Geodesic
Matrices and their Kirchhoff graphs
Ars Mathematica Contemporanea, Tome 9 (2015) no. 1, pp. 125-144.

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The fundamental relationship between matrices over the rational numbers and a newly defined type of graph, a Kirchhoff graph, is established. For a given matrix, a Kirchhoff graph represents the orthogonal complementarity of the null and row spaces of that matrix. A number of basic results are proven, and then a relatively complicated Kirchhoff graph is constructed for a matrix that is the transpose of the stoichiometric matrix for a reaction network for the production of sodium hydroxide from salt. A Kirchhoff graph for a reaction network is a circuit diagram for that reaction network. Finally it is conjectured that there is at least one Kirchhoff graph for any matrix with rational elements, and a process for constructing an incidence matrix for a Kirchhoff graph from a given matrix is discussed.
DOI : 10.26493/1855-3974.329.82d
Keywords: Kirchhoff graphs, fundamental theorem of linear algebra, reaction networks. 
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Joseph D. Fehribach. Matrices and their Kirchhoff graphs. Ars Mathematica Contemporanea, Tome 9 (2015) no. 1, pp. 125-144. doi : 10.26493/1855-3974.329.82d. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.329.82d/

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