### Signature bond compatibility

So, given my previous posts on what Faulon's signatures are, here is an explanation of how they are used in the structure enumeration algorithm that I am almost finished implementing.

The core test in this algorithm is for compatible bonds. Two atoms are only joined if : a) they have compatible target signatures and b) there are less than the target number of bonds already. A target signature here is just a signature that is set on the atom for it to match, like a pattern.

The first of these tests is illustrated here:

Another (overly) complex diagram! But the formula here is a bit difficult to interpret otherwise. In the top left corner is a graph G (slightly resembling hexane without the hydrogens) which is, by convention, composed of vertices (V) and edges (E).

The equation to the right of the graph defines part of the condition for a compatible bond. The tau terms are just target signatures, as shown on the upper right. The tricky term is h-1ÏƒÏ„(y)(z) which means 'a signature starting from the neighbour z of y in the subgraph defined by Ï„(y)'. This requires using a target signature (Ï„(y)) as if it was a subgraph - shown on the bottom left for the target b and the neighbour n.

The same process is repeated on the bottom right of the figure for b and m - which matches the height - 1 target signature for c. This should make sense, since the atoms labelled with b in the graph are attached to both a and c - so the signature for b must be compatible with both. It is easy to check that a and c are not compatible, and cannot therefore be bonded.

### 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…