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Uniform Bounds for Non-negativity of the Diffusion Game

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I will discuss a variant of the chip-firing game known as the diffusion game. In the diffusion game, we begin with some integer labelling of the vertices of a graph, interpreted as a number of chips on each vertex, and then for each subsequent step every vertex simultaneously fires a chip to each neighbour with fewer chips. In general, this could result in negative vertex labels. In this talk I will answer the following question: do there exist values f(n), for each n, such that whenever we have a graph on n vertices and an initial allocation with at least f(n) chips on each vertex, then the number of chips on each vertex will remain non-negative. I will also consider the possibility of a similar bound g(d) for each d, where d is the maximum degree of the graph.

This talk is part of the Trinity Mathematical Society series.

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