21-25 August 2012
Portorož, Slovenia
UTC timezone
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What Kirchhoff Really Did

Presented by Dr. Roger MALLION
Type: Oral presentation

Content

In 1845, as a 21-year-old undergraduate, Gustav Robert Kirchhoff (writing under the pseudonym ‘Studiosus’ Kirchhoff) published, as ‘Mitglied des physikalischen Seminars zu Königsberg’ at the Albertina University, a classic paper1 in which he stated his Laws I and II of electrical circuits — laws which, for more than a century, have been ‘well-known to every schoolboy’. Two years later, in a second paper2 in the Analen der Physik und Chemie, he examined the mathematical foundations of these laws and it is this paper which is still of interest to modern Graph Theorists. This is because it has deep implications for the theory of linear equations3, and it was also the first paper to consider, if only implicitly, the idea of the spanning trees of a graph.4,5 The speaker has long had an interest the enumeration of spanning trees6,7 and this talk evaluates what Kirchhoff actually did in his much-cited paper of 1847, and how this relates to modern work.5–7 1. ‘Studiosus’ Kirchhoff. Ueber den Durchgang eines elektrischen Stromes durch eine Ebene, insbesondere durch eine kreisförmige. Annalen der Physik u. Chemie (‘Poggendorff’s Annalen’) 1845, 64, 497–514. 2. G. Kirchhoff. Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanische Ströme geführt wird. Annalen der Physik u. Chemie (‘Poggendorff’s Annalen’) 1847, 72, 497–508. 3. W. Ahrens. Ueber das Gleichungssystem einer Kirchhoff’schen galvanischen Stromverzweigung, Mathematische Annalen 1897 49, 311–324. 4. C. W. Borchardt. Ueber eine der Interpolation entsprechende Darstellung der Eliminations-Resultante, Journal für reine u. angewandte Mathematik (Borchardt’s Journal) 1860, 57, 111–121. 5. R. L. Brooks, C. A. B. Smith, A. H. Stone & W. T. Tutte. The dissection of rectangles into squares. Duke Mathematical Journal 1940, 7, 312–340. 6. E. C. Kirby, D. J. Klein, R. B. Mallion, P. Pollak & H. Sachs. A theorem for counting spanning trees in general chemical graphs and its particular application to toroidal fullerenes. Croatica Chemica Acta 2004, 77, 263–278. 7. I. Gutman, R. B. Mallion & J. W. Essam. Counting the spanning trees of a labelled molecular-graph. Molecular Physics 1983, 50, 859–877.

Place

Location: Portorož, Slovenia
Address: University of Primorska, Faculty of Tourism Studies, Obala 11a, SI-6320 Portorož - Portorose, Slovenia
Room: VP1

Primary authors

  • Dr. Roger MALLION School of Physical Sciences, University of Kent, Canterbury, England, United Kingdom
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Co-authors

  • Mr. Paul POLLAK The King's School, Canterbury, Kent, CT1 2ES, England, United Kingdom
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