On the Sandpile Model of Modified Wheels I

Raza, Zahid; Waheed, Seemal Abdul

doi:10.1515/ijnsns-2012-0064

Cited by 3 publications

(4 citation statements)

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“…Its most celebrated contribution is perhaps its role in developping discrete Riemann-Roch and Abel-Jacobi theory on graphs [5]. The structure of the sandpile group has also been explicitly computed on a variety of graph families, including complete graphs [18], nearly-complete graphs (complete graphs with a cycle removed) [41], wheel graphs [9] (see also [35,36]), Cayley graphs of the dihedral group D n [20], and many more (see e.g. [15] and the references therein for a more complete list).…”

Section: Introductionmentioning

confidence: 99%

Combinatorial aspects of sandpile models on wheel and fan graphs

Selig¹

2022

Preprint

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We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations of the model's recurrent configurations on these families. For wheel graphs, we exhibit a bijection between these recurrent configuations and the set of subgraphs of the cycle graph which maps the level of the configuration to the number of edges of the subgraph. This bijection relies on two key ingredients. The first consists in considering a stochastic variant of the standard Abelian sandpile model (ASM), rather than the ASM itself. The second ingredient is a mapping from a given recurrent state to a canonical minimal recurrent state, exploiting similar ideas to previous studies of the ASM on complete bipartite graphs and Ferrers graphs. We also show that on the wheel graph with 2n vertices, the number of recurrent states with level n is given by the first differences of the central Delannoy numbers. Finally, we prove using similar tools that the recurrent configurations of the ASM on fan graphs are in bijection with certain lattice paths, which we name Kimberling paths after the author of the corresponding entry in the Online Encyclopedia of Integer Sequences.

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Section: Introductionmentioning

confidence: 99%

Combinatorial aspects of sandpile models on wheel and fan graphs

Selig¹

2022

Preprint

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“…In [1], it has been shown that the set of critical configurations can be used to give the structure of an abelian group by giving critical configurations for a wheel graph W n . For general references on the ASM (denoted by SP (G,q)), see, e.g., [5][6][7][8][9][10][11][12][13][14] and on the critical group (denoted by K(G)), see for instance [15][16][17][18][19][20][21][22][23][24]. The definitions and statements of theorems presented in this section have been taken from [1].…”

Section: Introductionmentioning

confidence: 99%

On the sandpile model of modified wheels II

et al. 2020

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We investigate the abelian sandpile group on modified wheels {\hat{W}}_{n} by using a variant of the dollar game as described in [N. L. Biggs, Chip-Firing and the critical group of a graph, J. Algebr. Comb. 9 (1999), 25–45]. The complete structure of the sandpile group on a class of graphs is given in this paper. In particular, it is shown that the sandpile group on {\hat{W}}_{n} is a direct product of two cyclic subgroups generated by some special configurations. More precisely, the sandpile group on {\hat{W}}_{n} is the direct product of two cyclic subgroups of order {a}_{n} and 3{a}_{n} for n even and of order {a}_{n} and 2{a}_{n} for n odd, respectively.

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“…When we delete, from ∆Γ, the row and column corresponding to the sink, we obtain the reduced graph Laplacian ∆ Γ . The sandpile group G Γ is then defined as the cokernel of ∆ Γ acting on Z Γ [25,12,45,1], i.e. faces, as these morphisms directly induce epimorphisms between the respective sandpile groups [9] (see also [46,2]).…”

Section: Introduction 1backgroundmentioning

confidence: 99%

“…Only for few infinite families of graphs, the structure of the respective sandpile groups has been (partly) determined, including complete graphs [39], complete multipartite graphs [29], cycles (equivalent to domains of Z 1 ) [39], thick cycles [1] (see also [26]), wheels [12,43], modified wheels [45], wired regular trees [38], thick trees [19], polygon flowers [17], nearly complete graphs [43], threshold graphs [43], Möbius ladders [18,24], prism graphs/graphs D n of the dihedral group [23,24], and n-cubes [6,2]. While this list is certainly not complete, the decomposition of the sandpile group on domains of Z 2 , for which the sandpile model was originally defined [7], is-to our knowledge-yet unknown.…”

Section: Introduction 1backgroundmentioning

confidence: 99%

Sandpile monomorphisms and limits

Lang¹,

Shkolnikov²

2019

Preprint

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We introduce a tiling problem between bounded open convex polyforms P ⊂ R 2 with directed and uniquely colored edges. If there exists a tiling of the polyform P2 by P1, we show that one can construct a monomorphism from the sandpile group GΓ 1 = Z Γ 1 /∆(Z Γ 1 ) on the domain (graph) Γ1 = P1 ∩ Z 2 to the respective group on Γ2 = P2 ∩ Z 2 . We provide several examples of infinite series of such tilings with polyforms converging to R 2 , and thus the first definition of scaling-limits for the sandpile group on the plane. Additional results include an exact sequence relating sandpile configurations to harmonic functions, an alternative formula for the order of the sandpile group based on a basis for the module of integer-valued harmonic functions, and three examples of how to prove the existence of (cyclic) subgroups for infinite families of sandpile groups by constructing appropriate integer-valued harmonic functions. The main open question concerns if the scaling-limits of the sandpile group for different sequences of polyforms converging to R 2 are isomorphic.

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On the Sandpile Model of Modified Wheels I

Cited by 3 publications

References 5 publications

Combinatorial aspects of sandpile models on wheel and fan graphs

Combinatorial aspects of sandpile models on wheel and fan graphs

On the sandpile model of modified wheels II

Sandpile monomorphisms and limits

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