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.
The sandpile model on the complete split graph, Motzkin words, and tiered parking functions
