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2024 | OriginalPaper | Chapter

A Matrix Criterion for Harmonic Morphisms of Graphs with Applications to Graph Products

Authors : Caroline G. Melles, David Joyner

Published in: Combinatorics, Graph Theory and Computing

Publisher: Springer Nature Switzerland

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Abstract

Urakawa, and later Baker and Norine, developed the notion of a harmonic morphism as a type of graph morphism with similar properties to holomorphic maps of Riemann surfaces. We give a matrix criterion for harmonic graph morphisms which allows us to translate combinatorial questions about harmonic morphisms to linear algebra questions. We illustrate its use by finding conditions under which certain maps of NEPS graph products (for example, tensor, Cartesian, and strong products) are harmonic, and calculating their vertical and horizontal multiplicity matrices and degrees. We also apply the matrix criterion to lexicographic products, the special case of graph blow-ups, and graph joins.

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previous chapter The Spectrum Problem for the 4-Uniform 4-Colorable 3-Cycles with Maximum Degree 2
next chapter Applications of Topological Graph Theory to 2-Manifold Learning
Footnotes
1
This definition of multicover differs from the concept of a set multicover in theoretical computer science and also from the concept of a multicover in [7] which is a special case of a cover.
 
2
The disjunctive product has an edge between vertices \((v,v^\prime )\) and \((w,w^\prime )\) whenever v is adjacent to w, or \(v^\prime \) is adjacent to \(w^\prime \), or both.
 
Literature
1.
2.
Cvetković, D., Lučić, R.: A new generalization of the concept of the p-sum of graphs. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 302–319, 67–71 (1970).MathSciNet
3.
4.
Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs (2nd ed.). CRC Press, Boca Raton (2011).CrossRef
5.
6.
Joyner, D., Melles, C.: Adventures in Graph Theory. Birkhaüser, Basel (2017).
7.
8.
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Metadata
Title
A Matrix Criterion for Harmonic Morphisms of Graphs with Applications to Graph Products
Authors
Caroline G. Melles
David Joyner
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_19

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