Chip-firing games, potential theory on graphs, and spanning trees

Baker, Matthew; Shokrieh, Farbod

doi:10.1016/j.jcta.2012.07.011

Cited by 97 publications

(158 citation statements)

References 41 publications

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“…12 The results of [Bac14] can be used to prove that the inverse of the natural map B(G) → P ic g (G) is efficiently computable. See [BS13] for an explanation of how such a bijection can be used to find random spanning trees. (1) Let e 1 , e 2 be edges incident to v, and let β 1 = β (v,e 1 ) and β 2 = β (v,e 2 ) be the corresponding Bernardi bijections.…”

Section: G) From This We Get Amentioning

confidence: 99%

“…If G is a connected graph on n vertices, the Picard group Pic 0 (G) of G (also called the sandpile group, critical group, or Jacobian group) is a finite abelian group whose cardinality is the determinant of any (n − 1) × (n − 1) principal sub-minor of the Laplacian matrix of G. By Kirchhoff's Matrix-Tree Theorem, this quantity is equal to the number of spanning trees of G. There are several known families of bijections between spanning trees and elements of Pic 0 (G), which we think of as giving bijective proofs of Kirchhoff's Theorem -see for example [BS13] and the references therein. However, such bijections depend on various auxiliary choices, and there is no canonical bijection in general between Pic 0 (G) and the set S(G) of spanning trees of G (see [CCG15,p.…”

Section: Introductionmentioning

confidence: 99%

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The Bernardi Process and Torsor Structures on Spanning Trees

Baker

Wang

2017

International Mathematics Research Notices

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Abstract. Let G be a ribbon graph, i.e., a connected finite graph G together with a cyclic ordering of the edges around each vertex. By adapting a construction due to Olivier Bernardi, we associate to any pair (v, e) consisting of a vertex v and an edge e adjacent to v a bijection β (v,e) between spanning trees of G and elements of the set Pic g (G) of degree g divisor classes on G, where g is the genus of G in the sense of Baker-Norine. We give a new proof that the map β (v,e) is bijective by explicitly constructing an inverse. Using the natural action of the Picard group Pic 0 (G) on Pic g (G), we show that the Bernardi bijection β (v,e) gives rise to a simply transitive action β v of Pic 0 (G) on the set of spanning trees which does not depend on the choice of e. A plane graph has a natural ribbon structure (coming from the counterclockwise orientation of the plane), and in this case we show that β v is independent of v as well. Thus for plane graphs, the set of spanning trees is naturally a torsor for the Picard group. Conversely, we show that if β v is independent of v then G together with its ribbon structure is planar. We also show that the natural action of Pic 0 (G) on spanning trees of a plane graph is compatible with planar duality.These findings are formally quite similar to results of Holroyd et al. and ChanChurch-Grochow, who used rotor-routing to construct an action r v of Pic 0 (G) on the spanning trees of a ribbon graph G, which they show is independent of v if and only if G is planar. It is therefore natural to ask how the two constructions are related. We prove that β v = r v for all vertices v of G when G is a planar ribbon graph, i.e. the two torsor structures (Bernardi and rotor-routing) on the set of spanning trees coincide. In particular, it follows that the rotor-routing torsor is compatible with planar duality. We conjecture that for every non-planar ribbon graph G, there exists a vertex v with β v = r v .We thank Spencer Backman, Melody Chan, and Dan Margalit for helpful discussions and feedback on an earlier draft. We also thank Chi Ho Yuen for catching a sign error in Theorem 6.1, and the anonymous referees for their extraordinarily careful proofreading and numerous useful suggestions. This work began at the American Insitute of Mathematics workshop "Generalizations of chip firing and the critical group" in July 2013, and we would like to thank AIM as well as the organizers of that conference (L. Levine, J. Martin, D. Perkinson, and J. Propp) for providing a stimulating environment.

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Section: G) From This We Get Amentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Bernardi Process and Torsor Structures on Spanning Trees

Baker

Wang

2017

International Mathematics Research Notices

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“…For compact metric graphs, a fundamental solution of the Laplacian is given by j-functions. We follow the notation of [CR93] (see also [Zha93,BR07,BR10,Sho10,BS13]). The main result in this section is an explicit formula in Proposition 3.17 for j z (x, y), which we will need later and it might also be of independent interest.…”

Section: Hodge Decompositionmentioning

confidence: 99%

Canonical measures on metric graphs and a Kazhdan’s theorem

Shokrieh

2018

Invent. math.

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We extend the notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of "hyperbolic measures" on universal covers of metric graphs. Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. We will prove a generalized version of this theorem for metric graphs, allowing any infinite Galois cover to replace the universal cover. We will show all such limiting measures satisfy a version of Gauss-Bonnet formula which, using the theory of von Neumann dimensions, can be interpreted as a "trace formula". In the special case where the infinite cover is the universal cover, we will provide explicit methods to compute the corresponding limiting (hyperbolic) measure. Our ideas are motivated by non-Archimedean analytic and tropical geometry.Contents 2. Metric graphs, models, and Galois covers 2.1. Metric graphs.Definition 2.1. A metric graph (or a locally finite abstract tropical curve) Γ is a connected metric space such that every point p ∈ Γ has a neighborhood isometric to a star-shaped set of finite valence and of radius at least ǫ, for some fixed ǫ > 0, endowed with the path metric.By a star-shaped set of finite valence n and radius r we mean a set of the form S(n, r) = {z ∈ C : z = te k 2πi n for some 0 ≤ t < r and some k ∈ Z} , where distance between two points in S(n, r) is the length of the shortest path connecting the two points.In tropical geometry, metric graphs are usually compact. We do not require them to be compact in our definition because we will also be working with infinite covers of (compact) metric graphs. Models.Definition 2.2. A weighted graph G is a connected combinatorial graph (i.e. 1-dimensional, locally finite simplicial complex) with a positive R-valued length function ℓ defined on its edge set.

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“…We refer the reader to [4] for a proof that Algorithm 1 terminates and that the resulting divisor is indeed Red v (D). As a corollary of Lemma 2.3, we have the following result.…”

Section: Algorithm 1: Dhar's Burning Algorithmmentioning

confidence: 99%

Graphs of gonality three

Aidun¹,

Dean²,

Morrison³

et al. 2019

Algebraic Combinatorics

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In 2013, Chan classified all metric hyperelliptic graphs, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge-and vertex-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three.

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Chip-firing games, potential theory on graphs, and spanning trees

Cited by 97 publications

References 41 publications

The Bernardi Process and Torsor Structures on Spanning Trees

The Bernardi Process and Torsor Structures on Spanning Trees

Canonical measures on metric graphs and a Kazhdan’s theorem

Graphs of gonality three

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