### Orbit Saturation of Degree Sequences

Degree sequences and signatures are related, so it seems reasonable that Faulon's orbit-saturation algorithm for enumerating molecules should also work for just degree sequences. For example, height-1 signatures for a carbon skeleton are practically identical to a degree sequence. See this image for details:

On the left is a target signature, with its corresponding carbon skeleton below. On the right is the equivalent degree sequence, and its simple graph. Of course, signatures can handle multiple bonds, and different elements, but the principle is the same.

So the major problem with Király's method for enumerating graphs from their degree sequences is that it produces isomorphic solutions. The problem I had with Faulon's method was that I couldn't get it to work! A functional implementation for the (much simpler) degree sequence problem should help.

Encouragingly, the initial crude implementation does work for simple examples. The key seems to be to  check that a solution is canonical by permuting within the original orbits, and not the final ones. The example I have been using is the sequence [3, 3, 2, 2, 1, 1] and one of its solutions is instructive here:

These two graphs are isomorphic, but the one on the left is the canonical form according to its adjacency matrix (below each graph). The matrix can be converted to a string by reading left-to-right and top-to-bottom (below each matrix). These strings can be compared to find a maximum one - which is the left hand one in this case as 11100... is greater than 10011... (etc).

To efficiently generate graphs, it is necessary to be able to test these strings without storing all the already generated isomorphs. A simple way to test for a canonical graph is to try all permutations of labels on the vertices; in other words, renumber the graph. Now clearly the only way to renumber the left hand graph here to get the right is to permute labels on vertices 0 and 1.

However, in the final graphs, these are in different orbits! So it may be that it is necessary to use the original orbits - which is just that induced by the degree sequence, or [{0, 1}, {2, 3}, {4, 5}]. Time will tell...

One final problem is that this simple canonical checking procedure is very expensive for large regular graphs. For [3, 3, 3, 3, 3, 3, 3, 3] (38) we get various graphs, including a cube and a cuneane-like graph, but the canonical check has to try 8 factorial (8!) permutations for each successful solution.

Luckily there is a solution - the good old partition refiner :) It seems to work so far.

### 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…

### 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…

### Generating Trees

Tree generation is a well known (and solved!) problem in computer science. On the other hand, it's pretty important for various problems - in my case, making tree-like fusanes. I'll describe here the slightly tortuous route I took to make trees.

Firstly, there is a famous theorem due to Cayley that the number of (labelled) trees on n vertices is nn - 2 which can be proved by using Prüfer sequences. That's all very well, you might well say - but what does all this mean?

Well, it's not all that important, since there is a fundamental problem with this approach : the difference between a labelled tree and an unlabelled tree. There are many more labeled trees than unlabeled :

There is only one unlabeled tree on 3 vertices, but 3 labeled ones
this is easy to check using the two OEIS sequences for this : A000272 (labeled) and A000055 (unlabeled). For n ranging from 3 to 8 we have [3, 16, 125, 1296, 16807, 262144] labeled trees and [1, 2, 3, 6, 11, 23] unlabeled ones. Only 23 …