Kirchhoff Graphs

Vector Dependencies and Implications

Joseph D. Fehribach

• Publisher: De Gruyter
• Year: 2025
• eBook ISBN: 9783111409399
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

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.

Table of contents

• Title Page
• Copyright
• Contents
• Frontmatter
• Dedication
• Notes to the reader
• Contents
• 1 Introduction
• 2 Basic results
• 3 Kirchhoff graph uniformity
• 4 Kirchhoff graph construction
• 5 Matroids and Kirchhoff graphs over finite fields
• 6 Equitable Kirchhoff graphs
• 7 Kirchhoff graph duality and Maxwell reciprocal diagrams
• 8 Kirchhoff graphs and reaction networks
• Index