Skip to main content

Square Grids, Cylinders, Spheres, and Toruses

This is straying from the point; but if any graph can be described by canonized trees made from its subgraphs, then what are the properties of very large (regular) graphs? A grid, for example?

Starting with a square, and fusing squares together results in this situation:


two and three fused squares are similar, at the height-two tree level. With four rings, a new type c appears (b becomes b' with three rings). Beyond this point, any number of rings fused in a row like this has the same structure.

Moving to a second dimension of growth, a square grid (Gnn) has the following structure:


On the left are 'snapshots' of the advancing wave of the tree. Looks a lot like a breadth-first search, I suppose. On the right are the trees for each snapshot, with increasing heights. Although this is not shown, 8 of the 20 leaves of the third tree are duplicates. This should be clear from the third snapshot, which has only 12 (filled) circles on the 'wavefront'.

These trees can even be extended to grids wrapped around three-dimensional objects. Or, to put it another way, grids wrapped up as surfaces:

The diamond shapes with lines radiating out are the expanding trees. To the left of each surface is a very rough sketch of what the spanning tree would be like. The dashed lines indicate cutpoints where the wavefronts meet - these are duplicated on the trees. This is similar to the idea of 'gluing' surfaces in topology.

Comments

Popular posts from this blog

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 …