Commuting graphs of groups

Over the summer I have had the pleasure of supervising a vacation student Aedan Pope. This is a program funded by the Australian Mathematical Sciences Institute which allows maths students to spend 6 weeks working on a project in the summer vacation (usually after third year) and also funds a trip to Sydney for the student to go to the CSIRO‘s Big Day In.

Aedan Pope’s research project focussed on commuting graphs of groups and we have just submitted a paper  with the results. It is available on the arxiv. I will outline the work here.

Given a group G, the commuting graph of G is the graph whose vertices are the elements of G which do not lie in the centre of G and two vertices are adjacent if they commute.  The commuting graph of a group appears to have first been studied in Brauer and Fowler’s paper `On groups of even order‘. Though they don’t actually use the words “commuting graph” they do study the distance between two elements of the group in what would be the commuting graph.

My interest in commuting graphs was piqued by the paper `On the commuting graph associated with the symmetric and alternating groups‘ by Iranmanesh and Jafarzadeh. This paper shows that when the commuting graph of the alternating group is connected it has diameter at most 5.  They also make the following conjecture:

There is an absolute constant c such that if the commuting graph of a finite group is connected, it has diameter at most c.

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