A wired tree is a graph obtained from a tree by collapsing the leaves to a single vertex. We describe a pair of short exact sequences relating the sandpile group of a wired tree to the sandpile groups of its principal subtrees. In the case of a regular tree these sequences split, enabling us to compute the full decomposition of the sandpile group as a product of cyclic groups. This resolves in the affirmative a conjecture of E. Toumpakari concerning the ranks of the Sylow p-subgroups.
We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpile group of LG by its k-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs.
Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., th…
We answer a question of Laszlo Babai concerning the abelian sandpile model. Given a graph, the model yields a finite abelian group of recurrent configurations which is closely related to the combinatorial Laplacian of the graph. We explicitly describe the group elements and operations in the case of thick trees with loops--that is, graphs which are obtained from trees by setting arbitrary edge multiplicities and adding loops at vertices. We do this both concretely (by describing the so-called recurrent and identity configurations) and abstractly (by computing the group's abstract structure), a…
The sandpile group of a connected graph is the group of recurrent configurations in the abelian sandpile model on this graph. We study the structure of this group for the case of regular trees. A description of this group is the following: Let T(d,h) be the d-regular tree of depth h and let V be the set of its vertices. Denote the adjacency matrix of T(d,h) by A and consider the modified Laplacian matrix D:=dI-A. Let the rows of D span the lattice L in Z^V. The sandpile group of T(d,h) is Z^V/L. We compute the rank, the exponent and the order of this abelian group and find a cyclic Hall-subgro…
A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the tiered trees introduced by Dugan et al. [10]. This bijection allows certain parameters of the recurrent configurations to be read on the corresponding tree. In particular, we show that the level of a recurrent configuration can be interpreted as the external activity of the corresponding tree, so that the bijection exhibited provides a new proof of a famous …
A EW-tableau is a certain 0/1-filling of a Ferrers diagram, corresponding uniquely to an acyclic orientation, with a unique sink, of a certain bipartite graph called a Ferrers graph. We give a bijective proof of a result of Ehrenborg and van Willigenburg showing that EW-tableaux of a given shape are equinumerous with permutations with a given set of excedances. This leads to an explicit bijection between EW-tableaux and the much studied Le-tableaux, as well as the tree-like tableaux introduced by Aval, Boussicault and Nadeau. We show that the set of EW-tableaux on a given Ferrers diagram are i…
We compute the sandpile groups of families of planar graphs having a common weak dual by evaluating the indeterminates of the critical ideals of the weak dual at the lengths of the cycles bounding the interior faces. This method allow us to determine the algebraic structure of the sandpile groups of outerplanar graphs, and can be used to compute the sandpile groups of many other planar graph families. Finally, we compute the identity element for the sandpile groups of the dual graphs of many outerplane graphs.
The article considers the procedure of connection of graphs to the edges of a cyclic graph and its influence on the sandpile group of the graph thus obtained. A series of classes of graphs CH_n(a_1,...,a_n) is defined. Recurrent and non-recurrent formulas for calculating the sandpile groups of all graphs of classes CH_n (a_1, ..., a_n) are proposed.
In this article, we give a partial description of the sandpile group of the cone of the cartesian product of graphs in function of the sandpile group of the cone of their factors. Also, we introduce the concept of uniform homomorphism of graphs and prove that every surjective uniform homomorphism of graphs induces an injective homomorphism between their sandpile groups. As an application of these result we obtain an explicit description of a set of generators of the sandpile group of the cone of the hypercube of dimension d.
We determine the asymptotic distribution of the p-rank of the sandpile groups of random bipartite graphs. We see that this depends on the ratio between the number of vertices on each side, with a threshold when the ratio between the sides is equal to 1/p. We follow the approach of Melanie Wood and consider random graphs as a special case of random matrices, and rely on a variant the definition of min-entropy given by Maples, in order to obtain useful results about these random matrices. Our results show that unlike the sandpile groups of Erdos-Renyi random graphs, the distribution of the sandp…
The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graph's sandpile group is part of a larger algebraic structure on the graph, known as its sandpile monoid. Most of the work on sandpiles so far has focused on the sandpile group rather than the sandpile monoid of a graph, and has also assumed the underlying graph to be undirected. A notable exception is the recent work of Babai and Toumpakari, which builds up the theory of sandpile monoids on directed graphs from scratch and provide…
The majority of graphs whose sandpile groups are known are either regular or simple. We give an explicit formula for a family of non-regular multi-graphs called thick cycles. A thick cycle graph is a cycle where multi-edges are permitted. Its sandpile group is the direct sum of cyclic groups of orders given by quotients of greatest common divisors of minors of its Laplacian matrix. We show these greatest common divisors can be expressed in terms of monomials in the graph's edge multiplicities.
We consider random directed graphs, and calculate the distribution of the cokernels of their laplacian, following the methods used by Wood. As a corollary, we show that the probability that a random digraph is coeulerian is asymptotically upper bounded by a constant around $0.43$.
We define two general classes of nonabelian sandpile models on directed trees (or arborescences) as models of nonequilibrium statistical phenomena. These models have the property that sand grains can enter only through specified reservoirs, unlike the well-known abelian sandpile model. In the Trickle-down sandpile model, sand grains are allowed to move one at a time. For this model, we show that the stationary distribution is of product form. In the Landslide sandpile model, all the grains at a vertex topple at once, and here we prove formulas for all eigenvalues, their multiplicities, and the…
Let $C_{k_1}, \ldots, C_{k_n}$ be cycles with $k_i\geq 2$ vertices ($1\le i\le n$). By attaching these $n$ cycles together in a linear order, we obtain a graph called a polygon chain. By attaching these $n$ cycles together in a cyclic order, we obtain a graph, which is called a polygon ring if it can be embedded on the plane; and called a twisted polygon ring if it can be embedded on the Möbius band. It is known that the sandpile group of a polygon chain is always cyclic. Furthermore, there exist edge generators. In this paper, we not only show that the sandpile group of any (twisted) polygon …
We classify all recurrent configurations of the Abelian sandpile model (ASM) on Ferrers graphs. The classification is in terms of decorations of EW-tableaux, which undecorated are in bijection with the minimal recurrent configurations. We introduce decorated permutations, extending to decorated EW-tableaux a bijection between such tableaux and permutations, giving a direct bijection between the decorated permutations and all recurrent configurations of the ASM. We also describe a bijection between the decorated permutations and the intransitive trees of Postnikov, the breadth-first search of w…
Let $C_t$ be a cycle of length $t$, and let $P_1,\ldots,P_t$ be $t$ polygon chains. A polygon flower $F=(C_t; P_1,\ldots,P_t)$ is a graph obtained by identifying the $i$th edge of $C_t$ with an edge $e_i$ that belongs to an end-polygon of $P_i$ for $i=1,\ldots,t$. In this paper, we first give an explicit formula for the sandpile group $S(F)$ of $F$, which shows that the structure of $S(F)$ only depends on the numbers of spanning trees of $P_i$ and $P_i/ e_i$, $i=1,\ldots,t$. By analyzing the arithmetic properties of those numbers, we give a simple formula for the minimum number of generators o…
In the abelian sandpile model, recurrent chip configurations are of interest as they are a natural choice of coset representatives under the quotient of the reduced Laplacian. We investigate graphs whose recurrent identities with respect to different sinks are compatible with each other. The maximal stable configuration is the simplest recurrent chip configuration, and graphs whose recurrent identities equal the maximal stable configuration are of particular interest, and are said to have the complete maximal identity property. We prove that given any graph $G$ one can attach trees to the vert…
Biggs proposed the sandpile group of certain modified wheel graphs for cryptosystems relying on the difficulty of the discrete logarithm problem. Blackburn and independently Shokrieh showed that the discrete logarithm problem is efficiently solvable. We study Shokrieh's method in cases of graphs such that the sandpile group is not cyclic, namely the square cycle graphs and the wheel graphs. Knowing generators of the group or the form of the pseudoinverse of the Laplacian matrix makes the problem more vulnerable. We also consider the discrete logarithm problem in case of the so-called subdivide…
We classify recurrent states of the Abelian sandpile model (ASM) on the complete split graph. There are two distinct cases to be considered that depend upon the location of the sink vertex in the complete split graph. This characterisation of decreasing recurrent states is in terms of Motzkin words and can also be characterised in terms of combinatorial necklaces. We also give a characterisation of the recurrent states in terms of a new type of parking function that we call a tiered parking function. These parking functions are characterised by assigning a tier (or colour) to each of the cars,…
The (univariate) avalanche polynomial of a graph, introduced by Cori, Dartois and Rossin in 2006, captures the distribution of the length of (principal) avalanches in the abelian sandpile model. This polynomial has been used to show that the avalanche distribution in the sandpile model on a multiple wheel graph does not follow the expected power law function. In this article, we introduce the (multivariate) avalanche polynomial that enumerates the toppling sequences of all principal avalanches. This polynomial generalizes the univariate avalanche polynomial and encodes more information. In par…
We determine the distribution of the sandpile group (a.k.a. Jacobian) of the Erdős-Rényi random graph G(n,q) as n goes to infinity. Since any particular group appears with asymptotic probability 0 (as we show), it is natural ask for the asymptotic distribution of Sylow p-subgroups of sandpile groups. We prove the distributions of Sylow p-subgroups converge to specific distributions conjectured by Clancy, Leake, and Payne. These distributions are related to, but different from, the Cohen-Lenstra distribution. Our proof involves first finding the expected number of surjections from the sandpile …
We consider the subgroup of the abelian sandpile group of the grid graph consisting of configurations of sand that are symmetric with respect to central vertical and horizontal axes. We show that the size of this group is (i) the number of domino tilings of a corresponding weighted rectangular checkerboard; (ii) a product of special values of Chebyshev polynomials; and (iii) a double-product whose factors are sums of squares of values of trigonometric functions. We provide a new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of …
We introduce two operators on stable configurations of the sandpile model that provide an algorithmic bijection between recurrent and parking configurations. This bijection preserves their equivalence classes with respect to the sandpile group. The study of these operators in the special case of the complete bipartite graph ${K}_{m,n}$ naturally leads to a generalization of the well known Cyclic Lemma of Dvoretsky and Motzkin, via pairs of periodic bi-infinite paths in the plane having slightly different slopes. We achieve our results by interpreting the action of these operators as an action …
The aim of the current work is to investigate structural properties of the sandpile group of a special class of self-similar graphs. More precisely, we consider Abelian sandpiles on Sierpinski gasket graphs and for the choice of normal boundary conditions, we give a characterization of the identity element and a recursive description of the sandpile group. Finally, we consider Abelian sandpile Markov chains on the aforementioned graphs and we improve the existing bounds on the speed of convergence to stationarity.
A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $Γ$ gives rise to a finite group $\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M$ known as the sandpile (or critical) group $S(Γ)$ of $Γ$. We determine $S(Γ)$ of the generalized de Bruijn graphs $Γ=\mathrm{DB}(n,d)$ with vertices $0,\dots,n-1$ and arcs $(i,di+k)$ for $0\leq i\leq n-1$ and $0\leq k\leq d-1$, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ and an $n$-cycle permutation matrix $X\in\mathrm{GL}_n(p)$ we show that $…
We consider a stochastic variant of the Abelian Sandpile Model (ASM) on a finite graph, introduced by Chan, Marckert and Selig. Even though it is a more general model, some nice properties still hold. We show that on a certain probability space, even if we lose the group structure due to topplings not being deterministic, some operators still commute. As a corollary, we show that the stationary distribution still does not depend on how sand grains are added onto the graph in our model, answering a conjecture of Selig.
Let $p$ and $q$ be distinct primes, and let $X_{p,q}$ be the $(q+1)$-regular graph whose nodes are supersingular elliptic curves over $\overline{\mathbb{F}}_p$ and whose edges are $q$-isogenies. For fixed $p$, we compute the distribution of the $\ell$-Sylow subgroup of the sandpile group (i.e.\ Jacobian) of $X_{p,q}$ as $q \to \infty$. We find that the distribution disagrees with the Cohen-Lenstra heuristic in this context. Our proof is via Galois representations attached to modular curves. As a corollary of our result, we give an upper bound on the probability that the Jacobian is cyclic, whi…
We study the distribution of the sandpile group of random d-regular graphs. For the directed model, we prove that it follows the Cohen-Lenstra heuristics, that is, the limiting probability that the $p$-Sylow subgroup of the sandpile group is a given $p$-group $P$, is proportional to $|Aut(P)|^{-1}$. For finitely many primes, these events get independent in the limit. Similar results hold for undirected random regular graphs, where for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne. This answers an open question of Frieze and Vu whether the adjacency matrix …
We identify the asymptotic distribution of $p$-rank of the sandpile group of a random directed bipartite graphs which are not too imbalanced. We show this matches exactly that of the Erd{ö}s-R{é}nyi random directed graph model, suggesting the Sylow $p$-subgroups of this model may also be Cohen-Lenstra distributed. Our work builds on results of Koplewitz who studied $p$-rank distributions for unbalanced random bipartite graphs, and showed that for sufficiently unbalanced graphs, the distribution of $p$-rank differs from the Cohen-Lenstra distribution. Koplewitz \cite{K} conjectured that for ran…
The avalanche polynomial on a graph captures the distribution of avalanches in the abelian sandpile model. Studied on trees, this polynomial could be defined by simply considering the size of the subtrees of the original tree. In this article, we study some properties of this polynomial on plane trees. Previously it has been proved that two different trees could have the same avalanche polynomial. We show here that the problem of finding a tree with a prescribed polynomial is NP-complete. In a second part, we study the average and the variance of the avalanche distribution on trees and give a …
The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of this primer is to apply the theory of lattice ideals from algebraic geometry to the Laplacian matrix, drawing out connections with the ASM. An extended summary of the ASM and of the required algebraic geometry is provided. New results include a characterization of graphs whose Laplacian lattice ideals are complete intersection ideals; a new construction of arithmetically Gorenstein ideals; a generalization to directed multigraphs of a dualit…
Let $D$ be a connected weighted digraph. The relation between the vertex weighted complexity (with a fixed root) of the line digraph of $D$ and the edge weighted complexity (with a fixed root) of $D$ has been given in (L. Levine, Sandpile groups and spanning trees of directed line graphs, J. Combin. Theory Ser. A 118 (2011) 350-364) and, independently, in (S. Sato, New proofs for Levine's theorems, Linear Algebra Appl. 435 (2011) 943-952). In this paper, we obtain a relation between the vertex weighted complexity of the middle digraph of $D$ and the edge weighted complexity of $D$. Particularl…
We study the scaling limits of three different aggregation models on the integer lattice Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a determini…
Let G be a finite graph, and let G_n be the n-th iterated cone over G. We study the structure of the critical group of G_n arising in divisor and sandpile theory.
The sandpile group Pic^0(G) of a finite graph G is a discrete analogue of the Jacobian of a Riemann surface which was rediscovered several times in the contexts of arithmetic geometry, self-organized criticality, random walks, and algorithms. Given a ribbon graph G, Holroyd et al. used the "rotor-routing" model to define a free and transitive action of Pic^0(G) on the set of spanning trees of G. However, their construction depends a priori on a choice of basepoint vertex. Ellenberg asked whether this action does in fact depend on the choice of basepoint. We answer this question by proving that…
Kirchhoff's matrix-tree theorem states that the number of spanning trees of a graph G is equal to the value of the determinant of the reduced Laplacian of $G$. We outline an efficient bijective proof of this theorem, by studying a canonical finite abelian group attached to $G$ whose order is equal to the value of same matrix determinant. More specifically, we show how one can efficiently compute a bijection between the group elements and the spanning trees of the graph. The main ingredient for computing the bijection is an efficient algorithm for finding the unique $G$-parking function (reduce…
Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree --- namely acyclicity and connectedness --- can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-…
The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Department of Mathematics, Middle East Technical University（中东理工大学数学系）；Guangdong Technion Israel Institute of Technology (GTIIT)（广东以色列理工学院）；Technion-Israel Institute of Technology（以色列理工学院）；Center for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV)（国立理工学院高级研究中心）；Laboratory of Combinatorial and Geometric Structures, Moscow Institute of Physics and Technology（莫斯科物理技术学院组合与几何结构实验室）
Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study properties of {\em maximum} $G$-parking functions and provide a new bijection between them and the set of spanning trees of $G$ with no broken circuit. As a case study, we specialize some of our results to the graph corresponding to the disc…
Hereditary chip-firing models generalize the Abelian sandpile model and the cluster firing model to an exponential family of games induced by covers of the vertex set. This generalization retains some desirable properties, e.g. stabilization is independent of firings chosen and each chip-firing equivalence class contains a unique recurrent configuration. In this paper we present an explicit bijection between the recurrent configurations of a hereditary chip-firing model on a graph and its spanning trees.
