Chip-firing Games on Graphs
نویسندگان
چکیده
منابع مشابه
Chip-firing Games on Graphs
We analyze the following (solitaire) game: each node of a graph contains a pile of chips, and a move consists of selecting a node with at least as many chips on it as its degree, and let it send one chip to each of its neighbors. The game terminates if there is no such node. We show that the finiteness of the game and the terminating configuration are independent of the moves made. If the numbe...
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We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors (G-parking functions) on graphs as the solution to an energy (or potential) minimization problem and providing an algorithm to efficiently compute reduced divisors. Applications include an “efficient bijective” proof of Kirchhoff’s matrix-tree theorem and a new algorithm for finding ...
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Chip firing is a one-player game where piles start with an initial number of chips and any pile with at least two chips can send one chip to the piles on either side of it. When all of the piles have no more than a single chip, the game ends. In this paper we review fundamental theorems related to this game on a two dimensional number line, including the fact that termination and final configur...
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Let q be a power of a prime and E be an elliptic curve defined over Fq. In [17], the present author examined a sequence of polynomials which express the Nk’s, the number of points on E over the field extensions Fqk , in terms of the parameters q and N1 = #E(Fq). These polynomials have integral coefficients which alternate in sign, and a combinatorial interpretation in terms of spanning trees of...
متن کاملElliptic Curve Groups and Chip-Firing Games
Abstract. The author illustrates several results from the theory of elliptic curves, as well as the theory of chip-firing games on graphs. More specifically, in both of these cases, we obtain analogues of cyclotomic polynomials with several combinatorial and number theoretic properties. We also provide an analysis of zeta functions which highlights the connections between these two disparate fi...
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ژورنال
عنوان ژورنال: European Journal of Combinatorics
سال: 1991
ISSN: 0195-6698
DOI: 10.1016/s0195-6698(13)80111-4