Balance in gain graphs - A spectral analysis

A gain graph is a graph where the edges are given some orientation and labeled with the elements (called gains) from a group so that gains are inverted when we reverse the direction of the edges. A signed graph with labels from the multiplicative group {1,-1} on the edges can be taken as a particular case of a gain graph. In this article, we initiate a matrix analysis of gain graphs by defining the adjacency matrices of gain graphs when the underlying group for labeling the edges is the multiplicative group of a field and characterize balance in such gain graphs using the characteristic polynomials. We also establish recurrence relations for the characteristic polynomials of a gain graph and discuss consequences of these relations.