Kirchhoff-Type Laws for Signed Graphs - Josephine Reynes

Title: Kirchhoff-Type Laws for Signed Graphs

Abstract: Bidirected incidence orientations can be used to study signed graphs. Each edge can be assigned a value of -1 or +1 depending on the orientation of the incidences. If the incidences have the same sign the edge is considered negative, and if the incidences have opposite signs then the edge is considered positive. An oriented hypergraph allows for more than two incidences to appear within edges. This allows for more than two vertices to be adjacent through the same edge. The sign of an adjacency can be determined by the sign of the relevant incidences observed locally within an edge. The hypergraph technique of signing adjacencies, instead of edges, through the incidence structure can be applied to sign graphs. Tutte showed that graphs follow Kirchhoff’s laws by using transpedances, which are a comparison between two 2-arborescences. Kirchhoff-type Laws for signed graphs are characterized by generalizing transpedances using the incidence-oriented, bidirected structure and techniques of signed graphs. The classical 2-arborescence interpretation of Tutte is equivalent to single-element Boolean classes of reduced incidence-based cycle covers, call contributors. A generalization of contributor-transpedances can be obtained using entire Boolean classes that are naturally cancellative in a graph. The contributor-transpedances on signed graphs produce non-conservative Kirchhoff-type Laws, where every contributor has a unique source-sink path property. Finally, the maximum values of a contributor-transpedance can be calculated from the signless Laplacian.