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2024 | OriginalPaper | Chapter

Prime, Composite and Fundamental Kirchhoff Graphs

Authors : Jessica Wang, Joseph D. Fehribach

Published in: Combinatorics, Graph Theory and Computing

Publisher: Springer Nature Switzerland

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Abstract

A Kirchhoff graph is a vector graph with orthogonal cycles and vertex cuts. An algorithm has been developed that constructs all the Kirchhoff graphs up to a fixed edge multiplicity. This algorithm is used to explore the structure of prime Kirchhoff graph tilings. The existence of infinitely many prime Kirchhoff graphs for a given set of fundamental Kirchhoff graphs is established, as is the existence of a minimal multiplicity for Kirchhoff graphs to exist.

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previous chapter A New Upper Bound for the Site Percolation Threshold of the Square Lattice
next chapter On the Minimum Locating Number of Graphs with a Given Order
Literature
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J.D. Fehribach, Vector-space methods and Kirchhoff graphs for reaction networks, SIAM Journal of Applied Mathematics70 (2009) 543–562.MathSciNetCrossRef
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J. D. Fehribach, J. J. McDonald, Matrices and Kirchhoff graphs, a rank-two, nullity-two construction, Congressus Numerantium230 (2018) 199–207.MathSciNet
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T. Reese, J.D. Fehribach & R. Paffenroth, Duality in geometric graphs: vector graphs, Kirchhoff graphs and Maxwell reciprocal figures, Symmetry8 (2016) 1–28.MathSciNetCrossRef
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T. Reese, J.D. Fehribach, R. Paffenroth & B. Servatius, Matrices over finite fields and their Kirchhoff graphs, Linear Algebra and its Applications547 (2018) 128–147.MathSciNetCrossRef
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T. Reese, J.D. Fehribach, R. Paffenroth & B. Servatius, Uniform Kirchhoff graphs, Linear Algebra and its Applications566 (2019) 1–16.MathSciNetCrossRef
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J. D. Fehribach, Kirchhoff graph uniformity, Congressus Numerantium233 (2019) 143–150.MathSciNet
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T. Reese, J.D. Fehribach & R. Paffenroth, Equitable edge partitions and Kirchhoff graphs, Linear Algebra and its Applications639 (2022) 225–242.MathSciNetCrossRef
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J. D. Fehribach, Kirchhoff graphs and stoichiometry, Journal of the Electrochemical Society170 (2023) 083504.CrossRef
Metadata
Title
Prime, Composite and Fundamental Kirchhoff Graphs
Authors
Jessica Wang
Joseph D. Fehribach
Copyright Year
2024
Book
Combinatorics, Graph Theory and Computing

Print ISBN: 978-3-031-62165-9

Electronic ISBN: 978-3-031-62166-6

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-62166-6_10

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