A gain graph is a graph with edges orientably labelled from a group. It is balanced if the product of gains (the labels) around any circle is the identity (we call the circle balanced). Under set sum the circles, regarded as edge sets, generate a binary vector space, the binary cycle space. For which graphs Gamma is it true that, whenever you have a gain graph based on Gamma such that there exists a basis of the binary cycle space that is composed of balanced circles, then the gain graph is balanced? The answer is intriguing in that it is complicated but exactly solvable and it gives a special role to the group Z3.
This is joint work with Konstantin Rybnikov.