  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.
