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arXiv · 2302.00199

Adjacency matrices over a finite prime field and their direct sum decompositions

Akihiro Higashitani·Yuya Sugishita
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Abstract

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field $\mathbb{F}_p$. We apply symmetric (row and column) elementary transformations to the adjacency matrix over $\mathbb{F}_p$ in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over $\mathbb{F}_p$ for any odd prime $p$. Our key tool is quadratic residues of $\mathbb{F}_p$.

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Akihiro Higashitani, Yuya Sugishita. 2023-02-01. Adjacency matrices over a finite prime field and their direct sum decompositions. https://arxiv.org/abs/2302.00199

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