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.
Further supporting evidence for this conjecture is in the paper `Anisotropic groups of type and the commuting graph of finite simple groups‘ by Yoav Segev and Gary Seitz, which shows that for all finite classical simple groups over a field of size at least 5, if the commuting graph is connected it has diameter at most 10. The bound on the field size is mainly there so that they can give necessary and sufficient conditions for when the graph is connected. Looking at their proof, it is possible to show that the commuting graph of GL(n,2) has diameter at most 8.
Moreover, in their paper `On the diameters of commuting graphs‘, Akbari, Ghandehari, Hadian and Mohammadian, show that for a field F of size at least 3, if the commuting graph of GL(n,F) is connected it has diameter at most 6. This is not sharp and the authors conjecture that it is actually at most 5. It is shown in `Commuting graphs of matrix algebras‘ by Akbari, Bidkorii and Mohammadian that the commuting graph of GL(n,q) for is connected if and only if n is not a prime, while for GL(n,2) it is connected if and only if n and n-1 are both not primes.
Instead of looking at matrix groups over fields, we looked at matrix groups over the integers modulo m. The main result is that for a composite integer m, the commuting graph of is connected and has diameter 3.
The commuting graph of a ring can be similarly defined and the commuting graph of is also connected and has diameter 3.
Aedan also conducted some computer calculations of the diameters of commuting graphs for small groups using the small group database which is available in Magma. Despite searching through all groups of order up to 2000 except for those of the form for k composite and not equal to 9 or 4, no graph of diameter larger than 6 was found. We do not know whether or not the upper bound on 8 for the diameter of the commuting graph of GL(n,2) is sharp, so it would be interesting to see if we can find examples of groups which have a commuting graph with diameter greater than 6.
Interestingly, no graph of diameter 2 was found either. In fact Segev, in `The commuting graph of minimal nonsolvable groups‘ has shown that the commuting graph of a finite minimal nonsoluble group has diameter at least 3. It is not possible to have a group whose commuting graph is complete (as the vertices of the commuting graph are not in the centre of the group) but is there one for which the diameter is 2?