Sandpile Group Research Articles

On the sandpile group of [formula omitted

The sandpile group of a graph is a refinement of the number of spanning trees of the graph and is closely connected to the graph Laplacian. In this note the sandpile group of P 4 × C n is proved to be always the direct sum of three cyclic groups with even orders and an exact formula for the order of these cyclic groups is also given.

On the sandpile group of the square cycle [formula omitted

The sandpile group of a graph is a refinement of the number of spanning trees of the graph and is closely connected with the graph Laplacian. In this paper, the structure of the sandpile group on the square cycle C n 2 is determined and it is shown that the Smith normal form of the sandpile group is always the direct sum of two or three cyclic groups.

On the sandpile group of regular trees

The sandpile group of a connected graph is the group of recurrent configurations in the abelian sandpile model on this graph. We study the structure of this group for the case of regular trees. A description of this group is the following: Let T ( d , h ) be the complete d -regular tree of depth h and let V be the set of its vertices. Denote the adjacency matrix of T ( d , h ) by A and consider the modified Laplacian matrix Δ ≔ d I − A . Let the rows of Δ span the lattice Λ in Z V . The sandpile group G ( d , h ) of T ( d , h ) is Z V / Λ . We compute the rank, the exponent, the order, and other structural parameters of the abelian group G ( d , h ) . We find a cyclic Hall-subgroup of order ( d − 1 ) h . We prove that the rank of G ( d , h ) is ( d − 1 ) h and that G ( d , h ) contains a subgroup isomorphic to Z d ( d − 1 ) h ; therefore, for all primes p dividing d , the rank of the Sylow p -subgroup is maximal (equal to the rank of the entire group). We find that the base - ( d − 1 ) logarithm of the exponent and of the order are asymptotically 3 h 2 / π 2 and c d ( d − 1 ) h , respectively. We conjecture an explicit formula for the ranks of all Sylow subgroups.

Sandpile group on the graph [formula omitted] of the dihedral group

In this paper we study the structure of the Abelian sandpile group on the Cayley graph D n of the dihedral group D n =〈 a, b∣ a n = b 2=( ab) 2=1〉. We prove that the Smith normal form of the sandpile group is not cyclic as one can generally expect but is always the direct product of two or three cyclic groups. We conclude by considering some particular cases.

A chip-firing game and Dirichlet eigenvalues

We consider a variation of the chip-firing game in an induced subgraph S of a graph G. Starting from a given chip configuration, if a vertex v has at least as many chips as its degree, we can fire v by sending one chip along each edge from v to its neighbors. Chips are removed at the boundary δ S. The game continues until no vertex can be fired. We will give an upper bound, in terms of Dirichlet eigenvalues, for the number of firings needed before a game terminates. We also examine the relations among three equinumerous families, the set of spanning forests on S with roots in the boundary of S, a set of “critical” configurations of chips, and a coset group, called the sandpile group associated with S.

On the identity of the sandpile group

The sandpile automaton was introduced by physicists in 1987. This cellular automaton presents the same critical exponents as some natural systems. Moreover, this is the simplest model known. Dhar et al. show that the recurrent configurations of this automaton form an finite abelian group. We study here the identity of this group if the automaton is defined on rectangular grids. We prove that the identity on such grids of size p× q with p> 2 q can be decomposed into three parts. This proof is based on the study of the toppling of a single pile of chips on an infinite grid filled uniformly. Résumé L'automate cellulaire du Tas de Sable, créé en 1987 par les physiciens Bak, Tang et Wiesenfeld est le système le plus simple rendant compte de certains phénomènes physiques critiques tels les tremblements de terre. En 1991 Dhar et al. montrent que l'on peut munir les configurations récurrentes de ce système d'une structure de groupe abélien fini. Dans cet article nous étudions la structure de l'identité de ce groupe. Nous démontrons que pour des grilles de taille p× q où q> 2 p , alors l'identité de ce groupe peut être décomposée en trois zones dont la centrale est uniforme. Cette étude repose sur le problème auxiliaire de l’éboulement d'une pile unique sur une grille infinie uniforme.

On the Sandpile Group of Dual Graphs

The group of recurrent configurations in the sandpile model, introduced by Dhar , may be considered as a finite abelian group associated with any graph G; we call it the sandpile group of G. The aim of this paper is to prove that the sandpile group of planar graph is isomorphic to that of its dual. A combinatorial point of view on the subject is also developed.