Peter Cameron's homepage

Welcome to my new homepage on GitHub. This page is under construction (and probably always will be!)

I am an Emeritus 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 My official page My diary zbMATH profile, MathSciNet profile, Google Scholar ORCID record: orcid.org/0000-0003-3130-9505 Mathematical genealogy, Wikipedia page CV and publications (PDF) Biographical stuff, travel diaries On this site Talks, publications, conjectures, coauthors, students Books: Introduction to Algebra, Combinatorics, Notes on Counting, Permutation Groups, Sets, Logic and Categories, ADE: Patterns in Mathematics, The Shrikhande graph (some dead links to be sorted out) Problems from my QMUL homepage

Elsewhere My homepage at QMUL Papers on arXiv Algebra and Combinatorics at St Andrews St Andrews pure mathematics colloquia CIRCA ORCiD Lecture notes Australasian Journal of Combinatorics Cameron Counts Mathematical quotes The Infinite Quest, a short course from the IAI Academy Mathematics: the next generation (LMS–Gresham lecture)

Research snapshots

Recognising the commuting graph

There are many graphs defined on groups which have been studied recently, including the power graph, generating graph, and independence graph. The questions posed here for the commuting graph could be repeated for any of the others.

The commuting graph of G has vertex set G, with two vertices adjacent if and only if they commute. Given a graph, how hard is it to decide if it is the commuting graph of some group (and to construct the group if the answer is yes)?

This is a slightly strange question since a finite group can be defined with polylogarithmically many bits but a graph takes polynomially many bits to describe.

In a recent paper with V. Arvind, X. Ma and N. Maslova, which can be found here, we showed that there is a quasi-polynomial time algorithm for this. The proof involved finding a short description of a finite group which could be translated quickly into the multiplication table and vice versa.

We also conjectured that, for certain nice classes of graphs such as cographs and chordal graphs, the computation might be possible in polynomial time.

Old research snapshots are kept here.

I am Honorary Editor-In-Chief of the Australasian Journal of Combinatorics, an international open-access journal published by the Combinatorial Mathematics Society of Australasia.

A problem

Consider the following three graphs defined on a finite group G:

• The power graph: x and y are joined if one is a power of the other.
• The enhanced power graph: x and y are joined if they are each powers of some element w.
• The intersection power graph: x and y are joined if some non-identity element z is a power of each of them (and, by convention, the identity is joined to all other elements).

It is known that the power graph and enhanced power graph are equal if and only if every element of G has prime power order. After preliminary work by Higman and Suzuki, such groups were all determined by Brandl.

Problem: For which groups are the power graph and the intersection power graph equal?

It is known that groups in which all elements have prime order have this property, and that it implies that the power graph is a cograph. But an exact characterisation is unknown.

What we know about this question is in my paper with Sudip Bera, arXiv 2509.03919.

Old poblems are kept here.