Integral Flow and Cycle Chip-Firing on Graphs

Abstract

Motivated by the notion of chip-firing on the dual graph of a planar graph, we consider ‘integral flow chip-firing’ on an arbitrary graph G. The chip-firing rule is governed by \({\mathcal {L}}^*(G)\), the dual Laplacian of G determined by choosing a basis for the lattice of integral flows on G. We show that any graph admits such a basis so that \({\mathcal {L}}^*(G)\) is an M-matrix, leading to a firing rule on these basis elements that is avalanche finite. This follows from a more general result on bases of integral lattices that may be of independent interest. Our results provide a notion of z-superstable flow configurations that are in bijection with the set of spanning trees of G. We show that for planar graphs, as well as for the graphs \(K_5\) and \(K_{3,3}\), one can find such a flow M-basis that consists of cycles of the underlying graph. We consider the question for arbitrary graphs and address some open questions.

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References

  1. 1.

    R. Bacher, P. De La Harpe, and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France 125, no. 2 (1997), pp. 167–198.

    MathSciNet  Article  Google Scholar 

  2. 2.

    P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality, Phys. Rev. A, 38 (1988), pp. 364–374.

    MathSciNet  Article  Google Scholar 

  3. 3.

    M. Baker and S. Norine, Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math., 215 (2007), pp. 766–788.

    MathSciNet  Article  Google Scholar 

  4. 4.

    M. Baker, F. Shokrieh, Chip-firing games, potential theory on graphs, and spanning trees, J. Combin. Theory Ser. A, 120, Issue 1 (2013), pp. 164–182.

    MathSciNet  Article  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

    A. Björner, L. Lovász, P. W. Shor, Chip-firing games on graphs, European J. Combin., 12 (1991), pp. 283–291.

    MathSciNet  Article  Google Scholar 

  7. 7.

    D. Chebikin and P. Pylyavskyy, A family of bijections between G-parking functions and spanning trees, J. Combin. Theory Ser. A, 110, Issue 1 (2005), pp. 31–41.

    MathSciNet  Article  Google Scholar 

  8. 8.

    R. Cori and D. Rossin, On the Sandpile Group of Dual Graphs, Europ. J. Combinatorics, 21 (2000), pp. 447–459.

    MathSciNet  Article  Google Scholar 

  9. 9.

    S. Corry and D. Perkinson, Divisors and Sandpiles: An Introduction to Chip-Firing, American Mathematical Society, 2018.

    Book  Google Scholar 

  10. 10.

    D. Dhar, Self-organized critical state of sandpile automaton models, Phys. Rev. Lett., 64 (14) (1990), pp. 1613–1616.

    MathSciNet  Article  Google Scholar 

  11. 11.

    Andrei Gabrielov, Abelian avalanches and Tutte polynomials, Phys. A, 195 (1993), no. 1-2, pp. 253–274.

    MathSciNet  Article  Google Scholar 

  12. 12.

    J. Guzmán, C. Klivans, Chip-firing and energy minimization on M-matrices, J. Combin. Theory Ser. A, 132, (2015), pp. 14–31.

    MathSciNet  Article  Google Scholar 

  13. 13.

    B. Jacobson, Critical groups of graphs, unpublished thesis, University of Minnesota.

  14. 14.

    C. Klivans, The Mathematics of Chip-Firing, Chapman & Hall / CRC Press, 2018.

  15. 15.

    C. Merino, Chip firing and the tutte polynomial, Ann. Comb., 1 (1997) pp. 253–259.

    MathSciNet  Article  Google Scholar 

  16. 16.

    R.J. Plemmons, M-matrix characterizations. I. Nonsingular M-matrices, Linear Algebra Appl., 18 (2) (1977), pp. 175–188.

  17. 17.

    W. A. Stein et al., Sage Mathematics Software (Version 9.0), The Sage Development Team, 2020, http://www.sagemath.org.

  18. 18.

    C. H. Yuen, N. Zelesko, personal communication, 2021.

  19. 19.

    M. Wood, The distribution of sandpile groups of random graphs, J. Amer. Math. Soc., 30 (2017), pp. 915–958.

    MathSciNet  Article  Google Scholar 

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Correspondence to Anton Dochtermann.

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Dochtermann, A., Meyers, E., Samavedam, R. et al. Integral Flow and Cycle Chip-Firing on Graphs. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00542-7

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