Chip-Firing and the Critical Group of a Graph
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In this article, a variant of the chip-firing game on a graph is defined and the set of configurations that are stable and recurrent for this game can be given the structure of an abelian group, and the order of the group is equal to the tree number of the graph.Abstract:
A variant of the chip-firing game on a graph is defined. It is shown that the set of configurations that are stable and recurrent for this game can be given the structure of an abelian group, and that the order of the group is equal to the tree number of the graph. In certain cases the game can be used to illuminate the structure of the group.read more
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Journal ArticleDOI
Riemann–Roch and Abel–Jacobi theory on a finite graph
Matthew Baker,Serguei Norine +1 more
TL;DR: In this paper, a graph-theoretic analogue of the Riemann-Roch theorem is presented, and the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph is characterized.
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Tropical curves, their Jacobians and Theta functions
Grigory Mikhalkin,Ilia Zharkov +1 more
TL;DR: In this article, the authors studied Jacobian varieties for tropical curves, which are real tori equipped with integral affine structure and symmetric bilinear form, and defined tropical counterpart of the theta function and established tropical versions of the Abel-Jacobi and Riemann theta divisor theorems.
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Algebraic Potential Theory on Graphs
TL;DR: A motley collection of ideas from several areas of mathematics, including, in no particular order, random walks, the Picard group, exchange rate networks, chip-firing games, cohomology, and the conductance of an electrical network, have been discussed in this paper.
Book ChapterDOI
Chip-Firing and Rotor-Routing on Directed Graphs
TL;DR: In this article, the authors give a self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them.
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On the Sandpile Group of Dual Graphs
Robert Cori,Dominique Rossin +1 more
TL;DR: It is proved that the sandpile group of planar graph is isomorphic to that of its dual, and a combinatorial point of view on the subject is developed.
References
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Book
Distance-Regular Graphs
TL;DR: In this paper, a connected simple graph with vertex set X of diameter d is considered, and the authors define Ri X2 by (x, y) Ri whenever x and y have graph distance.
Journal ArticleDOI
Chip-firing games on graphs
TL;DR: The number of steps in a finite game is related to the least positive eigenvalue of the Laplace operator of the graph to show that the finiteness of the game and the terminating configuration are independent of the moves made.
Journal ArticleDOI
Algebraic Potential Theory on Graphs
TL;DR: A motley collection of ideas from several areas of mathematics, including, in no particular order, random walks, the Picard group, exchange rate networks, chip-firing games, cohomology, and the conductance of an electrical network, have been discussed in this paper.
Journal ArticleDOI
Recursive families of graphs
TL;DR: A recursive family of graphs is defined as a sequence of graphs whose Tutte polynomials satisfy a homogeneous linear recurrence relation as discussed by the authors, and necessary conditions for a family to be recursive are proved, and the theory is applied to the families of graphs known as the prisms and the Mobius ladders.