Generalizing Kirchhoff laws for Signed Graphs

TL;DR

This paper extends Kirchhoff's laws to signed graphs by introducing generalized contributor-transpedances, revealing non-conservative behaviors and linking maximum values to the signless Laplacian, thus broadening the theoretical framework for signed graph analysis.

Contribution

It introduces a novel generalized contributor-transpedance concept for signed graphs, connecting Boolean classes, cycle covers, and the signless Laplacian, expanding Kirchhoff law applications.

Findings

Contributor-transpedances produce non-conservative Kirchhoff-type laws.

Maximum contributor-transpedance is computed via the signless Laplacian.

Classical conservation corresponds to trivial Boolean classes.

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.