 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:
1. Given N vertices, partition 2(N - 1) into N parts of at most 3 -> D = {d0, d1, ... }
2. For each d_i in D, connect the degrees in all possible ways that make trees.
3. 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) and at the bottom the desired output of "lattice-trees" (a kind of polyomino, apparently). I should really have a consistent degree color scheme...

Anyway, it's probably not the neatest approach for this particular problem, but I think it would work. Since the number of trees generated from each degree sequence is only a fraction of the space, it seems reasonable to do all-v-all checking for isomorphism in this case.

### 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…