تقرير
Commuting graphs of $p$-adic matrices
| العنوان: | Commuting graphs of $p$-adic matrices |
|---|---|
| المؤلفون: | Morrison, Ralph |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Rings and Algebras, Mathematics - Combinatorics, Mathematics - Number Theory, 16S50, 15A27, 05C50, 11S99 |
| الوصف: | 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 $n$. Comment: 9 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2407.13848 |
| رقم الأكسشن: | edsarx.2407.13848 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |