The day started badly. I woke at 3am to a sick and miserable one-year-old, I got stuck in a traffic jam on the way to work, and by lunchtime I was feeling down about the usual whackacademic innuendo. But at 3pm, Eric Swartz knocked on my door. Eric has stunningly proved an outstanding conjecture on generalised quadrangles which I’ll report on when I have his blessing.

Rather than let this blog die off due to lack of posts, I thought I’d write about a small but annoying problem that I’ve been thinking about recently, but without success, and a natural conjecture that has arisen from it.

[Note: On some browsers there appears to be a sizing problem with the images WordPress uses for LaTeX formulas, and they appear ten times the right size. If this happens, reloading the webpage seems to fix it]

The problem arose in the context of binary matroids, but because a binary matroid is really just a set of points in a binary vector space, it can be phrased entirely as a linear algebra problem.

So let ${M}$ be a set of non-zero vectors in the vector space ${V = GF(2)^r}$ such that ${M}$ spans ${V}$ and, for reasons to be clarified later, no vector in ${M}$ is independent of the others. In matroid terminology, this just says that ${M}$ is a simple binary matroid of rank ${r}$ with no coloops.

Then define a  basis of ${M}$ to be a linearly independent subset of ${M}$ of rank ${r}$ (in other words, just a basis of ${V}$ and a circuit of ${M}$ to be a minimally dependent set of vectors, i.e. a set of vectors that is linearly dependent but any proper subset of which is linearly independent.

As an example, take ${M = PG(2,2)}$ (a.k.a the Fano plane) — this means to take all the non-zero vectors in ${GF(2)^3}$. This has 28 bases (7 choices for a first vector ${v}$, 6 for a second vector ${w}$ and then 4 for the third vector which cannot be ${v}$, ${w}$ or ${v+w}$ , then all divided by 6 because this counts each basis ${6}$ times). It has 7 circuits of size 3 (each being of the form ${v}$, ${w}$, ${v+w}$) and 7 circuits of size 4, being the complements of the circuits of size 3, for a total of 14 circuits.

Letting ${b(M)}$, ${c(M)}$ denote the numbers of bases and circuits of ${M}$ respectively, the question is about the ratio of these two numbers. More precisely,

Determine a lower bound, in terms of the rank ${r}$, for the ratio ${\frac{b(M)}{c(M)}}$?

Harald Helfgott, a friend of SymOmega, has only yesterday uploaded an article on the arxiv claiming a proof of the “Ternary Goldbach Conjecture”:

Every odd integer $n>5$ is the sum of three primes.

This is big news in number theory since it is one of the best results on the Goldbach Conjecture to date. The “Ternary GC” also had its origins in the letters of Goldbach and Euler in 1742, and is sometimes referred to as Goldbach’s weak conjecture, since if every even number greater than 4 is the sum of two primes, then by adding the number 3 to the two primes will result in every odd number greater than 5 being the sum of three primes.

I’ve just posted a paper entitled “The Merino-Welsh Conjecture holds for Series-Parallel Graphs” to the arxiv which proves precisely the assertion given in the paper’s title.  This paper resulted from Steve Noble’s visit here last year, where we worked quite hard on this conjecture, and in fact I blogged about the problem then. The conjecture is that the number of spanning trees of a graph is dominated either by the number of acyclic orientations or by the number of totally cyclic orientations.

Previous results on the Merino-Welsh conjecture had proved the result for either very sparse graphs (where the number of spanning trees is dominated by the number of acyclic orientations) or very dense graphs (where the number of spanning trees is dominated by the number of totally cyclic orientations) regardless of the graph’s structure. However for intermediate-sized graphs — in particular those with $m = 2(n-1)$ — the graph structure counts, and there is a subtle interplay between the numbers of the two types of orientations.

Series-parallel graphs are always a great test case, because there is a natural decomposition of a series-parallel graph into smaller series-parallel graphs that provides an ideal inductive framework for a vast range of applications. For us, it provides enough structure to be able to keep track of the numbers of spanning trees and both types of orientation. Not entirely precisely, but enough to identify certain series-parallel graphs that cannot form part of a minimal counterexample to the conjecture. Working systematically, it was possible to ultimately exhaust all the possibilities for graphs that might be part of a minimal counterexample, thereby showing that this hypothetical counterexample cannot actually exist.

On one hand, I rather like this paper because the approach to coping with the interaction between the parameters seems novel and interesting, but on the other hand it relies heavily on the very constrained construction steps allowed in building series-parallel graphs and so will be almost impossible to generalise to larger classes!

A position here in the School of Maths and Stats at UWA has just been advertised.  It is a full-time continuing position as either an Assistant or Associate Professor (Level B/C)  and the school is looking for ` a generalist mathematician with excellent skills in teaching and learning to develop and deliver a range of first year units to students from across the University.’  More details are available at the link above and queries should be directed to the Head of School, Andrew Bassom.

Over the summer break, John and I each supervised a 2nd year undergraduate student in a research project, and the previous post summarised Michael Martis’ project. I supervised Melissa Lee, who learned about the energy of graphs and this is what she did

Hello everyone! My name is Melissa Lee and I’m about to start my third year of Chemical Engineering and Pure Mathematics at UWA. Over these summer holidays, I’ve been given the opportunity to take on a six week research project with Professor Gordon Royle. The area that I have researched is the energy of graphs, with a particular focus on extremal cases. It’s been a new and exciting time for me, being my first experience of research and also my first taste of graph theory.

The idea of the energy of a graph was first introduced by Serbian mathematician Ivan Gutman at a conference in Austria in 1978. Its origins lie in chemistry, where it is defined as “the total $\pi$ electron energy of a conjugated hydrocarbon as calculated with the Hückel Molecular Orbital method”. The concept didn’t receive much attention from mathematicians for many years. However, sometime around the turn of the century, mathematicians realised its value and since then, increasing numbers of papers have been published each year about the energy of graphs.

Over the summer break, Gordon and I each supervised a 2nd year undergraduate student in a research project. At UWA, the first course in group theory is in 3rd year, and we do not teach any combinatorics, so we needed to give each student a crash course before they could sink their teeth into a research problem. One of their outcomes was a blog post, and the first of these is by my student, who was also supervised by Sylvia Morris. His project was on “Sets of type $(m,n)$ in projective spaces”.