  The commuting graph of a group $G$ is the simple undirected graph whose
vertices are the non-central elements of $G$ and two distinct vertices are
adjacent if and only if they commute. It is conjectured by Jafarzadeh and
Iranmanesh that there is a universal upper bound on the diameter of the
commuting graphs of finite groups when the commuting graph is connected. In
this paper we determine upper bounds on the diameter of the commuting graph for
some classes of groups to rule them out as possible counterexamples to this
conjecture. We also give an example of an infinite family of groups with
trivial centre and diameter 6, the previously largest known diameter for an
infinite family was 5 for $S_n$.
