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### Graph Layout

Finally, after much effort, the planar embedder is working.

Along the way, a couple of other things have come out of this work : one is an implementation of a fusane generator roughly related to Brinkmann, Caporossi, and Hansen's paper. I say 'roughly' - actually, it is nothing like it! (To be described in a later post.)

I now also understand better algorithms for spanning tree generations, and cycle finding. These will also be described in later posts, if necessary. For now, though : what about those planar graphs, eh? Got any images...

Fusanes! Or, well, graphs that could model fusanes (polyhexes), to be exact. The colors here are vertex equivalence classes, calculated using signatures. These are quite easy examples, however - they are all outerplanar graphs. So what about something more tricky? How about this :

Not very nicely laid out, but looks a lot like the top picture here, I think. One lesson I have learnt is that an embedding (which is a combinatorial object) is not at all like a drawing (which is a geometric object). the embedding only tells you which vertices are part of which face, while the drawing has actual 2D coordinates.

There is an implementation of Plestenjak's spring layout algorithm - but I must have messed up somewhere, as it doesn't work so well. Take a look at this 'before and after' image of fullerene-26 :

It's a bit hard to see at this size, but the one on the left is the 'before' picture, where no refinement of the coordinates has been performed, and the one on the right has been refined. Well 'coarsened' in some places - notably the vertex at the top, which happens to be the central one.

Code for this is here.

### 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…