Kirchhoff graphs are a new type of graph, one whose edges are vectors. They depict the dependencies in sets of vectors, and in this regard, they are akin to matroids. Indeed every binary matroid is Kirchhoff graphic. If the vectors in the set represent the stoichiometry of a reaction network (chemical, electrochemical or otherwise), a Kirchhoff graph is a circuit diagram for that reaction network. This book discusses the basic definitions for Kirchhoff graphs, their connections to matroids and finite fields, their connections to reaction networks and use as circuit diagrams, Kirchhoff graph uniformity, prime, composite and fundamental Kirchhoff graphs, and the structure of families of Kirchhoff graphs. They been studied by the author and his collaborators since 2005, and this book presents what we have learned in the past twenty years.

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Joseph Fehribach (PhD Duke, 1985) is a Professor of Mathematics at Worcester Polytechnic Institute. His research interests include linear algebra and graph theory as well in differential equations and numerical analysis. He teaches courses in (advanced) calculus, vector and tensor calculus, linear algebra, and differential equations. He is the author of Multivariable and Vector Calculus, De Gruyter, 2020 and Sequences and Series in Calculus, De Gruyter 2023.