Matrices over finite fields and their Kirchhoff graphs
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.
scientific article; zbMATH DE number 6865288
scientific article