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 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 ∆ := dI − A. Let the rows of ∆ span the lattice Λ in Z V . The sandpile group of We compute the rank, the exponent and the order of this abelian group and find a cyclic Hall-subgroup of order (d − 1) h . We find that the base (d−1)-logarithm of the exponent and of the order are asymptotically 3h 2 /π 2 and c d (d − 1) h , respectively. We conjecture an explicit formula for the ranks of all Sylow subgroups.
Toumpakari REF discusses some interesting properties of sandpile groups of regular trees where questions related to group rank are studied and the paper is concluded with an interesting conjecture on the rank of all Sylow subgroups of the sandpile group.
On the sandpile group of regular trees
Eur. J. Comb.
Mathematics Computer Science