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 un
iversal 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 $S_n$.
The commuting graph of a group G, denoted by Gamma(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. Let Z_m be the commutative ring of equivalence
classes of integers modulo m. In this paper we investigate the connectivity and diameters of the commuting graphs of GL(n,Z_m) to contribute to the conjecture that there is a universal upper bound on diam(Gamma(G)) for any finite group G when Gamma(G) is connected. For any composite m, it is shown that Gamma(GL(n,Z_m)) and Gamma(M(n,Z_m)) are connected and diam(Gamma(GL(n,Z_m))) = diam(Gamma(M(n,Z_m))) = 3. For m a prime, the instances of connectedness and absolute bounds on the diameters of Gamma(GL(n,Z_m)) and Gamma(M(n,Z_m)) when they are connected are concluded from previous results.
