Matrices over finite fields and their Kirchhoff graphs

Given a stoichiometric matrix A with integer entries, we seek to visualize the information by a reaction network N, whose edges are labeled by the columns of A, such that all column dependencies are realized as cycles in N. Moreover, the vector edges of N are assigned integer multiplicities so that...

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Bibliographic Details
Published in:Linear algebra and its applications 2018-06, Vol.547, p.128-147
Main Authors: Reese, Tyler M., Fehribach, Joseph D., Paffenroth, Randy C., Servatius, Brigitte
Format: Article
Language:English
Subjects:
Cayley color graphs
Finite fields
Finite volume method
Graph theory
Graphs
Integers
Kirchhoff graphs
Linear algebra
Matroids
Sparsity
Surge protectors
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Summary:Given a stoichiometric matrix A with integer entries, we seek to visualize the information by a reaction network N, whose edges are labeled by the columns of A, such that all column dependencies are realized as cycles in N. Moreover, the vector edges of N are assigned integer multiplicities so that every vertex of N corresponds to a vector in the row space of A. A finite such N is called a Kirchhoff graph. We establish the existence of nontrivial Kirchhoff graphs over finite fields, but the general problem over the integers is still open.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2018.02.020