Graph Coverings and Critical Groups of Signed Graphs and Voltage Graphs

Given a graph covering, there is an induced surjection of critical groups. We will try to determine the structure of the kernel and give results at different levels of generality. A main tool will be dualizing the short exact sequence and working instead with the cokernel. In the case of signed graphs and two-sheeted coverings, we will show that the kernel of the induced surjection can be described using the critical group of another signed graph. This, in particular, gives an interpretation of H. Bai’s computation of the p-group component of the critical group of the n-cube, for p odd.