The diameters of commuting graphs of linear groups and matrix rings over the integers modulo m

The commuting graph of a group G, denoted by Γ(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 Zm 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,Zm) to contribute to the conjecture that there is a universal upper bound on diam(Γ(G)) for any finite group G when Γ(G) is connected. For any composite m, it is shown that Γ(GL(n,Zm)) and Γ(M(n,Zm)) are connected and diam(Γ(GL(n,Zm))) = diam(Γ(M(n,Zm))) = 3. For m a prime, the instances of connectedness and absolute bounds on the diameters of Γ(GL(n,Zm)) and Γ(M(n,Zm)) when they are connected are concluded from previous results.