Spanning forests on the Sierpinski gasket

Combinatorics International audience We study the number of spanning forests on the Sierpinski gasket SGd(n) at stage n with dimension d equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket SGd;b(n) with d = 2 and b = 3 ; 4 are obtained. We also derive upper bounds for the asymptotic growth constants for both SGd and SG2,b.

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ACYCLIC ORIENTATIONS ON THE SIERPINSKI GASKET

International Journal of Modern Physics B ◽

10.1142/s0217979212501287 ◽

2012 ◽

Vol 26 (24) ◽

pp. 1250128 ◽

Cited By ~ 3

Author(s):

SHU-CHIUAN CHANG

Keyword(s):

Sierpinski Gasket ◽

Upper Bounds ◽

Two Dimensional ◽

Sierpiński Gasket ◽

Asymptotic Behaviors ◽

Asymptotic Growth ◽

Acyclic Orientations ◽

Growth Constants

We study the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket SG 2,b(n) at stage n with b equal to two and three, and determine the asymptotic behaviors. We also derive upper bounds for the asymptotic growth constants of SG 2,b and d-dimensional Sierpinski gasket SG d.

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Number of connected spanning subgraphs on the Sierpinski gasket

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.470 ◽

2009 ◽

Vol Vol. 11 no. 1 (Combinatorics) ◽

Author(s):

Shu-Chiuan Chang ◽

Lung-Chi Chen

Keyword(s):

Lower Bounds ◽

Sierpinski Gasket ◽

Upper And Lower Bounds ◽

Sierpiński Gasket ◽

Growth Constant ◽

Asymptotic Growth ◽

Spanning Subgraphs ◽

International Audience

Combinatorics International audience We study the number of connected spanning subgraphs f(d,b) (n) on the generalized Sierpinski gasket SG(d,b) (n) at stage n with dimension d equal to two, three and four for b = 2, and layer b equal to three and four for d = 2. The upper and lower bounds for the asymptotic growth constant, defined as zSG(d,b) = lim(v ->infinity) ln f(d,b)(n)/v where v is the number of vertices, on SG(2,b) (n) with b = 2, 3, 4 are derived in terms of the results at a certain stage. The numerical values of zSG(d,b) are obtained.

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Structure of spanning trees on the two-dimensional Sierpinski gasket

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.476 ◽

2011 ◽

Vol Vol. 12 no. 3 (Combinatorics) ◽

Author(s):

Shu-Chiuan Chang ◽

Lung-Chi Chen

Keyword(s):

Probability Distribution ◽

Spanning Tree ◽

Spanning Trees ◽

Sierpinski Gasket ◽

Rational Numbers ◽

Limiting Distribution ◽

Two Dimensional ◽

Sierpiński Gasket ◽

Average Probability ◽

International Audience

Combinatorics International audience Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.

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Asymptotic behavior of spanning forests and connected spanning subgraphs on two-dimensional lattices

International Journal of Modern Physics B ◽

10.1142/s0217979220502495 ◽

2020 ◽

Vol 34 (27) ◽

pp. 2050249

Author(s):

Shu-Chiuan Chang ◽

Robert Shrock

Keyword(s):

Asymptotic Behavior ◽

Exponential Growth ◽

Upper Bounds ◽

Great Length ◽

Two Dimensional ◽

Lower And Upper Bounds ◽

Spanning Subgraphs ◽

Spanning Forests ◽

Growth Constants ◽

Limiting Values

We calculate exponential growth constants [Formula: see text] and [Formula: see text] describing the asymptotic behavior of spanning forests and connected spanning subgraphs on strip graphs, with arbitrarily great length, of several two-dimensional lattices, including square, triangular, honeycomb, and certain heteropolygonal Archimedean lattices. By studying the limiting values as the strip widths get large, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for these exponential growth constants, with fractional uncertainties ranging from [Formula: see text] to [Formula: see text]. We show that [Formula: see text] and [Formula: see text] are monotonically increasing functions of vertex degree for these lattices.

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Exponential growth constants for spanning forests on Archimedean lattices: Values and comparisons of upper bounds

International Journal of Modern Physics B ◽

10.1142/s0217979221500855 ◽

2021 ◽

Vol 35 (06) ◽

pp. 2150085

Author(s):

Shu-Chiuan Chang ◽

Robert Shrock

Keyword(s):

Asymptotic Behavior ◽

Exponential Growth ◽

Upper Bounds ◽

Growth Constant ◽

Spanning Forests ◽

Growth Constants

We compare our upper bounds on the exponential growth constant [Formula: see text] characterizing the asymptotic behavior of spanning forests on Archimedean lattices [Formula: see text] with recently derived upper bounds.Our upper bounds on [Formula: see text], which are very close to the respective values of [Formula: see text] that we have calculated, are shown to be significantly better for these lattices than the new upper bounds.

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