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.
Similar content being viewed by others
References
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.
P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality, Phys. Rev. A, 38 (1988), pp. 364–374.
M. Baker and S. Norine, Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math., 215 (2007), pp. 766–788.
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.
N. Biggs, Chip-firing and the critical group of a graph, J. Algebraic Combin., 9 (1999), pp. 25–45.
A. Björner, L. Lovász, P. W. Shor, Chip-firing games on graphs, European J. Combin., 12 (1991), pp. 283–291.
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.
R. Cori and D. Rossin, On the Sandpile Group of Dual Graphs, Europ. J. Combinatorics, 21 (2000), pp. 447–459.
S. Corry and D. Perkinson, Divisors and Sandpiles: An Introduction to Chip-Firing, American Mathematical Society, 2018.
D. Dhar, Self-organized critical state of sandpile automaton models, Phys. Rev. Lett., 64 (14) (1990), pp. 1613–1616.
Andrei Gabrielov, Abelian avalanches and Tutte polynomials, Phys. A, 195 (1993), no. 1-2, pp. 253–274.
J. Guzmán, C. Klivans, Chip-firing and energy minimization on M-matrices, J. Combin. Theory Ser. A, 132, (2015), pp. 14–31.
B. Jacobson, Critical groups of graphs, unpublished thesis, University of Minnesota.
C. Klivans, The Mathematics of Chip-Firing, Chapman & Hall / CRC Press, 2018.
C. Merino, Chip firing and the tutte polynomial, Ann. Comb., 1 (1997) pp. 253–259.
R.J. Plemmons, M-matrix characterizations. I. Nonsingular M-matrices, Linear Algebra Appl., 18 (2) (1977), pp. 175–188.
W. A. Stein et al., Sage Mathematics Software (Version 9.0), The Sage Development Team, 2020, http://www.sagemath.org.
C. H. Yuen, N. Zelesko, personal communication, 2021.
M. Wood, The distribution of sandpile groups of random graphs, J. Amer. Math. Soc., 30 (2017), pp. 915–958.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Kolja Knauer.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dochtermann, A., Meyers, E., Samavedam, R. et al. Integral Flow and Cycle Chip-Firing on Graphs. Ann. Comb. 25, 595–616 (2021). https://doi.org/10.1007/s00026-021-00542-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-021-00542-7