scholarly journals Structure of spanning trees on the two-dimensional Sierpinski gasket

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.

scholarly journals ACYCLIC ORIENTATIONS ON THE SIERPINSKI GASKET

2012 ◽  
Vol 26 (24) ◽  
pp. 1250128 ◽  
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.

Acyclic Orientations on the Generalized Two-Dimensional Sierpinski Gasket

2011 ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
Zacharias Anastassi
Keyword(s):  
Sierpinski Gasket ◽  
Two Dimensional ◽  
Sierpiński Gasket ◽  
Acyclic Orientations

scholarly journals The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket

2013 ◽  
Vol 392 (8) ◽  
pp. 1776-1787 ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen ◽  
Hsin-Yun Lee
Keyword(s):  
Sierpinski Gasket ◽  
Two Dimensional ◽  
Vertex Model ◽  
Sierpiński Gasket ◽  
Ice Model

scholarly journals Analysis of the total costs for variants of the Union-Find algorithm

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Method Of Moments ◽  
Spanning Tree ◽  
Limiting Distribution ◽  
Tree Model ◽  
Average Behavior ◽  
Finite Set ◽  
International Audience

International audience We study the average behavior of variants of the UNION-FIND algorithm to maintain partitions of a finite set under the random spanning tree model. By applying the method of moments we can characterize the limiting distribution of the total costs of the algorithms "Quick Find Weighted'' and "Quick Find Biased'' extending the analysis of Knuth and Schönhage, Yao, and Chassaing and Marchand.

scholarly journals Spanning forests on the Sierpinski gasket

2008 ◽  
Vol Vol. 10 no. 2 (Combinatorics) ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Sierpinski Gasket ◽  
Upper Bounds ◽  
Sierpiński Gasket ◽  
Asymptotic Behaviors ◽  
Asymptotic Growth ◽  
Spanning Forests ◽  
International Audience ◽  
Growth Constants

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.

scholarly journals Spanning Trees on the Sierpinski Gasket

2007 ◽  
Vol 126 (3) ◽  
pp. 649-667 ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen ◽  
Wei-Shih Yang
Keyword(s):  
Spanning Trees ◽  
Sierpinski Gasket ◽  
Sierpiński Gasket

scholarly journals Weighted Spanning Trees on some Self-Similar Graphs

10.37236/503 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Daniele D'Angeli ◽  
Alfredo Donno
Keyword(s):  
Generating Functions ◽  
Spanning Trees ◽  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Finite Graphs ◽  
Sierpiński Graphs ◽  
Schreier Graphs ◽  
Finite Approximations ◽  
Self Similar ◽  
Ternary Tree

We compute the complexity of two infinite families of finite graphs: the Sierpiński graphs, which are finite approximations of the well-known Sierpiński gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we study the weighted generating functions of the spanning trees, associated with several natural labellings of the edge sets.

scholarly journals Label-based parameters in increasing trees

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Probability Distribution ◽  
Random Variable ◽  
Growth Behavior ◽  
Limiting Distribution ◽  
Factorial Moments ◽  
International Audience ◽  
Recursive Trees ◽  
Insertion Process

International audience Grown simple families of increasing trees are a subclass of increasing trees, which can be constructed by an insertion process. Three such tree families contained in the grown simple families of increasing trees are of particular interest: $\textit{recursive trees}$, $\textit{plane-oriented recursive trees}$ and $\textit{binary increasing trees}$. Here we present a general approach for the analysis of a number of label-based parameters in a random grown simple increasing tree of size $n$ as, e.g., $\textit{the degree of the node labeled j}$, $\textit{the subtree-size of the node labeled j}$, etc. Further we apply the approach to the random variable $X_{n,j,a}$, which counts the number of size-$a$ branches attached to the node labeled $j$ (= subtrees of size $a$ rooted at the children of the node labeled $j$) in a random grown simple increasing tree of size $n$. We can give closed formulæ for the probability distribution and the factorial moments. Furthermore limiting distribution results for $X_{n,j,a}$ are given dependent on the growth behavior of $j=j(n)$ compared to $n$.

scholarly journals On the number of spanning trees of K_n^m #x00B1 G graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Stavros D. Nikolopoulos ◽  
Charis Papadopoulos
Keyword(s):  
Complete Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Star Graph ◽  
Complete Multipartite Graph ◽  
The Family ◽  
International Audience ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Number Of Spanning Trees

International audience The K_n-complement of a graph G, denoted by K_n-G, is defined as the graph obtained from the complete graph K_n by removing a set of edges that span G; if G has n vertices, then K_n-G coincides with the complement øverlineG of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K_n^m #x00b1 G, where K_n^m is the complete multigraph on n vertices with exactly m edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K_n^m; the graph K_n^m + G (resp. K_n^m - G) is obtained from K_n^m by adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result from K_n^m by adding and removing edges of multigraphs spanned by sets of edges of the graph K_n^m. We also prove closed formulas for the number of spanning tree of graphs of the form K_n^m #x00b1 G, where G is (i) a complete multipartite graph, and (ii) a multi-star graph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of their spanning trees.

scholarly journals Number of connected spanning subgraphs on the Sierpinski gasket

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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