Matrices over finite fields and their Kirchhoff graphs
type: Item
instance of: scholarly article
title: Matrices over finite fields and their Kirchhoff graphs
zbMATH Open document ID: 1394.15022
author: Tyler M. Reese
author: Joseph D. Fehribach
author: Randy C. Paffenroth
author: Brigitte Servatius
published in: Linear Algebra and its Applications
publication date: 2018-04-27T00:00:00Z
review text: Kirchhoff's laws are well known for electrical networks. They can be characterized by the circuit matrix and the cutset matrix of the associated network graph. Likewise, an electrochemical reaction network can be encoded by an integer-valued matrix called the stoichiometric matrix, the columns of which correspond to reaction steps. In particular, given this matrix, what is a suitable graphic rendering of a network that properly visualizes the underlying chemical reactions? This paper extends previous work on Kirchhoff graphs for matrices over the rationals to matrices over finite fields. It is shown that for any integer-valued matrix, there exists a nontrivial \(\mathbb{Z}_p\)-Kirchhoff graph for the matrix (mod \(p\)) for sufficiently large prime \(p\). Moreover, nonzero \(\mathbb{Z}_p\)-Kirchhoff graphs are explicitly constructed for \(\mathbb{Z}_p\)-valued matrices with an entry-wise nonzero vector in the row space. Finally, the authors also demonstrate that every binary matrix has a nonzero Kirchhoff graph, and they consider what this result means in the context of matroids.
reviewed by: Wai-Kai Chen
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 6865288
zbMATH Keywords: finite field
zbMATH Keywords: Cayley color graph
zbMATH Keywords: Kirchhoff graph
zbMATH Keywords: matroid
MaRDI profile type: Publication
full work available at URL: j.laa.2018.02.020
OpenAlex ID: W2790648052
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Wikidata QID: Q130093351
DOI: 10.1016/J.LAA.2018.02.020
description: scientific article; zbMATH DE number 6865288