### Open Molecule Generator's Algorithm

With the recent release of the Open Molecule Generator (OMG) I thought it would be nice to add to (or augment) the description of the algorithm in the paper. The description here will be in terms of graphs, but the principle is largely the same.

OMG's algorithm is a variant of McKay's canonical path augmentation algorithm, as mentioned before here. However, instead of augmenting by vertex it augments by edges. To illustrate this, consider a diagram for graphs with up to 4 vertices:
The graphs are grouped by vertex count (boxes), and each box is sorted into columns. Along the top are the number of edges for graphs in that column, and along the bottom are number of vertices for graphs in the box. Two graphs are connected by an arrow if the larger can be made from the smaller by adding a single edge.

A couple of important things to note about edge-addition are : 1) two paths can lead to the same graph, and 2) at least one of the graphs is disconnected. The first of these is solved by canonical augmentation; a bond is only added if it is the inverse of a canonical deletion. For example:

The image above shows two parent graphs on the left, and an extension of each on their right. The extensions are isomorphic, and have the same canonical form on the far right. The canonical deletion edge is 2:3 - which maps to 0:3 in both graphs. Only one extension is equivalent to the canonical deletion edge.

The second problem - that of disconnected graphs - is only a problem if the canonicalisation routine cannot handle such graphs. Unfortunately, this currently applies to signatures; luckily OMG uses nauty for this part, which doesn't suffer from this limitation. An example of extending a disconnected graph is shown below:

Clearly connected graphs can be constructed from disconnected ones, although it is possible that a canonicalisation method could be designed that avoided disconnected parents. In other words, one that only chose edges that didn't separate the graph.

One final important point is that the OMG algorithm cannot proceed by extending every member of the set of graphs G(n) on n vertices to get non-redundant G(n + 1). For example, extending from both of the members of G(3) will produce multiple copies of one of G(4) - at least, in my test code. This might rule out one easy way to run the program in parallel (running separate subtrees in different processes), but possibly could be solved by using a subset of G(n) to generate the next level.

### How many isomers of C4H11N are there?

One of the most popular queries that lands people at this blog is about the isomers of C4H11N - which I suspect may be some kind of organic chemistry question on student homework. In any case, this post will describe how to find all members of a small space like this by hand rather than using software.

Firstly, lets connect all the hydrogens to the heavy atoms (C and N, in this case). For example:

Now eleven hydrogens can be distributed among these five heavy atoms in various ways. In fact this is the problem of partitioning a number into a list of other numbers which I've talked about before. These partitions and (possible) fragment lists are shown here:

One thing to notice is that all partitions have to have 5 parts - even if one of those parts is 0. That's not strictly a partition anymore, but never mind. The other important point is that some of the partitions lead to multiple fragment lists - [3, 3, 2, 2, 1] could have a CH+NH2 or an NH+CH2.

The final step is to connect u…

### 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) matrixA iff p*dominatesq 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 the column sum vector below.

Finally, the matrix can be interpreted as a bi…

### Havel-Hakimi Algorithm for Generating Graphs from Degree Sequences

A degree sequence is an ordered list of degrees for the vertices of a graph. For example, here are some graphs and their degree sequences:

Clearly, each graph has only one degree sequence, but the reverse is not true - one degree sequence can correspond to many graphs. Finally, an ordered sequence of numbers (d1 >= d2 >= ... >= dn > 0) may not be the degree sequence of a graph - in other words, it is not graphical.

The Havel-Hakimi (HH) theorem gives us a way to test a degree sequence to see if it is graphical or not. As a side-effect, a graph is produced that realises the sequence. Note that it only produces one graph, not all of them. It proceeds by attaching the first vertex of highest degree to the next set of high-degree vertices. If there are none left to attach to, it has either used up all the sequence to produce a graph, or the sequence was not graphical.

The image above shows the HH algorithm at work on the sequence [3, 3, 2, 2, 1, 1]. Unfortunately, this produce…