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Chip-Firing Games and Critical Groups

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The critical group—a finite abelian group hidden in every connected graph—governs spanning trees, random graph statistics, and algebraic geometry, and the paper shows undergraduates how to explore it.

desk verdict A useful, well-written survey of critical groups for undergraduates, but it states a false theorem about C_n(1,2) and has a sign error in the random-graph Laplacian construction. read the letter →

arxiv 1908.04395 v1 pith:PE4V6FY3 submitted 2019-08-12 math.CO

Darren Glass , Nathan Kaplan This is my paper · ORCID
classification math.CO MSC 05C5005C2505C8015A36
keywords chip-firingcriticalgroupsandpilegraphLaplacianSmithnormalformspanningtreesrandomgraphsarithmeticalstructures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper makes a teaching-and-research case for a single object: every finite connected graph carries a finite abelian group, its critical group, which can be built from a simple chip-firing game on its vertices. The authors develop the group from first principles—as the torsion part of the Laplacian's cokernel, as degree-zero divisors modulo chip-firing, and via Smith normal form—and then show how much information it condenses. Its order counts the graph's spanning trees, it behaves predictably under duality, wedging, and subdivision, and for large random graphs its Sylow $p$-subgroups converge to a universal distribution independent of edge probability. The paper also surveys arithmetical structures, a generalization with its own critical groups, and closes with open problems pitched at undergraduates. The sympathetic reader takes away that the critical group is a real organizing idea connecting combinatorics, algebraic geometry, and probability, not just a classroom curiosity.

What carries the argument

The central object is the critical group $K(G) = \operatorname{tors}(\operatorname{cok}(L(G)))$, where $L(G) = D - A$ is the combinatorial Laplacian; equivalently, $K(G)$ is the group of degree-zero divisors on the graph modulo chip-firing moves. The identity doing the load-bearing work is the Smith normal form of $L(G)$—the diagonal form obtained by unimodular row and column operations—which makes the invariant factors of $K(G)$ explicit and, together with the Matrix Tree Theorem, identifies the group's order with the number of spanning trees. Supporting machinery includes $q$-reduced divisors as unique representatives of each class, verified efficiently by the burning algorithm, and the monodromy pairing, a perfect symmetric bilinear pairing on $K(G)$ used in the random-graph distribution formulas.

What would settle it

For any connected graph, compute the Smith normal form of the reduced Laplacian and compare the order of the torsion with the number of spanning trees; a single graph where these differ would break the chain that identifies $|K(G)|$ with the spanning tree count. A second check: simulate many random graphs on $n$ vertices, estimate the probability that $K(G)$ has a trivial Sylow $2$-subgroup, and compare with the claimed constant $\prod_{k\ge 0}(1 - 2^{-2k-1}) \approx 0.4194$; a substantial discrepancy at large $n$ would falsify the stated universal distribution.

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Extended reading notes

Core claim

The paper claims that the critical group $K(G)$, defined as the torsion subgroup of the cokernel of the graph Laplacian $L(G) = D - A$ and equivalently as degree-zero divisors modulo chip-firing equivalence, is a finite abelian group that encodes essential information about the graph. The surveyed results include: the order of $K(G)$ equals the number of spanning trees; the group is invariant under planar duality, decomposes as a direct sum over wedge sums, and has rank at most the graph's genus; subdividing every edge by $k$ multiplies each invariant factor by $k$. Every finite abelian group occurs as some critical group when multiple edges are allowed, while simple graphs are much more restrictive. For random graphs with a fixed edge probability, the asymptotic distribution of each Sylow $p$-subgroup of $K(G)$ is a universal constant that does not depend on the edge probability. The paper also introduces arithmetical structures, a generalization in which the diagonal entries of the Laplacian vary, and derives a spanning-tree-sum formula for the order of their critical groups together with exact counts on paths and cycles. The chapter is organized around exercises and open 'research projects,' with the explicit thesis that undergraduates can contribute to the subject.

Load-bearing premise

The paper's promise that a student with only linear algebra and group theory can follow the entire exposition, since several later sections depend on advanced number theory and probability that the chapter does not develop.

Editorial extensions

If this is right

  • The order of the critical group equals the spanning tree count, so the two quantities must agree for every connected graph, giving a fast check in any computation.
  • Critical groups are invariant under planar duality and decompose over wedge sums, so graph constructions can be studied through their effect on this group.
  • For random graphs with a fixed edge probability, the Sylow $p$-subgroup distribution approaches a universal limit independent of the edge probability, so different random graph models share the same asymptotic prime-power statistics.
  • Every finite abelian group occurs as a critical group when multiple edges are allowed, while simple graphs exclude groups like $(\mathbb{Z}/2\mathbb{Z})^k$ for large $k$, leaving sharp realizability questions open.
  • Arithmetical structures produce a family of critical groups per graph, and on paths and cycles their orders have closed formulas (Catalan and binomial), showing the enumeration is tractable for structured families.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, this suggests a testable course design: the critical group could serve as a running example linking linear algebra, group theory, and probability in one semester, with the p-adic sections deferred to a follow-up course.
  • Beyond the paper, the universality result invites computational experiments in other random graph models—bipartite, regular, or threshold graphs—to see whether the same Sylow $p$-subgroup distribution appears; some deviations are already conjectured.
  • Beyond the paper, the smoothing operations for arithmetical structures hint that exact enumeration may extend to graph families with tree-like skeletons or small treewidth, where recursive smoothing could yield closed forms beyond paths and cycles.
  • Beyond the paper, the realizability questions suggest an algorithmic companion problem: given a finite abelian group, find the smallest simple graph whose critical group is that group, which could be attacked by search over bounded-genus families.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This expository survey introduces the critical group of a finite connected graph, defined as the torsion subgroup of the cokernel of the Laplacian and equivalently as degree-zero divisors modulo chip-firing. It develops basic definitions, Smith normal form computations, spanning trees, graph operations, realizability questions, generators, random graph results, the monodromy pairing, divisor rank and gonality, directed graphs, and arithmetical structures. The paper is aimed at undergraduates with linear algebra and group theory, and it contains many worked examples, exercises, research projects, and an explicit highlighting of undergraduate contributions.

Significance. If its statements are corrected, this would be a valuable and unusual resource: it collects a coherent set of results, gives reproducible Smith normal form computations, and carefully attributes theorems to the literature. The pedagogical design, with concrete worked examples, exercises, and open research projects, is a real strength. However, the survey's reliability as a reference for its target audience is compromised by at least one false stated theorem and an incorrect random-matrix model. These issues need to be fixed before the survey can be recommended without qualification.

major comments (2)
  1. [§1.3, Theorem 3] Theorem 3 is false as stated. For n = 4, C4(1,2) is the complete graph K4, whose critical group is Z/4 ⊕ Z/4 (Smith normal form diag(1,4,4), order 16). The formula in Theorem 3, with F4 = 3 and d = gcd(4,3) = 1, gives Z/3 ⊕ Z/12, which has order 36. For n = 3, C3(1,2) is K3 and has critical group Z/3, while the formula predicts Z/2 ⊕ Z/6. Thus the theorem needs a corrected statement or an explicit hypothesis, and the attribution to the cited references should be rechecked. Because this is a stated result in a survey whose purpose is to present accurate mathematics to undergraduates, the error is load-bearing.
  2. [§1.9, random-matrix model] The claimed equivalence between random graphs and the described random matrix process is incorrect. With A a 0/1 symmetric zero-diagonal adjacency matrix, setting D = −diag(row sums) makes D − A = −Δ − A, which is not the Laplacian Δ − A and is not unimodularly equivalent to it. For example, if G = K4, the matrix obtained by deleting the last row and column of −Δ − A has determinant −20, so its cokernel has order 20, whereas K(K4) has order 16. The sign of D should presumably be positive, or an equivalent correction must be made; as written, the construction and the statements that depend on it, including the motivating discussion before Wood's theorem, are incorrect.
minor comments (3)
  1. [§1.7] The sentence 'Combining Theorem 4 and Theorem 7 implies...' should refer to Corollary 4 (K(Cn) = Z/nZ) and Theorem 7; Theorem 4, the uniqueness of q-reduced divisors, is not used in this implication.
  2. [Suggested prerequisites] Sections 1.9–1.11 invoke p-adic integers, Haar measure, universality of cokernels of random matrices, and Brill-Noether theory without development; the accessibility claim should be softened or these sections should be explicitly marked as requiring more background than linear algebra and group theory.
  3. [§1.9] In the display giving the probability that the Sylow 2-subgroup is trivial, the product mixes indices ('∏_{k≥0}(1 − 2^{−2i−1})'); the index should be uniform.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the survey is expository, and its substantive results are either proved in-line or attributed to external prior work.

full rationale

Walking the paper's derivation chain, the substantive claims are not predictions forced by fitted inputs or by self-referential definitions. The critical group is defined as the torsion of cok(L), and Proposition 2 identifies it with Div0 modulo chip-firing by a direct argument; Corollary 3 then combines this definition with the Matrix Tree Theorem, which is cited as an independent classical result. Later results are explicitly attributed to external sources: Theorem 4 to Baker and Norine, Theorem 6 to Cori and Rossin, Theorem 14 and the universality discussion to Wood, Theorem 16 to Shokrieh, Theorem 18 to Deveau–Jensen–Kainic–Mitropolsky, Theorem 19 to Dutta and Jensen, and Theorems 20–23 to Lorenzini, Braun et al., and related papers. The self-citations, such as [16], [22], and [39], point to genuine prior peer-reviewed publications rather than to the present text, and the survey does not present any fitted values as predictions. Sections that invoke p-adic measures, Haar measure, or Brill–Noether theory explicitly defer to the literature, so these are acknowledged background limitations rather than circular inputs. The apparent conflict between Theorem 3 and Exercise 6, including the n=4 counterexample to the printed formula, is a correctness issue in the survey's reporting, not a reduction of a claim to its own input, and therefore does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper contributes no free parameters or invented entities. Its content is assembled from standard theorems and cited literature; the only non-standard assumption is GRH, which the authors explicitly flag. The axioms listed are the principal external results on which the survey's statements rely.

assumptions (6)
  • standard math Smith normal form over Z: every integer matrix is equivalent to a diagonal matrix d with d_i dividing d_{i+1} and zero tail; the cokernel is read off from d.
    Used throughout Section 1.3 (Proposition 3) to compute critical groups; cited to [30, Theorem 2.33].
  • standard math Matrix-Tree theorem: the number of spanning trees of G equals |det(L_{i,j})| for the reduced Laplacian.
    Used in Section 1.5 (Theorem 5 and Corollary 3) to identify |K(G)| with the number of spanning trees; cited to Biggs [11] and related references.
  • domain assumption Baker-Norine theorem: every divisor class on a graph contains a unique q-reduced divisor.
    Used in Section 1.4 (Theorem 4) to justify canonical representatives for critical group elements; cited to [6, Prop 3.1].
  • domain assumption Wood's universality theorem for cokernels of random symmetric p-adic matrices.
    Used in Section 1.9 as Theorem 14 for the limiting distribution of Sylow p-subgroups of critical groups of Erdős-Rényi graphs; cited to [68].
  • domain assumption Generalized Riemann Hypothesis.
    Explicitly assumed in Theorem 17, Section 1.10, for the realization of finite abelian groups with pairing as Jacobians of graphs; this is an unproved conjecture.
  • standard math Lorenzini's theorem: a fixed finite connected graph has only finitely many arithmetical structures.
    Used in Section 2.2 as the basis for counting arithmetical structures; cited to [52].

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Pith. "Pith review of Chip-Firing Games and Critical Groups." pith.science (2026). https://pith.science/paper/PE4V6FY3

@misc{pith2026190804395,
  author       = {Pith},
  title        = {Pith review of: Chip-Firing Games and Critical Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PE4V6FY3}},
  note         = {Machine review of arXiv:1908.04395}
}
read the original abstract

In this expository article intended to be accessible to undergraduate students we introduce a finite abelian group that can be associated to any finite connected graph. This group can be defined in an elementary combinatorial way in terms of chip-firing operations, and has been an object of interest in combinatorics, algebraic geometry, statistical physics, and several other areas of mathematics. We will begin with basic definitions and examples and develop a number of properties that can be derived by looking at this group from different angles. Throughout, we will give exercises, some of which are straightforward and some of which are open questions. We will also attempt to highlight some of the many contributions to this area made by undergraduate students

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Reference graph

Works this paper leans on

69 extracted references · 67 canonical work pages

  1. [1]

    Alfaro and Carlos E

    Carlos A. Alfaro and Carlos E. V alencia, On the sandpile group of the cone of a graph , Linear Algebra Appl. 436 (2012), no. 5, 1154–1176

    work page 2012

  2. [2]

    Omid Amini and Janne Kool, A spectral lower bound for the divisorial gonality of metric graphs, Int. Math. Res. Not. IMRN (2016), no. 8, 2423–2450

    work page 2016

  3. [3]

    Sigma 2 (2014), e24, 25 pp

    Y ang An, Matthew Baker, Greg Kuperberg, and Farbod Shokri eh, Canonical representatives for divisor classes on tropical curves and the matrix-tree t heorem, Forum Math. Sigma 2 (2014), e24, 25 pp

    work page 2014

  4. [4]

    Kassie Archer, Abby Bishop, Alexander Diaz Lopez, Luis Da vid Garc´ ıa Puente, Darren Glass, and Joel Louwsma, Arithmetical structures on bidents, https://arxiv.org/abs/1903.01393, 2019

    work page Pith review arXiv 1903

  5. [5]

    Arash Asadi and Spencer Backman, Chip-firing and Riemann-Roch theory for directed graphs, https://arxiv.org/abs/1012.0287v2, (2011)

    work page Pith review arXiv 2011

  6. [6]

    Matthew Baker and Serguei Norine, Riemann-Roch and Abel-Jacobi theory on a finite graph , Adv. Math. 215 (2007), no. 2, 766–788

    2007

  7. [7]

    Matthew Baker and Serguei Norine, Harmonic morphisms and hyperelliptic graphs, Int. Math. Res. Not. IMRN (2009), no. 15, 2914–2955

    work page 2009

  8. [8]

    Matthew Baker and Farbod Shokrieh, Chip-firing games, potential theory on graphs, and spanning trees, J. Comb. Theory, Series A, 120 (2013), no. 1, 164–182

    work page 2013

Show all 69 references
  1. [9]

    Ryan Becker and Darren Glass, Cyclic Critical Groups of Graphs , Austral. Jour. of Comb. 64 (2016), 366–375

    2016

  2. [10]

    Andrew Berget, Andrew Manion, Molly Maxwell, Aaron Potechi n, and Victor Reiner, The critical group of a line graph , Ann. Comb. 16 (2012), no. 3, 449–488

    2012

  3. [11]

    Second edition

    Norman Biggs, Algebraic Graph Theory . Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1993. viii+205 pp

    1993

  4. [12]

    N. L. Biggs, Chip-firing and the critical group of a graph , J. Algebraic Combin. 9 (1999), no. 1, 25–45

    1999

  5. [13]

    Algebraic Com- bin

    Anders Bj¨ orner and L´ aszl´ o Lov´ asz,Chip-firing games on directed graphs , J. Algebraic Com- bin. 1 (1992), no. 4, 305–328

    1992

  6. [14]

    Siegfried Bosch and Dino Lorenzini, Grothendieck’s pairing on component groups of Jaco- bians, Invent. Math. 148 (2002), no. 2, 353–296

    2002

  7. [15]

    David Brandfonbrener, Pat Devlin, Netanel Friedenberg, Y u xuan Ke, Steffen Marcus, Henry Reichard, and Ethan Sciamma, Two-vertex generators of Jacobians of graphs, Electr. J. Comb. 25 (2018), P1.15

    2018

  8. [16]

    Martin, Gregg Musiker, and Carlos E

    Benjamin Braun, Hugo Corrales, Scott Corry, Luis David G arc´ ıa Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, and Carlos E. V alen cia, Counting arithmetical structures on paths and cycles , Discrete Math. 341 (2018), no. 10, 2949–2963

    2018

  9. [17]

    Brown, Jackson S

    Morgan V . Brown, Jackson S. Morrow, and David Zureick-Br own, Chip-firing groups of iter- ated cones, Linear Algebra Appl. 556 (2018), 46–54

    2018

  10. [18]

    Chandler, Peter Sin, and Qing Xiang, The Smith and critical groups of Paley graphs, J

    David B. Chandler, Peter Sin, and Qing Xiang, The Smith and critical groups of Paley graphs, J. Algebraic Combin. 41 (2015), no. 4, 1013–1022

    2015

  11. [19]

    1, 255 – 258

    Sheng Chen and Sheng Kui Y e, Critical groups for homeomorphism classes of graphs , Dis- crete Mathematics 309 (2009), no. 1, 255 – 258

    2009

  12. [20]

    Fan R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathemati cal Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997 . 42 Darren Glass and Nathan Kaplan

    1997

  13. [21]

    Mihai Ciucu, Weigen Y an, and Fuji Zhang, The number of spanning trees of plane graphs with reflective symmetry , J. Combin. Theory Ser. A 112 (2005), no. 1, 105–116

    2005

  14. [22]

    Algebraic Combin

    Julien Clancy, Nathan Kaplan, Timothy Leake, Sam Payne, and Melanie Matchett Wood, On a Cohen-Lenstra heuristic for Jacobians of random graphs , J. Algebraic Combin. 42 (2015), no. 3, 701–723

    2015

  15. [23]

    Julien Clancy, Timothy Leake, and Sam Payne, A note on Jacobians, Tutte polynomials, and two-variable zeta functions of graphs , Exp. Math. 24 (2015), no. 1, 1–7

    2015

  16. [24]

    Anna Comito, Jennifer Garcia, Josefina Alvarado Rivera, Nat alie L. F. Hobson, and Luis David Garcia Puente, On the sandpile group of circulant graphs , 2016

    2016

  17. [25]

    Le Borgne

    Robert Cori and Y . Le Borgne. The sand-pile model and Tutte polynomials . Adv. in Appl. Math., 30 (2003), no. 1, 44–52

    2003

  18. [26]

    Com- bin

    Robert Cori and Dominique Rossin, On the sandpile group of dual graphs , European J. Com- bin. 21 (2000), no. 4, 447–459

    2000

  19. [27]

    Cools, J

    F. Cools, J. Draisma, S. Payne, and E. Robeva, A tropical proof of the Brill–Noether theorem , Adv. Math. 230 (2012), no. 2, 759–776

    2012

  20. [28]

    V alencia, Arithmetical structures on graphs , Linear Algebra Appl

    Hugo Corrales and Carlos E. V alencia, Arithmetical structures on graphs , Linear Algebra Appl. 536 (2018), 120–151

    2018

  21. [29]

    Algebra Appl

    , Arithmetical structures on graphs with connectivity one , J. Algebra Appl. 17 (2018), no. 8, 1850147, 13

    2018

  22. [30]

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

    2018

  23. [31]

    Josse van Dobben de Bruyn and Dion Gijswijt, Treewidth is a lower bound on graph gonality, https://arxiv.org/abs/1407.7055, 2014

    2014 arXiv

  24. [32]

    4, 715–720

    Andrew Deveau, David Jensen, Jenna Kainic, and Dan Mitropol sky, Gonality of random graphs, Involve 9 (2016), no. 4, 715–720

    2016

  25. [33]

    Ducey, Jonathan Gerhard, and Noah Watson, The Smith and Critical Groups of the Square Rook’s Graph and its Complement , Electr

    Joshua E. Ducey, Jonathan Gerhard, and Noah Watson, The Smith and Critical Groups of the Square Rook’s Graph and its Complement , Electr. J. Comb. 23 (2016), no. 4, P4.9

    2016

  26. [34]

    341 (2018), no

    Neelav Dutta and David Jensen, Gonality of expander graphs , Discrete Math.. 341 (2018), no. 9, 2535–2543

    2018

  27. [35]

    Alan Frieze and Michał Karo´ nski, Introduction to Random Graphs , Cambridge University Press, Cambridge, 2016

    2016

  28. [36]

    Louis Gaudet, David Jensen, Dhruv Ranganathan, Nicholas Wa wrykow, and Theodore Weis- man, Realization of groups with pairing as Jacobians of finite gra phs, Ann. Comb. 22 (2018), no. 4, 781–801

    2018

  29. [37]

    Mark Giesbrecht, Fast computation of the Smith normal form of an integer matrix, Proceedings of the 1995 International Symposium on Symbolic and Algebra ic Computation (New Y ork, NY , USA), ISSAC ’95, ACM, 1995, pp. 110–118

    1995

  30. [38]

    Darren Glass and Criel Merino, Critical groups of graphs with dihedral actions , European J. Combin. 39 (2014), 95–112

    2014

  31. [39]

    567 (2019), 138–142

    Gopal Goel and David Perkinson, Critical groups of iterated cones, Linear Algebra Appl. 567 (2019), 138–142

    2019

  32. [40]

    Gouvˆ ea, p-adic Numbers: An introduction , second ed., Universitext, Springer- V erlag, Berlin, 1997

    Fernando Q. Gouvˆ ea, p-adic Numbers: An introduction , second ed., Universitext, Springer- V erlag, Berlin, 1997

    1997

  33. [41]

    Phillip A Griffiths and Joseph Harris, Principles of Algebraic Geometry, Wiley classics library, Wiley, New Y ork, NY , 1994

    1994

  34. [42]

    418 (2006), no

    Y aoping Hou, Chingwah Woo, and Pingge Chen, On the sandpile group of the square cycle cn2, Linear Algebra Appl. 418 (2006), no. 2, 457 – 467

    2006

  35. [43]

    3, 231–250

    Brian Jacobson, Andrew Niedermaier, and Victor Reiner, Critical groups for complete multi- partite graphs and Cartesian products of complete graphs, Journal of Graph Theory 44 (2003), no. 3, 231–250

    2003

  36. [44]

    Electron

    Sameer Kailasa, Vivian Kuperberg, and Nicholas Wawrykow, Chip-firing on trees of loops . Electron. J. Combin. 25 (2018), no. 1, Paper 1.19, 12 pp

    2018

  37. [45]

    Kirby, Roger B

    Edward C. Kirby, Roger B. Mallion, Paul Pollak, and Paweł J. Skrzy´ nski,What Kirchhoff actually did concerning spanning trees in electrical netwo rks and its relationship to modern graph-theoretical work, Croatica Chemica Acta 89 (2016). Chip-Firing Games and Critical Groups 43

    2016

  38. [46]

    Caroline Klivans, The Mathematics of Chip-Firing, Chapman and Hall/CRC, New Y ork, 2018

    2018

  39. [47]

    S. V . Konyagin, Double exponential lower bound for the number of representa tions of unity by Egyptian fractions, Math. Notes 95 (2014), no. 1-2, 277–281, Translation of Mat. Zametki 95 (2014), no. 2, 312–316

    2014

  40. [48]

    Shaked Koplewitz, Sandpile groups and the coeulerian property for random dire cted graphs, Adv. in Appl. Math. 90 (2017), 145–159

    2017

  41. [49]

    , Sandpile groups of random bipartite graphs, https://arxiv.org/abs/1705.07519, 2017

    2017 arXiv

  42. [50]

    Timothy Leake and Dhruv Ranganathan, Brill–Noether theory of maximally symmetric graphs, European J. Combin. 46 (2015), 115–125

    2015

  43. [51]

    Chang Mou Lim, Sam Payne, and Natasha Potashnik, A note on Brill–Noether theory and rank determining sets for metric graphs , Int. Math. Res. Not. IMRN (2012), no. 23, 5484–5504

    2012

  44. [52]

    Lorenzini, Arithmetical graphs, Math

    Dino J. Lorenzini, Arithmetical graphs, Math. Ann. 285 (1989), no. 3, 481–501

    1989

  45. [53]

    73 (1990), no

    , Groups of components of N´ eron models of Jacobians, Compositio Math. 73 (1990), no. 2, 145–160

    1990

  46. [54]

    91 (1991), no

    , A finite group attached to the Laplacian of a graph , Discrete Math. 91 (1991), no. 3, 277–282

    1991

  47. [55]

    , Smith normal form and Laplacians , J. Combin. Theory Ser. B 98 (2008), no. 6, 1271–1300

    2008

  48. [56]

    Jessie MacWilliams, Orthogonal matrices over finite fields , Amer. Math. Monthly 76 (1969), 152–164

    1969

  49. [57]

    Andr´ as M´ esz´ aros,The distribution of sandpile groups of random regular graph s, https://arxiv.org/abs/1806.03736v3, 2018

    2018 arXiv

  50. [58]

    Rick Miranda, Nondegenerate symmetric bilinear forms on finite abelian 2-groups, Trans. Amer. Math. Soc. 284 (1984), no. 2, 535–542

    1984

  51. [59]

    Hoi Nguyen and Melanie Matchett Wood, Random integral matrices: universality of surjec- tivity and the cokernel , https://arxiv.org/abs/1806.00596, 2018

    2018 arXiv

  52. [60]

    318 (2014), 10–40

    Victor Reiner and Dennis Tseng, Critical groups of covering, voltage and signed graphs , Discrete Math. 318 (2014), 10–40

    2014

  53. [61]

    Sedl´ aˇ cek,On the minimal graph with a given number of spanning trees , Canad

    J. Sedl´ aˇ cek,On the minimal graph with a given number of spanning trees , Canad. Math. Bull. 13 (1970), 515–517

    1970

  54. [62]

    Farbod Shokrieh, The monodromy pairing and discrete logarithm on the Jacobia n of finite graphs, J. Math. Cryptol. 4 (2010), no. 1, 43–56

    2010

  55. [63]

    Spielman, Graphs, vectors, and matrices, Bull

    Daniel A. Spielman, Graphs, vectors, and matrices, Bull. Amer. Math. Soc. (N.S.) 54 (2017), no. 1, 45–61

    2017

  56. [64]

    Stanley, Smith normal form in combinatorics, J

    Richard P . Stanley, Smith normal form in combinatorics, J. Combin. Theory Ser. A 144 (2016), 476–495

    2016

  57. [65]

    Arne Storjohann, Near optimal algorithms for computing Smith normal forms of integer matri- ces, Proceedings of the 1996 international symposium on Symbolic and algebraic computation (New Y ork, NY , USA), ISSAC ’96, ACM, 1996, pp. 267–274

    1996

  58. [66]

    Wagner, The critical group of a directed graph , https://arXiv:math/0010241, 2000

    David G. Wagner, The critical group of a directed graph , https://arXiv:math/0010241, 2000

    2000 arXiv

  59. [67]

    C. T. C. Wall, Quadratic forms on finite groups, and related topics , Topology 2 (1963), 281– 298

    1963

  60. [68]

    Melanie Matchett Wood, The distribution of sandpile groups of random graphs, J. Amer. Math. Soc. 30 (2017), no. 4, 915–958

    2017

  61. [69]

    , Random integral matrices and the Cohen-Lenstra heuristics , Amer. J. Math. 141 (2019), no. 2, 383–398. Note: We have marked papers that have at least one undergraduate co author in red

    2019

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