Commuting graphs of $p$-adic matrices
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911961746243584 |
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| author | Morrison, Ralph |
| author_facet | Morrison, Ralph |
| contents | 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13848 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Commuting graphs of $p$-adic matrices Morrison, Ralph Rings and Algebras Combinatorics Number Theory 16S50, 15A27, 05C50, 11S99 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 $n$. |
| title | Commuting graphs of $p$-adic matrices |
| topic | Rings and Algebras Combinatorics Number Theory 16S50, 15A27, 05C50, 11S99 |
| url | https://arxiv.org/abs/2407.13848 |