On the foundations of signed graphs I: chain groups, frame matroid, and bivariate flow polynomial
Beifang Chen
TL;DR
The paper establishes an intrinsic algebraic-topology framework for signed graphs with outer-edges, defining chain and cochain structures that yield natural constructions of the flow, tension, boundary, and homology theories with coefficients in any abelian group. It shows that the frame matroid of a signed graph arises from minimal supports of nonzero flows and that bonds correspond to minimal supports of nonzero tensions, providing a unified circuit/bond perspective. It then derives explicit descriptions of canonical bases, and the detailed structure of the flow, boundary, and homology groups, including a precise decomposition of the flow group as $\mathrm{F}(\Sigma,\mathbb{A}) \cong (\mathrm{Tor}_2(\mathbb{A}))^{u_c(\Sigma)} \oplus \mathbb{A}^{\mathrm{cr}(\Sigma)}$ and analogous results for $\mathrm{B}_0$ and $\mathrm{H}_0$. The centerpiece is the introduction of the bivariate flow polynomial $\varphi(\Sigma;t,x)$, which encodes the count of nowhere-zero flows over abelian groups and specializes to univariate odd/even flow polynomials, while obeying deletion-contraction and yielding a polynomial-or-quasi-polynomial structure reflecting the graph’s balance/negative-circle data. This framework clarifies longstanding questions about signed-graph flows and provides a robust tool for future combinatorial and topological investigations.
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.