Criteria for Balance in Abelian Gain Graphs

A gain graph (Γ, g, G) is a graph Γ = (V,E) together with a group G, the gain group, and a homomorphism g, the gain function, from the free group F (E) on the edge set E into G. We think of the edges of G as oriented in an arbitrary but fixed way, so that if e is an edge in one direction, then e−1 is the same edge in the opposite direction; thus g(e−1) = g(e)−1. A gain graph is balanced if for every simple closed walk w, g(w) = 1, the identity of G. The problem: how to tell whether or not a gain graph is balanced. One simple criterion is to examine the gains of a fundamental system of circles.[] A circle is the edge set of a nontrivial simple closed walk. If we take a spanning tree T of Γ, each edge e / ∈ T (a cotree edge) belongs to a unique circle in T ∪ e. These circles constitute the fundamental system of circles with respect to T . A circle C can be written as a simple closed walk and its gain taken; this gain is uniquely determined by C up to conjugation and inversion. In particular, it depends only on C, not on the choice of walk, whether g(C) = 1. If g(C) = 1 we say C is balanced. It is easy to show (and well known) that (Γ, g, G) is balanced if and only if every circle of a fundamental system (with respect to some spanning tree) is balanced (see for instance the generalization in [9, Corollary 3.2]). We propose here a much more powerful criterion for balance, whose principal difficulty is that it is not always valid. Our topic is the question of when the criterion is valid. To state it we must first define a ‘cycle’ in a graph. There are two kinds of cycle of concern to us: they are the topological 1-cycles with coefficients either integral (an integral cycle) or in Z2 = Z/2Z (a binary cycle). We write Z1(Γ) for the group of integral cycles and Z1(Γ; Z2) for that of binary cycles. (In graphical terms, a binary cycle is the indicator function of a finite edge set that has even degree at every vertex; thus we may identify Z1(Γ; Z2) with the class of all such edge sets, with symmetric difference as the addition operation.) There are