## Past Honours students, project students and coursework Masters students

- Michela Castagnone (2017) is working on an Honours project on methods for uniformly sampling graphs with given degrees, especially algorithms based on the switching method and rejection sampling.
- Dean Garden (2016 - 2017) is working on a coursework Masters project on enumeration of sparse graphs by degree sequence.
- Barton Lee (2015) wrote an Honours thesis on
*The theory and application of graph limits*. - Daniel Altman (2015) wrote an undergraduate project entitled
*The distribution of loose Hamilton cycles in random uniform regular hypergraphs*. -
Matteo Sfragara (2015) visited UNSW as a Practicum student and worked with me on
*The switch Markov chain for sampling irregular directed graphs*. -
Matthew Kwan (2014) wrote an Honours thesis on
*Stein's method*. Matthew is now a PhD student at ETH Zürich, working with Benny Sudakov. -
Arwa Al Munajam (2013) wrote a coursework Masters project report on
*Trees and their applications*. - Matthew Kwan (2013) wrote an undergraduate project entitled
*On the number of spanning trees in random regular graphs*. -
David Wind (2012), an exchange student from Denmark,
wrote an undergraduate project entitled
*On the distribution of spanning trees in random cubic graphs*. -
Corey Lewis (2011) wrote an Honours thesis on
*Analysis of the web graph*. -
Cris Koch (2006) wrote an Honours thesis on
*Term rewriting approach to polycyclic group collection*. -
Byron Ng (2005/2006) did his 4th year CSE project on
*Computer simulation for bounding the mixing time of Markov chains*. -
Le Anh Vinh (2005) did an Honours-level project on
*Ramsey theory*. -
Ka-shu Wong (2004) wrote an Honours thesis on
*The complexity of counting*. -
Michael Leeming (2004) did his 4th year CSE project on
*Analyzing a Markov chain for 7-colourings of 4-regular graphs*.

## Topics for Honours projects

Here is an outline of a couple of possible topics for Honours projects which I could supervise. In each case a fairly broad, survey-style project has been described. But this is just a jumping-off point - a student may find, over the course of the Honours year, that they have a particular interest in one part of the project and choose to focus on that.

(If you're worried that you don't have enough background in graph theory or in probability, then stop worrying. You can pick it up as you go along. That's the beauty of combinatorics, and for the most part the probability used is fairly basic.)

Of course if you have your own idea for a topic, feel free to run it by me.

You should also have a look at Honours topics in a different area of combinatorics, offered by Diana Combe. Diana is a member of the Department of Statistics who performs research in combinatorics and algebra.

### Combinatorial Markov chains

A Markov chain is a simple stochastic process which is "memoryless", in that the next transition depends only on the current state, and not on the history of the chain. We consider discrete time Markov chains on finite state spaces which have exponentially large size (with respect to some parameter). If the Markov chain is ergodic then it converges to a unique stationary distribution. But how quickly does this convergence occur?

There are a few methods for bounding this convergence rate, either to show that the chain "mixes rapidly" and hence can be used for efficient sampling, or that the chain "mixes torpidly" and converges exponentially slowly to the stationary distribution. The aim of the project is to survey the topic, learning about some of the results which have been proved and the methods used to prove them.

A couple of references:

**[1]** Mark Jerrum,
Counting, sampling and integrating: algorithms and complexity,
*Lectures in Mathematics - ETH Zürich*, Birkhaüuser, Basel, 2003.

**[2]**
Dana Randall,
Rapidly mixing Markov chains with applications in computer science and physics,
*Computing in Science & Engineering* **8** (2006), 30 - 41.

### Asymptotic enumeration of combinatorial structures

Given two vectors of positive integers with the same sum, how many different zero-one matrices are there with row sums given by the first vector and column sums given by the second vector? (Equivalently, how many bipartite graphs exist with degrees on the left given by the first vector and degrees on the right given by the second vector?) This is a fundamental question with many applications. Exact expressions seem unlikely to exist, so we look for highly accurate asymptotic expressions: that is, a formula which becomes more and more accurate as the row and column sums tend to infinity.

Different methods are used for asymptotic enumeration depending on whether the matrices are sparse (small matrix sum) or dense (fairly large matrix sum). In the sparse case, combinatorial arguments such as the switching method are employed. Perhaps surprisingly, in the dense case the problem is solved using complex analysis, by calculating a contour integral in many dimensions. The aim of the project is to survey the topic, learning about some of the results which have been proved and the methods used to prove them.

References for the sparse case (combinatorial arguments):

**[1]** B.D. McKay, Asymptotics for 0-1 matrices with prescribed
line sums, in *Enumeration and Design* (Academic Press, 1984),
225 - 238.
Available from
http://cs.anu.edu.au/~bdm/publications.html

**[2]** C. Greenhill and B.D. McKay,
Asymptotic enumeration of sparse nonnegative integer matrices with specified
row and column sums, *Advances in Applied Mathematics* **41** (2008),
459 - 481.
Available from
http://web.maths.unsw.edu.au/~csg/publications.html

References for the dense case (multidimensional complex integration):

**[3]** B.D. McKay and N.C. Wormald, Asymptotic enumeration by
degree sequence of graphs of high degree,
*European Journal of Combinatorics* **11** (1990), 565 - 580.
Available from
http://cs.anu.edu.au/~bdm/publications.html

** [4]**
E.R. Canfield, Z. Gao, C. Greenhill, B.D. McKay and R.W. Robinson,
Asymptotic enumeration of correlation-immune boolean functions,
*Cryptography and Communications* **2** (2010), 111-126.
Available from
http://web.maths.unsw.edu.au/~csg/publications.html

### Random graphs and random graph processes

The first random graph model was introduced by Erdős
in 1947. The idea is as follows:
to produce a random graph on *n* vertices, flip a fair
coin (one that comes up heads with probability exactly 1/2)
independently for each of the *n(n-1)/2* potential edges.
If the coin comes up
heads, then the edge is chosen for the graph, and otherwise
the (potential) edge is rejected. A generalisation of this
is the binomial model *G(n,p)*. This time we use a coin which
comes up heads with probability exactly *p*,
for a given value of *p* between 0 and 1 (which may depend on
*n*). This model has been extremely well studied.

Other models are for random graphs with a fixed number of edges, or random regular graphs where every vertex has the same number of neighbours. The most recent models to emerge are the so-called Web graphs, which aim to model the growth and structure of the World Wide Web. A related area is that of random graph processes, where a discrete-time stochastic process is used to "grow" a random graph, typically by adding an edge (or set of edges) at each step, according to some rule.

In all cases, we are interested in asymptotic behaviour,
which gives an idea of how these graphs behave as the number of
vertices *n* tends to infinity. The properties which
are studied include the connectivity, independence number,
chromatic number and diameter, to name a few.
The methods used are
a mixture of combinatorial and probabilistic. The aim
of the project is to survey the topic, learning about some
of the results which have been proved and the methods used to prove them.

A couple of references:

**[1]** Svante Janson, Tomasz Luczak, Andrzej Rucinski,
*Random Graphs*, Wiley, New York, 2000.

**[2]**
Bela Bollobás, *Random Graphs (2nd edn.)*, Cambridge
University Press, Cambridge, 2001.

**[3]**
Nicholas C. Wormald, *Models of random regular graphs*, in
*Surveys in Combinatorics 1999*
(J. D. Lamb and D A. Preece, eds.), London Mathematical Society Lecture
Note Series vol. 267, Cambridge University Press, Cambridge, 1999, pp. 239 - 298.

### The differential equations method for analysing combinatorial algorithms

The differential equations method is a way of analysing a discrete-time
combinatorial process. For instance, consider the following graph process
which is used to form graphs with maximum degree *d*.
The process
starts at time 0 with *n* vertices and no edges. At each time
step a pair of non-neighbouring vertices which both have current degree
less than *d* is chosen uniformly at random,
and this new edge is added to the graph. When analysing this process
it is very useful to know the number *Y*
of vertices of degree *d* at each step. (So *Y* is a non-negative
integer valued random variable.)
By finding the equation for the expected value
of the change of *Y* over one time step, and treating this
as the derivative of a related continuous function, we can approximate
*Y* by the solution of a differential equation.
(Similarly, a system of random variables can be approximated by the
solution to a system of differential equations.)

Wormald (1995, 1999) has presented theorems which give conditions under which this approximation is guaranteed to be close. (Often there is no need to actually solve the differential equations, or else it is sufficient to solve them numerically.) The method has been applied to algorithms for random graphs and random SAT instances.

The aim of the project is to survey this area, learning about the method and how it has been applied. The history of the differential equations method (which goes back to Kurtz in 1970) could also be looked into. The focus is not on calculus, but on the interplay between discrete and continuous mathematics.

A couple of references:

**[1]** Nicholas C. Wormald,
The differential equation method for random graph processes and greedy
algorithms, in
*Lectures on Approximation and Randomization Algorithms*
(M. Karonski and H.-J. Prömel, eds.), PWN, Warsaw, 1999.

**[2]** Dimitris Achlioptas,
Lower bounds for random 3-SAT via differential equations,
*Theoretical Computer Science* **265** (2001), 159 - 185.