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Entropy and enumeration of spanning connected unicyclic subgraphs in self-similar network

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  • Liang, Jing
  • Zhao, Haixing
  • Yin, Jun
  • Xie, Sun

Abstract

The number of spanning connected unicyclic subgraphs is a critical quantity to characterize the reliability of networks. In 1993, Colbourn in [Colbourn (1993)] proposed a problem about reliability polynomial: Can the number of spanning connected unicyclic subgraphs of a graph be computed efficiently? Up to now, there is no research about this problem. In this paper, we mainly study the entropy and the enumeration of spanning connected unicyclic subgraphs in self-similar networks. We propose a linear algorithm for computing the number of spanning connected unicyclic subgraphs in self-similar network (x, y)-flower. By the algorithm, the exact number of spanning connected unicyclic subgraphs is determined in (x, y)-flower networks. In addition, we define the entropy of spanning connected unicyclic subgraphs and obtain the entropies of spanning connected unicyclic subgraphs in (x, y)-flower networks.

Suggested Citation

  • Liang, Jing & Zhao, Haixing & Yin, Jun & Xie, Sun, 2022. "Entropy and enumeration of spanning connected unicyclic subgraphs in self-similar network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 590(C).
    • Handle: RePEc:eee:phsmap:v:590:y:2022:i:c:s0378437121009584
    DOI: 10.1016/j.physa.2021.126772
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    References listed on IDEAS

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    1. Xiao, Yuzhi & Zhao, Haixing, 2013. "New method for counting the number of spanning trees in a two-tree network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(19), pages 4576-4583.
    2. Zhang, Zhongzhi & Wu, Bin & Lin, Yuan, 2012. "Counting spanning trees in a small-world Farey graph," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(11), pages 3342-3349.
    3. Comellas, Francesc & Miralles, Alícia & Liu, Hongxiao & Zhang, Zhongzhi, 2013. "The number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(12), pages 2803-2806.
    4. Li, Tianyu & Yan, Weigen, 2019. "Enumeration of spanning trees of 2-separable networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 536(C).
    5. Lu, Zhe-Ming & Guo, Shi-Ze, 2012. "A small-world network derived from the deterministic uniform recursive tree," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 87-92.
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