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

Abstract

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 19 th century -is related to 20 th -and 21 st -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'.

Postscript on Viable Ground-States for Calculating Topological π-Electron Ring-Currents Using the Hückel–London–Pople–McWeeny Model

“…Figures in black in the centres of rings denote the corresponding ring-current intensities (expressed as a ratio to the benzene value); positive ring-currents are diamagnetic and are considered to flow in the anti-clockwise direction around their respective rings, whilst negative ring-currents are paramagnetic and flow in the clockwise direction. Red figures written along the bonds denote the bond-currents (in the directions indicated by the arrows); these bond currents are compliant with Kirchhoff's Law of currents at junctions [10,36] and are, thereby, consistent with these ring currents.…”

“…The answer to this question is materially connected with the fact that the 'ring current' in a microscopic conjugated system is the equivalent of the 'loop current' [36][37][38][39] that flows around an enclosed area in a macroscopic electrical network of the classic type to which Kirchhoff's Laws apply. [36][37][38][39] The 'bond currents' in the microscopic conjugated system correspond to the currents in the individual wires that comprise the traditional macroscopic electrical network.…”

Topological Ring Currents and Bond Currents in Some Neutral and Anionic Altans and Iterated Altans of Corannulene and Coronene

“…It should immediately be conceded, however, that the pair of zero currents (for bond b) are guaranteed to agree by symmetry, irrespective of the approximations and assumptions made in the calculation. Nevertheless, given that bond currents arising from ab initio calculations (unlike our HLPM bond currents) do not in general rigorously obey Kirchhoff's Conservation Law 38 at junctions (except in the idealized limit of a complete basis The Journal of Physical Chemistry A Article set), 48 the approximate equality between the above corresponding pairs of internal bonds in 1 and 1 2− is sufficiently encouraging to suggest that investigating the magnetic properties of the iterated altans 3−6 and their dianions by the ipso-centric ab intitio methods might prove to be worthwhile.…”

Some Observations on Triplet Ground-States in the Context of ‘Topological’ (HLPM) Ring-Currents in Conjugated Systems

“…Figures in black in the centres of rings denote the corresponding (diamagnetic) ring-current intensities (expressed as a ratio to the benzene value), whilst red figures written along the bonds denote the bond-currents (in the directions indicated by the arrows) that are consistent with them virtue of the validity of Kirchhoff's Law of Conservation of Currents at Junctions, [6] all bond-currents -the quantities depicted in red in Figure 3 -are automatically predetermined once the ring currents in all rings -the quantities in black, in Figure 3 -have been specified. [48][49][50] As we have previously pointed out, [17,18,50] the strict observance of the Kirchhoff Conservation Law in the case of the HLPM approach is essentially because, on the assumptions made, a conjugated hydrocarbon really is mimicked as if it were a microscopic version of a standard macroscopic Kirchhoff network, with EMFs and 'arms' consisting of macroscopic wires, loop currents (the macroscopic analogy of ring currents) around the closed regions, and currents (the macroscopic analogy of bond currents) in the individual wires.…”

“…OR a period of some 25 years, one of us (RBM) was privileged to work with the late Edward Cameron Kirby (1934Kirby ( -2019, the honorand of this Session of MATH/CHEM/COMP 31 (Dubrovnik, 2019). A considerable proportion of our collaboration was devoted to the idea of spanning trees in graphs, [1][2][3][4][5][6][7] especially of chemical graphs [8] that represent conjugated hydrocarbons (extant or hypothetical). A prominent application of these ideas in Mathematical Chemistry has been in the context of what the present authors have named [9,10] the Hückel [11,12] -London [13] -Pople [14] -McWeeny [15] (HLPM) formalism for calculating what are frequently called 'topological' ringcurrents and bond currents [9,10] -for reviews please see Refs.…”