Elmar Teufl ; Stephan Wagner

Spanning forests, electrical networks, and a determinant identity
dmtcs:2699 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2009,
DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)

https://doi.org/10.46298/dmtcs.2699
Spanning forests, electrical networks, and a determinant identity
Authors: Elmar Teufl ^{1}; Stephan Wagner ^{2}
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Elmar Teufl;Stephan Wagner
1 Fakultät für Mathematik = Faculty of Mathematics [Bielefeld]
2 Department of Mathematical Sciences [Matieland, Stellenbosch Uni.]
We aim to generalize a theorem on the number of rooted spanning forests of a highly symmetric graph to the case of asymmetric graphs. We show that this can be achieved by means of an identity between the minor determinants of a Laplace matrix, for which we provide two different (combinatorial as well as algebraic) proofs in the simplest case. Furthermore, we discuss the connections to electrical networks and the enumeration of spanning trees in sequences of selfsimilar graphs.
Gong, Helin; Li, Shuli, 2017, The Number Of Spanning Trees Of A Family Of 2Separable Weighted Graphs, Discrete Applied Mathematics, 229, pp. 154160, 10.1016/j.dam.2017.05.003.
Teufl, Elmar; Wagner, Stephan, 2010, Determinant Identities For Laplace Matrices, Linear Algebra And Its Applications, 432, 1, pp. 441457, 10.1016/j.laa.2009.08.028.