Flow polynomials of a signed graph

TL;DR

This paper investigates the structure and enumeration of group-flows in signed graphs, revealing how their counts depend on the group's properties and providing combinatorial interpretations of associated polynomials.

Contribution

It introduces a polynomial $F_d(G,x)$ for counting nowhere-zero group-flows in signed graphs and explains the influence of the group's subgroup structure on flow counts.

Findings

Flows can be generated by fundamental directed circuits.

Flow classes are determined by elements of order 2 in the group.

Coefficients of $F_d(G,x)$ relate to broken bonds in the graph.

Abstract

In contrast to ordinary graphs, the number of the nowhere-zero group-flows in a signed graph may vary with different groups, even if the groups have the same order. In fact, for a signed graph $G$ and non-negative integer $d$, it was shown that there exists a polynomial $F_d(G,x)$ such that the number of the nowhere-zero $\Gamma$-flows in $G$ equals $F_d(G,x)$ evaluated at $k$ for every Abelian group $\Gamma$ of order $k$ with $\epsilon(\Gamma)=d$, where $\epsilon(\Gamma)$ is the largest integer $d$ for which $\Gamma$ has a subgroup isomorphic to $\mathbb{Z}^d_2$. We focus on the combinatorial structure of $\Gamma$-flows in a signed graph and the coefficients in $F_d(G,x)$. We first define the fundamental directed circuits for a signed graph $G$ and show that all $\Gamma$-flows (not necessarily nowhere-zero) in $G$ can be generated by these circuits. It turns out that all $\Gamma$-flows in $G$ can be evenly classified into $2^{\epsilon(\Gamma)}$-classes specified by the elements of order 2 in $\Gamma$, each class of which consists of the same number of flows depending only on the order of the group. This gives an explanation for why the number of $\Gamma$-flows in a signed graph varies with different $\epsilon(\Gamma)$, and also gives an answer to a problem posed by Beck and Zaslavsky. Secondly, using an extension of Whitney's broken circuit theory we give a combinatorial interpretation of the coefficients in $F_d(G,x)$ for $d=0$, in terms of the broken bonds. As an example, we give an analytic expression of $F_0(G,x)$ for a class of the signed graphs that contain no balanced circuit. Finally, we show that the sets of edges in a signed graph that contain no broken bond form a homogeneous simplicial complex.