Tutte polynomial of the Apollonian network

Liao, Yunhua; Hou, Yaoping; Shen, Xiaoling

doi:10.1088/1742-5468/2014/10/p10043

Cited by 11 publications

(4 citation statements)

References 40 publications

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“…In Section 3, we study the Tutte polynomial T (M(t); x, y). Though determining the Tutte polynomial of a graph is generally an NP-hard problem, we shall find a recursive formula for the Tutte polynomial of a network model M(t) based on the subgraph decomposition method [21,27,28]. Section 4 is devoted to the applications of T (M(t); x, y).…”

Section: Introductionmentioning

confidence: 99%

Tutte polynomial of self-similar network models with applications

Aboutahoun,

El-Safty

2023

Ars Comb.

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Determining the Tutte polynomial T ( G ; x , y ) of a graph network G is a challenging problem for mathematicians, physicians, and statisticians. This paper investigates a self-similar network model M ( t ) and derives its Tutte polynomial. In addition, we evaluate exact explicit formulas for the number of acyclic orientations and spanning trees of it as applications of the Tutte polynomial. Finally, we use the derived T ( M ( t ) ; x , y ) to obtain the Tutte polynomial of another self-similar model N ( t ) presented in [1] and correct the main result discussed in [1] by Ma et al. and test our result numerically by using Matlab.

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Section: Introductionmentioning

confidence: 99%

Tutte polynomial of self-similar network models with applications

Aboutahoun,

El-Safty

2023

Ars Comb.

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“…They have already attracted significant attention of researchers both in graph theory and in statistical physics. For example, their Tutte polynomials and spanning trees were investigated in [8,14], while Ising and magnetic models on them were studied in [1,11] by employing transfer matrices and numerical methods. In this paper we study them from the graph-theoretical point of view and present explicit formulas for the number of close-packed dimer configurations by computing the number of perfect matchings in the corresponding graphs.…”

Section: Introductionmentioning

confidence: 99%

Solving the dimer problem on Apollonian gasket

Došlić¹,

Luka²

2023

Proceedings of the 4th Croatian Combinatorial Days

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For any three circles in the plane where each circle is tangent to the other two, the Descartes' theorem yields the existence of a fourth circle tangent to the starting three. Continuing this process by adding a new circle between any three tangent circles leads to Apollonian packings. The fractal structures resulting from infinite continuation of such processes are known as Apollonian gaskets. Close-packed dimer configurations on such structures are well modeled by perfect matchings in the corresponding graphs. We consider Apollonian gaskets for several types of initial configurations and present explicit expressions for the number of perfect matchings in such graphs.

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“…Particularly, in the physics literature a lot of effort has been devoted to enumerating spanning trees in specific self-similar networks by using different techniques. Examples include the Farey graph [16,17], (x, y)-flower [12], Apollonian network [18][19][20], Sierpinski gasket [21,22], pseudofractal scale-free web [23], and so on. However, very little work appears to have been done on counting the spanning trees in a non-selfsimilar graph.…”

Section: Introductionmentioning

confidence: 99%

The number of spanning trees of a hybrid network created by inner-outer iteration

2019

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We consider the number of spanning trees in a novel hybrid network created by inner-outer iteration. The hybrid network is small-world but not self-similar. Firstly, we introduce two particular electrically equivalent transformations: delta-edge transformation and Sierpinski-delta transformation. By using these two particular transformations, we find the changes of the conductances of corresponding electrically equivalent networks. Secondly, based on the inner-outer iteration structure characteristics, we obtain a closed-form formula for the number of spanning trees of the hybrid network, as well as its spanning tree entropy. Finally, we compare our result with those of previously investigated networks with the same average degree, and give an explanation for their differences.

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Tutte polynomial of the Apollonian network

Cited by 11 publications

References 40 publications

Tutte polynomial of self-similar network models with applications

Tutte polynomial of self-similar network models with applications

Solving the dimer problem on Apollonian gasket

The number of spanning trees of a hybrid network created by inner-outer iteration

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