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A general method for computing Tutte polynomials of self-similar graphs

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  • Gong, Helin
  • Jin, Xian’an

Abstract

Self-similar graphs were widely studied in both combinatorics and statistical physics. Motivated by the construction of the well-known 3-dimensional Sierpiński gasket graphs, in this paper we introduce a family of recursively constructed self-similar graphs whose inner duals are of the self-similar property. By combining the dual property of the Tutte polynomial and the subgraph-decomposition trick, we show that the Tutte polynomial of this family of graphs can be computed in an iterative way and in particular the exact expression of the formula of the number of their spanning trees is derived. Furthermore, we show our method is a general one that is easily extended to compute Tutte polynomials for other families of self-similar graphs such as Farey graphs, 2-dimensional Sierpiński gasket graphs, Hanoi graphs, modified Koch graphs, Apollonian graphs, pseudofractal scale-free web, fractal scale-free network, etc.

Suggested Citation

  • Gong, Helin & Jin, Xian’an, 2017. "A general method for computing Tutte polynomials of self-similar graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 483(C), pages 117-129.
    • Handle: RePEc:eee:phsmap:v:483:y:2017:i:c:p:117-129
    DOI: 10.1016/j.physa.2017.04.073
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    References listed on IDEAS

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    1. Chang, Shu-Chiuan & Shrock, Robert, 2001. "Exact Potts model partition functions on wider arbitrary-length strips of the square lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 296(1), pages 234-288.
    2. Chang, Shu-Chiuan & Shrock, Robert, 2001. "Exact Potts model partition functions on strips of the honeycomb lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 296(1), pages 183-233.
    3. Gong, Helin & Jin, Xian’an, 2014. "Potts model partition functions on two families of fractal lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 414(C), pages 143-153.
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    Cited by:

    1. Rasul Kochkarov, 2021. "Research of NP-Complete Problems in the Class of Prefractal Graphs," Mathematics, MDPI, vol. 9(21), pages 1-20, October.
    2. Rasul Kochkarov & Azret Kochkarov, 2022. "Introduction to the Class of Prefractal Graphs," Mathematics, MDPI, vol. 10(14), pages 1-17, July.
    3. Liao, Yunhua & Aziz-Alaoui, M.A. & Zhao, Junchan & Hou, Yaoping, 2019. "The behavior of Tutte polynomials of graphs under five graph operations and its applications," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.

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