On the foundations of signed graphs I: chain groups, frame matroid, and bivariate flow polynomial

TL;DR

This paper develops a comprehensive algebraic and matroidal framework for signed graphs with outer-edges, introducing new concepts like the bivariate flow polynomial and characterizing fundamental structures such as flow and tension groups.

Contribution

It standardizes concepts in signed graphs, introduces the bivariate flow polynomial, and links matroid theory with algebraic structures in signed graphs, filling gaps in existing literature.

Findings

Characterization of cuts and bonds in signed graphs with outer-edges

Structures of flow, boundary, and homology groups with arbitrary abelian coefficients

Introduction of the bivariate flow polynomial for signed graphs

Abstract

This paper studies signed graphs with possible outer-edges. We introduce and investigate the chain group, the boundary operator, the co-boundary operator, the flow group, the tension group, the homology group, the cohomology group, with coefficients in an abelian group. We also introduce and investigate the bivariate flow polynomial for signed graphs. The guiding principle is the correspondence between representable matroids over $\mathbb{R}$ on a ground set $E$ of edges and the subspaces of the vector space of real-valued chains on the same ground set. The frame matroid of signed graph emerges naturally by defining circuits as minimal supports of nonzero flows, rather than listing circuit patterns abruptly. Likewise, bonds, or co-circuits, can be obtained as minimal supports of nonzero tensions. In addition to standardizing the concepts and their meanings of signed graphs, we update the following results: (1) Characterization of cuts and bonds of signed graph with outer-edges. (2) The structures of flow group, boundary group, and homology group, with coefficients in an arbitrary abelian group. (3) Introduction of bivariate flow polynomial of signed graph, revealing the mystery of inexistence of univariate flow polynomial of signed graph in the literature.