Peter Cameron's homepage

Welcome to my St Andrews homepage. Under construction This page is under construction (and probably always will be!)

I am a half-time Professor in the School of Mathematics and Statistics at the University of St Andrews, and an Emeritus Professor of Mathematics at Queen Mary, University of London. In addition, I am an associate researcher at CEMAT, University of Lisbon, Portugal.

I am a Fellow of the Royal Society of Edinburgh.

About me

On this site

Me

Elsewhere


Research snapshots

The existential transversal property

The permutation group G has the k-existential transversal property, or k-et for short, if there is a k-subset A of the domain (the witnessing set) such that, for any k-partition P of the domain, there exists gG such that Ag is a transversal to P. With João Araújo and Wolfram Bentz, I determined all the finite permutation groups of degree n with the k-et property for 4 ≤ k ≤ n/2, apart from a few exceptional cases with k = 4 or k = 5. There are applications to regular semigroups. See arXiv 1808.06085.

Sylvester designs

The Sylvester graph is a remarkable distance-transitive graph of valency 5 on 36 vertices, which can be constructed using the even more remarkable outer automorphism of the symmetric group S6, as shown by Sylvester (and described in Chapter 6 of my book with Jack van Lint). Now there is no affine plane of order 6; but, with Rosemary Bailey, Leonard Soicher and Emlyn Williams, I have been investigating block designs for which there is a Sylvester graph on the set of treatments, and the concurrences are 2 for edges of the graph and 1 for all other pairs. They, and designs obtained by deleting resolution classes, are good substitutes for the missing lattice designs. There seem to be many such designs; but just one admits all the symmetries of the Sylvester graph.

Commuting graph, power graph, etc.

There are various graphs defined on the set of elements of a group, such as

For any group, the power graph is a subgraph of the enhanced power graph, which is a subgraph of the commuting graph; and, if the group is non-abelian and 2-generated, the commuting graph is a subgraph of the non-generating graph.

There are many questions about this sequence of graphs: for example, when are consecutive graphs equal (solved, see here for the first two pairs); connectedness, clique number, independence number, chromatic number, and so on (see arXiv 1910.06721). One interesting observation is that a random walk on the commuting graph has limiting distribution which is uniform on conjugacy classes: what about random walks on the other graphs?

Old research snapshots are kept here.


School of Mathematics and Statistics
University of St Andrews
North Haugh
St Andrews, Fife KY16 9SS
SCOTLAND
Tel.: +44 (0)1334 463769
Fax: +44 (0)1334 46 3748
Email: pjc20(at)st-arthurs(dot)ac(dot)uk
  [oops – wrong saint!]





Page revised 31 October 2019

A problem

The commuting graph of a group G is the graph with vertex set G in which two elements are joined whenever they commute. (This specification includes a loop at every vertex.) The uniform random walk on this graph has the property that its limiting distribution is uniform on conjugacy classes.

To say that two elements commute is equivalent to saying that they generate an abelian group. So we could generalise the above graph by putting different specifications on the group generated by the two elements: for example, it is cyclic (this gives the enhanced power graph), or it is the whole group (this gives the generating graph).

Problem: What can be said about the uniform random walk on one of these graphs?

Old problems are kept here.