TOPICS TO BE COVERED IN THIS SCHOOL

1. A quick tour of Algebraic Graph Theory

By Professor Peter J. Cameron, University of St Andrews

Peter Cameron is a professor of mathematics.

He was born in Australia and educated at the University of Queensland and at Oxford University in the UK, where he took his DPhil degree in 1971 under the supervision of Peter M. Neumann.

He subsequently taught at Oxford, then at Queen Mary University of London, and he is now a half-time professor of Mathematics at the University of St Andrews (Scotland’s oldest university).

His main area of interest is the relationship between groups and the structures they act on, especially combinatorial structures but also countably categorical structures in model theory.

The techniques of algebraic graph theory can be used for more practical purposes too; he has worked on optimal block designs in statistics, and synchronizing automata (which are used in industrial control and in genetics).

Apart from mathematics, Professor Cameron was also a runner, he became the Australian Universities cross country champion in 1968.

Algebraic graph theory has two main parts:

• Linear Algebra, properties of matrices associated with graphs, especially the adjacency and Laplacian matrices, and their eigenvalues, which give information especially about network properties (connectedness, spanning trees, electrical resistance, convergence of random walk, etc.) or bounds for clique number, chromatic number, etc.;
• Group Theory: automorphism group of a graph, graphs constructed from groups, such as Cayley graphs, graphs which give information about groups and other algebraic structures, such as the commuting graph of a group.

There are links between the two areas, beginning with the theorem of Mowshowitz asserting that if all the eigenvalues of the adjacency matrix of a graph are distinct, then the automorphism group of the graph is elementary abelian.

I will take you on a quick tour of some of these areas, with special reference to graphs on groups, an accessible area which has seen a lot of recent developments.

1. Some Aspects of Topological Graph Theory

By Professor Iain Moffatt, Royal Holloway University of London and Professor Eunice Mphako-Banda, University of the Witwatersrand.

Iain Moffatt is a professor of mathematics at the Royal Holloway, University of London in England (UK).

His main research interests lie in the interactions between topology and combinatorics.

He is especially interested in graph polynomials, topological graph theory, matroid theory, and knot theory.

He obtained his PhD in the Quantum Topology and under the supervision of S. Garoufalidis from the University of Warwick in 2005.