### Centrality as a Vertex Invariant (or 'Atom Descriptor')

EDIT: After some more tests, I now realise that this is not really as great a vertex label/descriptor as I thought it was. For example, see these four graphs on 7 vertices that fail to distinguish vertices properly:

The first one should have a central vertex in a different class than the other blue vertices. The green class in the second graph should be split, and same for the third graph. And so on.

So, in the last post I talked about the ideas of Randić et al for calculating the 'centrality' of vertices in a graph. Interestingly, the numbers calculated for each vertex act as a kind of equivalence class label or vertex invariant. This is similar in many ways to Morgan numbers (sorry, Egon's post doesn't actually explain them, but they are the sum of degrees across extended neighbourhoods).

For example, here is one of the examples from the previous post:

With the centrality matrix in the middle, and the 'label' made by sorting the row elements in descending order to the right of that. Finally, these labels are converted to more easily read alphabetic ones - classes in some sense ('a' = {0, 5} and 'b' = {1, 2, 3, 4}). These classes make sense, given that the middle vertices are in the same class, with the rest in another class.

Compare this to the graph with the same ORS of [6, 6, 5, 5, 5, 5]:

The graph here is nearly the same - with only the edge 1:3 missing - yet the labels don't distinguish between vertices {1, 3} and vertices {2, 4}. One possibility mentioned in the paper is to use combinations of descriptors (fairly common practice in cheminformatics, I suspect). The simplest one that occurs to me is just to add the degree of a vertex to the start of the label. That makes the label for {1, 3} into "1320000", distinguishing them from the one for {2, 4} which is "2320000".

Anyway, here is a picture of a number of pairs of graphs with the same ORS, colored by the label just described:

Note how some (but not all!) of these have the 'same' equivalence classes in different arrangements. Be aware that the colors may not be totally meaningful when compared between graphs. 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…

### 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 …