scholarly journals Minimal Recurrent Configurations of Chip Firing Games and Directed Acyclic Graphs

2010 ◽  
Vol DMTCS Proceedings vol. AL,... (Proceedings) ◽  
Author(s):  
Matthias Schulz
Keyword(s):  
Lower Bound ◽  
Close Relation ◽  
Directed Acyclic Graphs ◽  
Sandpile Model ◽  
Abelian Sandpile Model ◽  
Acyclic Graphs ◽  
Abelian Sandpile ◽  
Chip Firing ◽  
International Audience

International audience We discuss a very close relation between minimal recurrent configurations of Chip Firing Games and Directed Acyclic Graphs and demonstrate the usefulness of this relation by giving a lower bound for the number of minimal recurrent configurations of the Abelian Sandpile Model as well as a lower bound for the number of firings which are caused by the addition of two recurrent configurations on particular graphs.

scholarly journals Results and conjectures on the Sandpile Identity on a lattice

2003 ◽  
Vol DMTCS Proceedings vol. AB,... (Proceedings) ◽  
Author(s):  
Arnaud Dartois ◽  
Clémence Magnien
Keyword(s):  
Finite Lattice ◽  
Sandpile Model ◽  
Abelian Sandpile Model ◽  
Infinite Lattice ◽  
Abelian Sandpile ◽  
International Audience ◽  
Burning Algorithm

International audience In this paper we study the identity of the Abelian Sandpile Model on a rectangular lattice.This configuration can be computed with the burning algorithm, which, starting from the empty lattice, computes a sequence of configurations, the last of which is the identity.We extend this algorithm to an infinite lattice, which allows us to prove that the first steps of the algorithm on a finite lattice are the same whatever its size.Finally we introduce a new configuration, which shares the intriguing properties of the identity, but is easier to study.

scholarly journals On the sandpile model of modified wheels II

2020 ◽  
Vol 18 (1) ◽  
pp. 1531-1539
Author(s):  
Zahid Raza ◽  
Mohammed M. M. Jaradat ◽  
Mohammed S. Bataineh ◽  
Faiz Ullah
Keyword(s):  
Direct Product ◽  
Critical Group ◽  
Complete Structure ◽  
Sandpile Model ◽  
Cyclic Subgroups ◽  
Algebr Comb ◽  
Abelian Sandpile ◽  
Chip Firing ◽  
Sandpile Group

Abstract We investigate the abelian sandpile group on modified wheels {\hat{W}}_{n} by using a variant of the dollar game as described in [N. L. Biggs, Chip-Firing and the critical group of a graph, J. Algebr. Comb. 9 (1999), 25–45]. The complete structure of the sandpile group on a class of graphs is given in this paper. In particular, it is shown that the sandpile group on {\hat{W}}_{n} is a direct product of two cyclic subgroups generated by some special configurations. More precisely, the sandpile group on {\hat{W}}_{n} is the direct product of two cyclic subgroups of order {a}_{n} and 3{a}_{n} for n even and of order {a}_{n} and 2{a}_{n} for n odd, respectively.

scholarly journals Multiple and inverse topplings in the Abelian Sandpile Model

2012 ◽  
Vol 212 (1) ◽  
pp. 23-44 ◽  
Author(s):  
S. Caracciolo ◽  
G. Paoletti ◽  
A. Sportiello
Keyword(s):  
Sandpile Model ◽  
Abelian Sandpile Model ◽  
Abelian Sandpile

scholarly journals Avalanches and waves in the Abelian sandpile model

1997 ◽  
Vol 56 (4) ◽  
pp. R3745-R3748 ◽  
Author(s):  
Maya Paczuski ◽  
Stefan Boettcher
Keyword(s):  
Sandpile Model ◽  
Abelian Sandpile Model ◽  
Abelian Sandpile
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