Flow polynomials of a signed graph

For a signed graph G and non-negative integer d, it was shown by DeVos et al. 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 Z(2)(d). We define a class of particular directed circuits in G, namely the fundamental directed circuits, 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 partitioned 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 Gamma. Using an extension of Whitney's broken circuit theorem of Dohmen and Trinks, we give a combinatorial interpretation of the coefficients in F-d(G, x) for d = 0 in terms of broken bonds. Finally, we show that the sets of edges in a signed graph that contain no broken bond form a homogeneous simplicial complex.