Matroids of Gain Signed Graphs

TL;DR

This paper introduces a new class of matroids derived from gain signed graphs, combining properties of signed graphs and gain graphs, with a focus on abelian groups and their representations.

Contribution

It defines gain signed graphs for abelian groups, develops their elementary properties, and constructs associated matroids and their vector and hyperplanar representations.

Findings

Defined gain signed graphs with abelian group gains.

Established elementary properties and matroid structures.

Developed vector and hyperplanar representations.

Abstract

A signed graph has edge signs. A gain graph has oriented edge gains drawn from a group. We define the combination of the two for the abelian case, in which each oriented edge of a signed graph has a gain from an abelian group, concentrating on the case of the additive group of a field. We develop the elementary graph properties, the associated matroid, and the vector and hyperplanar representations.