Commuting graphs of p-adic matrices

We study the commuting graph of n× n matrices over the field of p-adics Qₚ, whose vertices are non-scalar n× n matrices with entries in Qₚ and whose edges connect pairs of matrices that commute under matrix multiplication. We prove that this graph is connected if and only if n≥ 3, with n neither prime nor a power of p. We also prove that in the case of $p=2$ and $n=2q$ for q a prime with q≥ 7, the commuting graph has the maximum possible diameter of $6$; these are the first known such examples independent of the axiom of choice. We also find choices of p and n yielding diameter $4$ and diameter $5$ commuting graphs, and prove general bounds depending on p and n.