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

中文翻译：

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改进型车轮沙堆模型研究 II

摘要 我们通过使用 [NL Biggs, Chip-Firing and the critical group of a graph, J. 代数。梳子。9 (1999), 25–45]。本文给出了一类图上沙堆群的完整结构。特别地，表明 W ˆ n {\hat{W}}_{n} 上的沙堆群是由一些特殊配置生成的两个循环子群的直积。更准确地说，W ˆ n {\hat{W}}_{n} 上的沙堆群是两个阶为 an {a}_{n} 和 3 an 3{a}_{n} 的循环子群的直积对于 n 个偶数和一个 {a}_{n} 和 2 个 2{a}_{n} 分别对于 n 个奇数。