Commuting graphs of $p$-adic matrices

TL;DR

This paper investigates the structure and properties of commuting graphs of matrices over the field of p-adic numbers, revealing conditions for connectivity and bounds on graph diameter.

Contribution

It characterizes when the commuting graph is connected and determines maximum diameters for specific p and n, providing new insights into p-adic matrix commutation.

Findings

The commuting graph is connected iff n ≥ 3 and n is neither prime nor a power of p.

Maximum diameter of 6 occurs for p=2 and n=2q with q ≥ 7 prime.

Bounds on diameter depend on p and n, with examples of diameters 4 and 5.

Abstract

We study the commuting graph of $n\times n$ matrices over the field of $p$-adics $\mathbb{Q}_p$, whose vertices are non-scalar $n\times n$ matrices with entries in $\mathbb{Q}_p$ and whose edges connect pairs of matrices that commute under matrix multiplication. We prove that this graph is connected if and only if $n\geq 3$, 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 $q\geq 7$, 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 $n$.