Spanning Trees on the Sierpinski Gasket

Chang, Shu Chiuan; Chen, Lung Chi; Yang, Wei

doi:10.1007/s10955-006-9262-0

Cited by 117 publications

(95 citation statements)

References 52 publications

(71 reference statements)

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“…Note:The result coincides with the result obtained in [30],and we give an alternative approach for explicitly determining the number of spanning trees for PSW.It is smaller than the the number of spanning trees for Sierpinski gasket [32].…”

Section: Theorem 4 For Any N ≥ 0the Number Of Spanning Trees Of Psw supporting

confidence: 84%

Tutte Polynomial of Pseudofractal Scale-Free Web

Peng

Xiong

2015

J Stat Phys

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The Tutte polynomial of a graph is a 2-variable polynomial which is quite important in both combinatorics and statistical physics. It contains various numerical invariants and polynomial invariants ,such as the number of spanning trees,the number of spanning forests , the number of acyclic orientations , the reliability polynomial,chromatic polynomial and flow polynomial . In this paper,we study and gain recursive formulas for the Tutte polynomial of pseudofractal scale-free web(PSW) which implies logarithmic complexity algorithm is obtained to calculate the Tutte polynomial of PSW although it is NP-hard for general graph.We also obtain the rigorous solution for the the number of spanning trees of PSW by solving the recurrence relations derived from Tutte polynomial ,which give an alternative approach for explicitly determining the number of spanning trees of PSW.Further more,we analysis the all-terminal reliability of PSW and compare the results with that of Sierpinski gasket which has the same number of nodes and edges with PSW. In contrast with the well-known conclusion that scale-free networks are more robust against removal of nodes than homogeneous networks (e.g., exponential networks and regular networks).Our results show that Sierpinski gasket (which is a regular network) are more robust against random edge failures than PSW (which is a scalefree network) .Whether it is true for any regular networks and scale-free networks ,is still a unresolved problem. Pseudofractal scale-free web(PSW) we studied is a deterministically growing network introduced by S.N. Dorogovtsev [27] which is used to model scale-free network with small-world effect.Lots of job was devoted to study its properties ,such as degree distribution ,degree correlation , clustering coefficient [27,28] ,diameter [28],average path length [29], the number of spanning trees [30] and mean first-passage time for random walk [31].As for its tutte polynomial and reliability polynomial ,to the best of our knowledge, related research was rarely reported .

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Section: Theorem 4 For Any N ≥ 0the Number Of Spanning Trees Of Psw supporting

confidence: 84%

Tutte Polynomial of Pseudofractal Scale-Free Web

Peng

Xiong

2015

J Stat Phys

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“…In the pseudofractal fractal web, the entropy is 0.8959 [20], a value less than 0.9458. For the square lattice and the two-dimensional Sierpinski gasket, their entropy of spanning trees are 1.16624 [15] and 1.0486 [18], respectively, both of which are greater than 0.9458. Therefore, the number of spanning trees in the Farey graph is larger than that of the pseudofractal fractal web, but is smaller than that corresponding to the square lattice or the two-dimensional Sierpinski gasket.…”

Section: Spanning Trees On Farey Graphmentioning

confidence: 97%

“…According to Eqs. (18)(19)(20), the three quantities p g (0), q g (0) and r g (0) obey the recursive relations:…”

Section: Spanning Trees On Farey Graphmentioning

confidence: 99%

Counting spanning trees in a small-world Farey graph

Zhang¹,

Wu²,

Lin³

2012

Physica A: Statistical Mechanics and its Applications

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The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is computationally intractable. In this paper, we perform a study on the enumeration of spanning trees in a specific small-world network with an exponential distribution of vertex degrees, which is called a Farey graph since it is associated with the famous Farey sequence. According to the particular network structure, we provide some recursive relations governing the Laplacian characteristic polynomials of a Farey graph and its subgraphs. Then, making use of these relations obtained here, we derive the exact number of spanning trees in the Farey graph, as well as an approximate numerical solution for the asymptotic growth constant characterizing the network. Finally, we compare our results with those of different types of networks previously investigated.

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“…For the graphs with average degree 3, the entropy of in nite outerplanar small-world graphs [26] is 0.657, the values of entropy in 3-12-12 and 4-8-8 lattices [27] are 0.721 and 0.787, and the honeycomb lattice [28] is 0.807. While for the graphs with average degree 4, the entropy of the pseudofractal fractal web [29] is 0.896, the fractal scale-free lattice [20] is 1.040, the values of the two-dimensional Sierpinski gasket [15] and the square lattice [28] are 1.049 and 1.166. The entropy of spanning trees in our graph is 0.860, which is larger than those of graphs with average degree 3, but smaller than those of graphs with average degree 4.…”

Section: Entropy Of Spanning Treesmentioning

confidence: 99%

“…Recently, counting the number of spanning trees has attracted increasing attention [13][14][15][16]. It is known that the number of spanning trees can be obtained by matrix-tree theorem [13].…”

Section: Introductionmentioning

confidence: 99%

Enumeration of spanning trees in the sequence of Dürer graphs

2017

Open Mathematics

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Abstract:In this paper, we calculate the number of spanning trees in the sequence of Dürer graphs with a special feature that it has two alternate states. Using the electrically equivalent transformations, we obtain the weights of corresponding equivalent graphs and further derive relationships for spanning trees between the Dürer graphs and transformed graphs. By algebraic calculations, we obtain a closed-form formula for the number of spanning trees with regard to iteration step. Finally we compare the entropy of our graph with other studied graphs and see that its value of entropy lies in the interval of those of graphs with average degree being 3 and 4.

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Spanning Trees on the Sierpinski Gasket

Cited by 117 publications

References 52 publications

Tutte Polynomial of Pseudofractal Scale-Free Web

Tutte Polynomial of Pseudofractal Scale-Free Web

Counting spanning trees in a small-world Farey graph

Enumeration of spanning trees in the sequence of Dürer graphs

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