Resistors in dual networks

Martina Furrer, Norbert Hungerbühler, Simon Jantschgi


Let G be a finite plane multigraph and G' its dual. Each edge e of G is interpreted as a resistor of resistance Re, and the dual edge e' is assigned the dual resistance Re':=1/Re. Then the equivalent resistance re over e and the equivalent resistance re' over e' satisfy re/Re+re'/Re'=1. We provide a graph theoretic proof of this relation by expressing the resistances in terms of sums of weights of spanning trees in G and G' respectively.


dual graphs, electrical networks, equivalent resistance

Full Text:




P. Ali, F. Atik, and R.B. Bapat, Identities for minors of the Laplacian, resistance and distance matrices of graphs with arbitrary weights, Linear Multilinear Algebra 68 (2) (2020), 323– 336.

R.B. Bapat, Graphs and Matrices, Universitext. Springer, London; Hindustan Book Agency, New Delhi, second edition, 2014.

N. Biggs, Spanning trees of dual graphs, J. Combin. Theory Ser. B 11 (1971), 127–131.

S. Chaiken, A combinatorial proof of the all minors matrix tree theorem, SIAM J. Algebraic Discrete Methods 3 (3) (1982), 319–329.

M. Furrer, Widerstandssumme in dualen Netzwerken, Mentorierte Arbeit, 2017, ETH Zurich.

J. Moody and P.K. Aravind, Resistor networks based on symmetrical polytopes, Electron. J. Graph Theory Appl. 3 (1) (2015), 56–69.

W.T. Tutte, Graph theory, volume 21 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001, With a foreword by Crispin St. J. A. Nash- Williams, Reprint of the 1984 original.

F.J. van Steenwijk, Equivalent resistors of polyhedral resistive structures, American Journal of Physics 66 (1) (1998), 90–91.

Y.J. Yang, An identity on resistance distances, In Material and Manufacturing Technology IV, volume 748 of Advanced Materials Research, pages 1024–1027. Trans Tech Publications, 10, 2013.


  • There are currently no refbacks.

ISSN: 2338-2287

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

View EJGTA Stats