Chip-Firing on Signed Graphs
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Graphical chip-firing is a process where ‘chips’ are exchanged between vertices of a graph, the dynamics of which are governed by the graph Laplacian. Chip-firing is a well-developed field with applications to physics, computer science, and many areas of mathematics.
Guzmán and Klivans introduced a generalization of the graphical chip-firing model such that the dynamics can be governed by any invertible matrix. In this model, the set of allowable configurations is described by the lattice points of a rational convex cone given by a choice of M-matrix and the notions of criticality and superstability from classical chip-firing have analogues. A signed graph is a generalization of a simple graph with edges assigned to be either positive or negative. Signed graphs were first introduced by Harary in the context of social psychology, further studied by Zaslavsky, who investigated their matroidal properties, and have uses in a wide range of fields from data science to ecology.
Here we study the chip-firing model on signed graphs that results from applying the Guzmán-Klivans theory to the invertible signed graph Laplacian and the M-matrix graph Laplacian of the underlying graph. We investigate the behavior of this model and develop tools to compute examples.