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

• Published in: Croatica chemica acta 2016-10, Vol.89 (4), p.1
• Main Authors: Kirby, Edward C., Mallion, Roger B., Pollak, Paul, Skrzyński, Paweł J.
• Format: Article
• Language: English
• Subjects: Autobiographies; chemical graph theory; cycle theorem; fundamental system of cycles; Kirchhoff; Mathematical research; Matrices; matrix tree theorem; spanning trees; Trees; Trees (Graph theory)
• ISSN: 0011-1643; 1334-417X
• DOI: 10.5562/cca2995

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'.