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Generalizations of chip-firing games


Chip-firing games first occurred independently in physics, economics and mathematics. In the graphical case, we consider a graph with integer number of chips placed on each vertex, allowing each vertex to fire, or give chips to its neighbors, under certain rules. The version that is more often studied is the chip-firing game with a sink vertex, normalizing the total number of chips to zero. Even in the graphical case, one can use a vector for representing the chip configuration and the graph laplacian for firings. Further generalizations include higher dimensional graphs, M-matrices and general invertible matrices. In each of these situations, we are concerned with various notions of stability and the relationships between them--this is related with a physical notion of energy minimization. Given that prior work hasn't been done on general invertible matrices, we worked to examine the extensions of the notions to this case and the consequences of these definitions.


Malvai, Harjasleen, "Generalizations of chip-firing games" (2015). Summer Research Symposium. Brown Digital Repository. Brown University Library.



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