Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T00:00:03.211Z Has data issue: false hasContentIssue false
  • English
  • Français

Signed Chip Firing Games and symmetric Sandpile Models on thecycles∗

Published online by Cambridge University Press:  31 October 2012

Robert Cori ,
Thi Ha Duong Phan  and
Thi Thu Huong Tran
Robert Cori
Affiliation:
Labri, University Bordeaux 1, 33405 Talence Cedex, France.. [email protected]
Thi Ha Duong Phan
Affiliation:
Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay district, Hanoi, Vietnam. ; [email protected]; [email protected]
Thi Thu Huong Tran
Affiliation:
Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay district, Hanoi, Vietnam. ; [email protected]; [email protected]
Get access

Abstract

We investigate the Sandpile Model and Chip Firing Game and an extension of these modelson cycle graphs. The extended model consists of allowing a negative number of chips ateach vertex. We give the characterization of reachable configurations and of fixed pointsof each model. At the end, we give explicit formula for the number of their fixedpoints.

Keywords

Sandpile ModelChip Firing Gamecircular distributionSymmetric Sandpile ModelSigned Chip Firing Gamelattice structure
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 47 , Issue 2 , April 2013 , pp. 133 - 146
Copyright
© EDP Sciences 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bak, P., Tang, C. and Wiesenfeld, K., Self-organized criticality : An explanation of 1/f noise. Phys. Rev. Lett. 59 (1987) 381–284. Google ScholarPubMed
Björner, A., Lovász, L. and Shor, P.W., Chip-firing games on graphs. Eur. J. Combin. 12 (1991) 283–291. Google Scholar
Cori, R. and Rossin, D., On the sandpile group of dual graphs. Eur. J. Combin. 21 (2000) 447–459. Google Scholar
Desel, J., Kindler, E., Vesper, T. and Walter, R., A simplified proof for the self-stabilizing protocol : A game of cards. Inf. Process. Lett. 54 (1995) 327–328. Google Scholar
Dhar, D., Ruelle, P., Sen, S. and Verma, D.-N., Algebraic aspects of abelian sandpile models. J. Phys. A 28 (1995) 805–831. Google Scholar
Formenti, E., Masson, B. and Pisokas, T., Advances in symmetric sandpiles. Fundam. Inf. 76 (2007) 91–112. Google Scholar
Goles, E. and Kiwi, M.A., Games on line graphes and sand piles. Theoret. Comput. Sci. 115 (1993) 321–349. Google Scholar
Goles, E., Morvan, M. and Phan, H.D., Lattice structure and convergence of a game of cards. Ann. Combin. 6 (2002) 327–335. Google Scholar
Goles, E., Morvan, M. and Phan, H.D.. Sandpiles and order structure of integer partitions. Discrete Appl. Math. 117 (2002) 51–64. Google Scholar
Goles, E., Morvan, M. and Phan, H.D., The structure of linear chip firing game and related models. Theoret. Comput. Sci. 270 (2002) 827–841. Google Scholar
Goles, É. and Latapy, M., Clémence Magnien, Michel Morvan and Ha Duong Phan. Sandpile models and lattices : a comprehensive survey. Theoret. Comput. Sci. 322 (2004) 383–407. Google Scholar
Huang, S.-T.. Leader election in uniform rings. ACM Trans. Program. Lang. Syst. 15 (1993) 563–573. Google Scholar
Karmakar, R. and Manna, S.S., Particle hole symmetry in a sandpile model, J. Stat. Mech. 2005 (2005) L01002. Google Scholar
Latapy, M. and Phan, H.D., The lattice structure of chip firing games. Physica D 115 (2001) 69–82. Google Scholar
Phan, H.D., Two sided sand piles model and unimodal sequences. RAIRO – Theor. Inf. Appl. 42 (2008) 631–646. Google Scholar