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arXiv · 2009.12680

Generalizing Kirchhoff laws for Signed Graphs

Lucas J. Rusnak·Josephine Reynes·Skyler J. Johnson·Peter Ye
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Abstract

Kirchhoff-type Laws for signed graphs are characterized by generalizing transpedances through the incidence-oriented structure of bidirected graphs. The classical $2$-arborescence interpretation of Tutte is shown to be equivalent to single-element Boolean classes of reduced incidence-based cycle covers, called contributors. A generalized contributor-transpedance is introduced using entire Boolean classes that naturally cancel in a graph; classical conservation is proven to be property of the trivial Boolean classes. The contributor-transpedances on signed graphs are shown to produce non-conservative Kirchhoff-type Laws, where every contributor possesses the unique source-sink path property. Finally, the maximum value of a contributor-transpedance is calculated through the signless Laplacian.

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Lucas J. Rusnak, Josephine Reynes, Skyler J. Johnson, Peter Ye. 2020-09-26. Generalizing Kirchhoff laws for Signed Graphs. https://arxiv.org/abs/2009.12680

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