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https://doi.org/10.5562/cca2995

What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work

Edward C. Kirby ; Resource Use Institute, Fishersview Court, Pitlochry PH16 5AN, Scotland, United Kingdom
Roger B. Mallion ; School of Physical Sciences, University of Kent, Canterbury CT2 7NH, England, United Kingdom
Paul Pollak ; The King’s School, Canterbury CT1 2ES, England, United Kingdom
Paweł J. Skrzyński ; The Canterbury High School, Canterbury CT2 8QA, England, United Kingdom

Puni tekst: engleski, pdf (887 KB) str. 403-417 preuzimanja: 2.218* citiraj
APA 6th Edition
Kirby, E.C., Mallion, R.B., Pollak, P. i Skrzyński, P.J. (2016). What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work. Croatica Chemica Acta, 89 (4), 403-417. https://doi.org/10.5562/cca2995
MLA 8th Edition
Kirby, Edward C., et al. "What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work." Croatica Chemica Acta, vol. 89, br. 4, 2016, str. 403-417. https://doi.org/10.5562/cca2995. Citirano 25.09.2021.
Chicago 17th Edition
Kirby, Edward C., Roger B. Mallion, Paul Pollak i Paweł J. Skrzyński. "What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work." Croatica Chemica Acta 89, br. 4 (2016): 403-417. https://doi.org/10.5562/cca2995
Harvard
Kirby, E.C., et al. (2016). 'What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work', Croatica Chemica Acta, 89(4), str. 403-417. https://doi.org/10.5562/cca2995
Vancouver
Kirby EC, Mallion RB, Pollak P, Skrzyński PJ. What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work. Croatica Chemica Acta [Internet]. 2016 [pristupljeno 25.09.2021.];89(4):403-417. https://doi.org/10.5562/cca2995
IEEE
E.C. Kirby, R.B. Mallion, P. Pollak i P.J. Skrzyński, "What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work", Croatica Chemica Acta, vol.89, br. 4, str. 403-417, 2016. [Online]. https://doi.org/10.5562/cca2995

Sažetak
What Kirchhoff actually did concerning spanning trees in the course of his classic paper in the 1847 Annalen der Physik und Chemie has, to some extent, long been shrouded in myth in the literature of Graph Theory and Mathematical Chemistry. In this review, Kirchhoff’s manipulation of the equations that arise from application of his two celebrated Laws of electrical circuits — formulated in the middle of the 19th century — is related to 20th- and 21st-century work on the enumeration of spanning trees. It is shown that matrices encountered in an analysis of what Kirchhoff really did include (a) the Kirchhoff (Laplacian, Admittance) matrix, K, that features in the well-known Matrix Tree Theorem, (b) the matrix G encountered in the theorem of Gutman, Mallion & Essam (1983), applicable only to planar graphs, and (c) the analogous matrix M that arises in the Cycle Theorem (Kirby et al. 2004), a theorem that applies to graphs of any genus. It is concluded that Kirchhoff himself was not interested in counting spanning trees, and, accordingly, he did not explicitly do so. Nevertheless, it is shown how the modulus of the determinant of a certain matrix (here denoted by the label C') — associated with the linear equations arising from application of Kirchhoff’s two Laws — is numerically equal to the number of spanning trees in the graph representing the connectivity of the electrical network being studied. Kirchhoff did, however, invoke the concept of spanning trees, introducing them in a complementary fashion by referring to the chords that must be removed from the original graph in order to form such trees. It is further emphasised that, in choosing the cycles in the network being studied, around which to apply his circuit Law, Kirchhoff explicitly selected what would now be called a ‘Fundamental System of Cycles’.

Creative Commons License This work is licensed under a Creative Commons Attribution 4.0 International License.

Ključne riječi
Kirchhoff; chemical graph theory; spanning trees; matrix tree theorem; cycle theorem; fundamental system of cycles

Hrčak ID: 172895

URI
https://hrcak.srce.hr/172895

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