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### Squashing Molecules Flat and Combinatorial Maps

So the traditional way to say it is probably 'planar embedding', but I've gone for the alternate terminology of 'squashing flat'. For many molecules, this is a fairly easy process : some decisions have to be made about rotatable bonds, but quite a few drugs are just things sticking off a benzene ring or two.

For fully 3-dimensional molecular graphs (for example) it is trickier. I don't yet know a simple algorithm for choosing different possible embeddings ... er ... squashings. Perhaps they are all complicated; they seem to involve tree data structures with names like "SPQR Tree" (see "Optimizing over all combinatorial embeddings of a planar graph").

Far easier, though, is the intermediate data structure between a graph and some 2D coordinates - known as a combinatorial map. These mathematical objects store the detail of the embedding by recording the order of 'half bonds' called flags (or darts?) around a vertex. Maybe a flag is the vertex and the half-bond. Anyway, here is a picture of K4 (or a squashed tetrahedrane skeleton):
Now the left hand side just shows the graph with labelled vertices, while the one on the right shows the flags (I'm going to call them that now, sorry). At the bottom is the permutation σ applied to the set of flags (f). To understand σ consider just the central vertex (1) : if we travel clockwise round from flag 1, we get the flags (1, 8, 6). In fact, this is how I constructed the permutation : clockwise round all the vertices. It can be helpful to consider σ in cycle notation as (0, 2, 4)(1, 6, 8)(5, 11, 9)(7, 10, 3) which has a cycle for each vertex.
This image shows two different embeddings of the same graph. Well, I suppose technically it is different embeddings of the same labelling. In any case, L and R are flipped : if there wasn't so much symmetry it might be better. The same procedure as before is used to get σL and σR (clockwise both times). Now check this out:

What is this crazy thing? Well, the other part of the combinatorial map is an 'involution' called α, which is really just another permutation that stores the pair of flags for each edge. In the images above, I have just ordered the edges using the vertices (as 0:1,0:2,0:3,1:2,1:3,2:3) then ordered the flags by vertex index : that is, α = [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10].

Anyway, using the formula φ = σ ⋅ α we get a permutation of what I actually am going to call 'darts'. This is all on the wikipedia page, but it should be clear that the labelled arrows on the image go in cycles. So there is (0, 6, 3) for example. Indeed φ = (0, 6, 3)(1, 4, 9)(2, 10, 5)(7, 8, 11) which is - of course - the four faces of the embedding, including the boundary face.

Phew! Fun stuff, and there is code here as well as some tests here. Well more like just System.out statements, but the beginning of tests...

### 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…