On the sandpile model of modified wheels II

We investigate the abelian sandpile group on modified wheels Wˆn{\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 Wˆn{\hat{W}}_{n} is a direct product of two cyclic subgroups generated by some special configurations. More precisely, the sandpile group on Wˆn{\hat{W}}_{n} is the direct product of two cyclic subgroups of order an{a}_{n} and 3an3{a}_{n} for n even and of order an{a}_{n} and 2an2{a}_{n} for n odd, respectively.