# Peter Cameron's homepage

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

I am a quarter-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.

### Research snapshots

#### Graphs on algebraic structures

At present I have a number of ongoing research projects on graphs defined on algebraic structures, mostly with colleagues in India.

For which groups is the power graph a cograph? The power graph of a group G has vertex set G and an edge from x to y if one is a power of the other. A cograph is a graph with no induced 4-vertex path. This question generalises the classification of groups in which every element has prime power order, first posed by Graham Higman in the 1950s. With Pallabi Manna and Ranjit Mehatari, I have made some progress on this. The class of groups in question is subgroup-closed; we have determined the nilpotent groups and the simple groups in the class, and have a partial description of the soluble groups.

Super graphs For various classes of graphs defined on groups, and various natural partitions on groups, such as "conjugacy" or "same order", one can define a graph in which x and y are joined if some elements in their equivalence classes are joined in the original graph. We (G. Arunkumar, Rajat Kanti Nath and Lavanya Selvaganesh) call these names on the pattern "conjugacy superpower graphs". Among a variety of results, we have shown that the conjugacy supercommuting graph coincides with the commuting graph if and only if the group is 2-Engel, that is, satisfies the identity [[y,x],x] = 1; while the groups for which the conjugacy superpower graph is equal to the power graph are just the Dedekind groups (those with all subgroups normal).

Commuting graphs Which pairs of groups have isomorphic commuting graphs? With Vikraman Arvind, I have examined this question. If two groups of the same order are isoclinic, then their commuting graphs are isomorphic. The converse is false, but we think it may be true for nilpotent groups of class 2; we can prove this in some cases.

Universality of zero divisor graphs of rings Rings here are finite commutative with identity; the zero divisor graph has vertices the zero divisors, two vertices joined if their product is 0. We (G. Arunkumar, T. Tamizh Chelvam abd T. Kavaskar) have shown that, for various classes of rings, the zero divisor graphs are universal, that is, contain every finite graph as an induced subgraph. There are some interesting classes where this is not true, for example, local rings whose maximal ideal is principal: the zero divisor graph is a threshold graph (and is universal for such graphs).

Other work I already mentioned the work with Swathi V. V. and M. S. Sunita on matchings: one of our striking results is that the matching numbers of the power graph and enhanced power graph of a finite group are equal, even though we cannot say what the value is in all cases. Also, with R. Raveendra Prathap and T. Tamizh Chelvam, we have generalised the prime sum graph of a finite abelian group to the "subgroup sum graph", and investigated its properties.

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.

 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 15 October 2022

### A problem

A (total) ordering of a finite vector space is called compatible if every order-preserving bijection between subspaces of the same dimension is linear.

It is straightforward to show that every finite vector space has a compatible ordering: for example, choose any ordering of the finite field F; then the lexicographic ordering of Fn is compatible, for any n.

Problem: How many compatible orderings does the n-dimensional vector space over the field of order q have? (Formulae are known for the cases n = 1 and n = 2.)

Old problems are kept here.