On the sandpile model of modified wheels II

Abstract

: We investigate the abelian sandpile group on modified wheels Wˆn 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 Wˆn is a direct product of two cyclic subgroups generated by some special configurations. More precisely, the sandpile group on Wˆn is the direct product of two cyclic subgroups of order an and 3an for n even and of order an and 2an for n odd, respectively.