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.

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

Recovering directed from undirected graphs

There are several examples of directed graphs defined on groups from the group structure. In each case one obtains an undirected graph by ignoring directions (and suppressing double edges), and can ask whether the undirected graph determines the directed graph up to isomorphism.

In the case of the power graph, the digraph has an arc x→y if y is a power of x. I showed that the answer to the question is "yes"; Daniela Bubboloni and Nicolas Pinzauti gave an algorithm; and further results are due to Samir Zahirović and coauthors.

Things are different for the endomorphism graph, where the digraph has an arc x→y if some endomorphism carries x to y. There are many negative examples. Perhaps the easiest are the two groups of order p³ and exponent p, for p an odd prime (one elementary abelian, the other non-abelian). The undirected graphs are both complete but the directed graphs are quite different. See arXiv 2511.15602.

The third case is the Engel graph, where the digraph has an arc x→y if [y,x,…,x]=1 (where the iterated commutator has any number of entries x). To my knowledge, the answer was not known until recently. It turns out that the undirected graph does not always determine the directed graph, though examples are not easy to find. There are just two group orders less than 100 for which examples exist (namely, 54 and 96), though several different groups realise each graph or digraph. A new version at arXiv 2408.93879 should be posted soon.

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 and a chordal graph. But an exact characterisation is unknown.

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

Old poblems are kept here.