Skip to main content

Tutte's Twist Operation on Cubic Graphs

There is an interesting book by W. T. Tutte called 'Graph Theory as I have known it' which is a cross between a normal mathematical text and a biography. So it's a description of the areas he was interested in, and his theorems. One thing that interested me was the use of a 'twist' operation on cubic graphs like so:

Where for the edge between vertices x and y labelled 'A' we reconnect the surrounding edges to form the arrangement on the right hand side. So detach edge D from y and connect it to x, and vice versa with edge C. The lower part of the picture shows what happens for a loop-edge - it transforms to a multi-edge.

This operation is used on a family of 'base' graphs looking like this:

with the first in the list is a vertexless loop graph - that is, it has no vertices and a single edge. From these base graphs, the twist operation can form any cubic graph. Note that all of Un are cubic with 2n vertices.

For example, from U3 we can get to both of the (simple) graphs with 6 vertices by the following sequence:


In this diagram, the twist is being applied to the red edge, then the blue, then the green, etc. The final step converts the prism (G6) to K3,3 (G7) while the other steps involve non-simple graphs with loops and multiple edges.

One of Tutte's uses for these transformations was to show that the number of 1-factors (perfect matching) J of a graph can be calculated by J(G) + J(GA) = J(H) + J(HA) where GA is a graph with the edge A deleted. So, starting with a base graph U - which has J = 1 except for U0 where J = 2. Then use that value to determine the number of 1-factors in the next graph in the sequence, and so on.

It does make me wonder if there is a way to generate cubic graphs from these base examples, by these twists. From a few simple examples it is clear that there would be a lot of redundancy at the leaves of the generated tree, but possibly that could be handled with canonical path augmentation in some way.

Comments

mario bianchi said…
Awesome work! This post has great reference value. It will absolutely help me in my projects. Thanks for sharing such informative content.

Popular posts from this blog

Listing Degree Restricted Trees

Although stack overflow is generally just an endless source of questions on the lines of "HALP plz give CODES!? ... NOT homeWORK!! - don't close :(" occasionally you get more interesting ones. For example this one that asks about degree-restricted trees. Also there's some stuff about vertex labelling, but I think I've slightly missed something there.

In any case, lets look at the simpler problem : listing non-isomorphic trees with max degree 3. It's a nice small example of a general approach that I've been thinking about. The idea is to:
Given N vertices, partition 2(N - 1) into N parts of at most 3 -> D = {d0, d1, ... }For each d_i in D, connect the degrees in all possible ways that make trees.Filter out duplicates within each set generated by some d_i. Hmm. Sure would be nice to have maths formatting on blogger....

Anyway, look at this example for partitioning 12 into 7 parts:

At the top are the partitions, in the middle the trees (colored by degree) …

Common Vertex Matrices of Graphs

There is an interesting set of papers out this year by Milan Randic et al (sorry about the accents - blogger seems to have a problem with accented 'c'...). I've looked at his work before here.

[1] Common vertex matrix: A novel characterization of molecular graphs by counting
[2] On the centrality of vertices of molecular graphs

and one still in publication to do with fullerenes. The central idea here (ho ho) is a graph descriptor a bit like path lengths called 'centrality'. Briefly, it is the count of neighbourhood intersections between pairs of vertices. Roughly this is illustrated here:


For the selected pair of vertices, the common vertices are those at the same distance from each - one at a distance of two and one at a distance of three. The matrix element for this pair will be the sum - 2 - and this is repeated for all pairs in the graph. Naturally, this is symmetric:


At the right of the matrix is the row sum (∑) which can be ordered to provide a graph invarian…

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…