Chip-firing and critical groups of signed graphs

We study chip-firing on a signed graph, employing a general
theory of chip-firing on invertible matrices introduced by Guzmán and
Klivans. This generalization for signed graphs gives rise to notions
of criticality as well as z-superstable configurations, both of which
are counted by the determinant of the signed Laplacian. We establish
general results regarding these critical configurations, focusing on
efficient methods of verifying the underlying properties. We then
study the critical groups of signed graphs in the context of vertex
switching and Smith normal forms. We use this to compute the critical
groups of various classes of signed graphs including signed cycles,
wheels, complete graphs, and fans, in the process generalizing results
of Biggs and others.