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Enumerating linear systems on graphs


The divisor theory of graphs views a finite connected graph G as a discrete version of a Riemann surface. Divisors on G are formal integral combinations of the vertices of G, and linear equivalence of divisors is determined by the discrete Laplacian operator for G. As in the case of Riemann surfaces, we are interested in the complete linear system |D| of a divisor D—the collection of nonnegative divisors linearly equivalent to D. Unlike the case of Riemann surfaces, the complete linear system of a divisor on a graph is always finite. We compute generating functions encoding the sizes of all complete linear systems on G and interpret our results in terms of polyhedra associated with divisors and in terms of the invariant theory of the (dual of the) Jacobian group of G. If G is a cycle graph, our results lead to a bijection between complete linear systems and binary necklaces. Our results also apply to a model in which the Laplacian is replaced by an invertible, integral M-matrix.

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  1. 1.

    For \(q'\in V\), writing \(D+kq = D+kq'+k(q-q')\) shows the dependence is “periodic” with period equal to the order of \([q-q']\in {{\,\mathrm{Jac}\,}}(G)\).

  2. 2.

    We often write \(\omega ^n\) instead of 1 for consistency of notation.

  3. 3.

    This switch in the placement of q was made in order to conform to the conventions for root systems considered in [6]. See Sect. 7.1.1, below.

  4. 4.

    Note that our convention for the Laplacian of a graph differs from that in [6] by a transpose.

  5. 5.

    For this section, in addition to [6], see the work by Gaetz [11].


  1. 1.

    Bacher, R., de la Harpe, P., Nagnibeda, T.: The lattice of integral flows and the lattice of integral cuts on a finite graph. Bull. Soc. Math. France 125(2), 167–198 (1997)

  2. 2.

    Bak, P., Tang, C., Weisenfeld, K.: Self-organized criticality: an explanation of \(1/f\) noise. Phys. Rev. Lett. 59(4), 381–384 (1987)

  3. 3.

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

  4. 4.

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

  5. 5.

    Beck, Matthias, Robins, Sinai: Computing the continuous discretely, second ed., Undergraduate Texts in Mathematics, Springer, New York, 2015, Integer-point enumeration in polyhedra, With illustrations by David Austin

  6. 6.

    Benkart, Georgia, Klivans, Caroline, Reiner, Victor: Chip firing on Dynkin diagrams and McKay quivers. Math. Z. 290(1–2), 615–648 (2018)

  7. 7.

    Björner, A., Lovász, L., Shor, P.W.: Chip-firing games on graphs. Eur. J. Combin. 12(4), 283–291 (1991)

  8. 8.

    Corry, Scott, Perkinson, David: Divisors and sandpiles, American Mathematical Society, Providence, RI, 2018, An introduction to chip-firing

  9. 9.

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

  10. 10.

    Gabrielov, A.: Asymmetric abelian avalanches and sandpile, preprint 93–65. Cornell University, MSI (1993)

  11. 11.

    Gaetz, Christian: Critical groups of group representations. Linear Algebra Appl. 508, 91–99 (2016)

  12. 12.

    Gathmann, Andreas, Kerber, Michael: A Riemann–Roch theorem in tropical geometry. Math. Z. 259(1), 217–230 (2008)

  13. 13.

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

  14. 14.

    Haase, Christian, Musiker, Gregg, Josephine, Yu.: Linear systems on tropical curves. Math. Z. 270(3–4), 1111–1140 (2012)

  15. 15.

    Holroyd, A.E., Levine, L., Mészáros, K., Peres, Y., Propp, J., Wilson, D.B.: Chip-firing and rotor-routing on directed graphs, In and Out of Equilibrium II (V. Sidoravicius and M. E. Vares, eds.), Progress in Probability, vol. 60, Birkhauser, pp. 331–364 (2008)

  16. 16.

    Klivans, Caroline J.: The mathematics of chip-firing, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, (2019)

  17. 17.

    Lorenzini, D.J.: Arithmetical graphs. Math. Ann. 285(3), 481–501 (1989)

  18. 18.

    Mikhalkin, Grigory, Zharkov, Ilia: Tropical curves, their Jacobians and theta functions, Curves and abelian varieties, Contemp. Math., vol. 465, Amer. Math. Soc., Providence, RI, pp. 203–230 (2008)

  19. 19.

    Oh, Suho, Park, Jina: Necklaces and slimes, arXiv:1904.11046, (2019)

  20. 20.

    Perkinson, David, Perlman, Jacob, Wilmes, John: Primer for the algebraic geometry of sandpiles, Tropical and non-Archimedean geometry, Contemp. Math., vol. 605, Am. Math. Soc., Providence, RI, pp. 211–256 ( 2013)

  21. 21.

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

  22. 22.

    Postnikov, Alexander, Shapiro, Boris: Trees, parking functions, syzygies, and deformations of monomial ideals. Trans. Am. Math. Soc. 356(8), 3109–3142 (2004)

  23. 23.

    Sloane, N.J.A.: The on-line encyclopedia of integer sequences, Accessed 7 Jan 2016

  24. 24.

    Richard, P.: Stanley, Invariants of finite groups and their applications to combinatorics. Bull. Am. Math. Soc. (N.S.) 1(3), 475–511 (1979)

  25. 25.

    Sturmfels, Bernd: Algorithms in invariant theory. Texts and monographs in symbolic computation, 2nd edn. Springer, Wien, NewYork, Vienna (2008)

  26. 26.

    The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.2), (2018), Accessed 7 Jan 2016

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This work was partially supported by a Reed College Science Research Fellowship and by the Reed College Summer Scholarship Fund. The first author is supported by the NSF Graduate Research Fellowship Program under Grant No. 00074041. We thank Gopal Goel, Gregg Musiker, and Vic Reiner for helpful discussions. We thank Scott Corry and an anonymous referee for their comments. We would also like to acknowledge our extensive use of the mathematical software SageMath [26] and the On-line Encyclopedia of Integer Sequences [23].

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Correspondence to Sarah Brauner.

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Brauner, S., Glebe, F. & Perkinson, D. Enumerating linear systems on graphs. Math. Z. (2020).

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  • Divisor theory of graphs
  • Complete linear system
  • Chip-firing
  • Graph Laplacian
  • Binary necklaces
  • M-matrix

Mathematics Subject Classification

  • Primary 05C30
  • Secondary 05C25