ON THE SANDPILE GROUP OF A FAMILY OF GRAPHS

2018 ◽  
Vol 40 (3) ◽  
pp. 229-240
Author(s):  
Zahid Raza ◽  
Sidra Iqbal
Keyword(s):  
Sandpile Group

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 On the sandpile group of the cone of a graph

2012 ◽  
Vol 436 (5) ◽  
pp. 1154-1176 ◽  
Author(s):  
Carlos A. Alfaro ◽  
Carlos E. Valencia
Keyword(s):  
Sandpile Group

scholarly journals Sandpile group on the graph Dn of the dihedral group

2003 ◽  
Vol 24 (7) ◽  
pp. 815-824 ◽  
Author(s):  
Arnaud Dartois ◽  
Francesca Fiorenzi ◽  
Paolo Francini
Keyword(s):  
Dihedral Group ◽  
Sandpile Group

scholarly journals Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Itamar Landau ◽  
Lionel Levine ◽  
Yuval Peres
Keyword(s):  
Spanning Trees ◽  
Initial Condition ◽  
Simple Process ◽  
Regular Tree ◽  
Single Vertex ◽  
Cyclic Subgroups ◽  
Chip Firing ◽  
International Audience ◽  
Number Of Spanning Trees ◽  
Sandpile Group

International audience The sandpile group of a graph $G$ is an abelian group whose order is the number of spanning trees of $G$. We find the decomposition of the sandpile group into cyclic subgroups when $G$ is a regular tree with the leaves are collapsed to a single vertex. This result can be used to understand the behavior of the rotor-router model, a deterministic analogue of random walk studied first by physicists and more recently rediscovered by combinatorialists. Several years ago, Jim Propp simulated a simple process called rotor-router aggregation and found that it produces a near perfect disk in the integer lattice $\mathbb{Z}^2$. We prove that this shape is close to circular, although it remains a challenge to explain the near perfect circularity produced by simulations. In the regular tree, we use the sandpile group to prove that rotor-router aggregation started from an acyclic initial condition yields a perfect ball.

scholarly journals On the sandpile group of 3×n twisted bracelets

2008 ◽  
Vol 429 (8-9) ◽  
pp. 1894-1904 ◽  
Author(s):  
Jin Shen ◽  
Yaoping Hou
Keyword(s):  
Sandpile Group

scholarly journals The sandpile group of a tree

2009 ◽  
Vol 30 (4) ◽  
pp. 1026-1035 ◽  
Author(s):  
Lionel Levine
Keyword(s):  
Sandpile Group

scholarly journals An Algebraic Analogue of a Formula of Knuth

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Lionel Levine
Keyword(s):  
Directed Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Torsion Subgroup ◽  
Directed Line ◽  
De Bruijn Graphs ◽  
International Audience ◽  
De Bruijn ◽  
Kautz Graphs ◽  
Sandpile Group

International audience We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph $G$ and its directed line graph $\mathcal{L} G$. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when $G$ is regular of degree $k$, we show that the sandpile group of $G$ is isomorphic to the quotient of the sandpile group of $\mathcal{L} G$ by its $k$-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs. Nous généralisons un théorème de Knuth qui relie les arbres couvrants dirigés d'un graphe orienté $G$ au graphe adjoint orienté $\mathcal{L} G$. On peut associer à tout graphe orienté un groupe abélien appelé groupe du tas de sable, et dont l'ordre est le nombre d'arbres couvrants dirigés enracinés en un sommet fixé. Lorsque $G$ est régulier de degré $k$, nous montrons que le groupe du tas de sable de $G$ est isomorphe au quotient du groupe du tas de sable de $\mathcal{L} G$ par son sous-groupe de $k$-torsion. Comme corollaire, nous déterminons les groupes de tas de sable de deux familles de graphes étudiées en informatique: les graphes de de Bruijn et les graphes de Kautz.

scholarly journals On the sandpile group of the square cycle Cn2

2006 ◽  
Vol 418 (2-3) ◽  
pp. 457-467 ◽  
Author(s):  
Yaoping Hou ◽  
Chingwah Woo ◽  
Pingge Chen
Keyword(s):  
Sandpile Group

scholarly journals On the sandpile group of regular trees

2007 ◽  
Vol 28 (3) ◽  
pp. 822-842 ◽  
Author(s):  
Evelin Toumpakari
Keyword(s):  
Sandpile Group
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