On the sandpile model of modified wheels II

Author:

Raza Zahid1,Jaradat Mohammed M. M.2,Bataineh Mohammed S.13,Ullah Faiz4

Affiliation:

1. Department of Mathematics, College of Sciences, University of Sharjah , Sharjah , UAE

2. Department of Mathematics, Statistics and Physics, Qatar University , Doha , Qatar

3. Department of Mathematics, Yarmouk University , Irbid , Jordan

4. Department of Mathematics, National University of Computer and Emerging Sciences , Lahore Campus , Pakistan

Abstract

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.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Reference24 articles.

1. N. L. Biggs, Chip-Firing and the critical group of a graph, J. Algebraic Combin. 9 (1999), 25–45.

2. D. Dhar, Self-organized critical state of sandpile automaton models, Phys. Rev. Lett. 64 (1990), no. 14, 1613–1616.

3. P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality: an explanation for the 1/f noise, Phys. Rev. Lett. 59 (1987), 381–384.

4. J. Spencer, Balancing vectors in the max norm, Combinatorica 6 (1986), 55–65.

5. W. Chen and T. Schedler, Concrete and abstract structure of the sandpile group for thick trees with loops, preprint, 2009. ArXiv:math/0701381.

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1. Combinatorial aspects of sandpile models on wheel and fan graphs;European Journal of Combinatorics;2023-05

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