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The number of connected sets in Apollonian networks

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  • Luo, Zuwen
  • Xu, Kexiang

Abstract

A vertex subset in a graph that induces a connected subgraph is referred to as a connected set. Counting the number of connected sets N(G) in a graph G is generally a #P-complete problem. In our recent work [Graphs Combin. (2024)], a linear recursive algorithm was designed to count N(G) in any Apollonian network. In this paper we extend our research by establishing a tight upper bound on N(G) in Apollonian networks with an order of n≥3, along with a characterization of the graphs that reach this upper bound. Our approach primarily utilizes linear programming techniques. Moreover, we propose a conjecture regarding the lower bound on N(G) in Apollonian networks with a given order.

Suggested Citation

  • Luo, Zuwen & Xu, Kexiang, 2024. "The number of connected sets in Apollonian networks," Applied Mathematics and Computation, Elsevier, vol. 479(C).
    • Handle: RePEc:eee:apmaco:v:479:y:2024:i:c:s0096300324003448
    DOI: 10.1016/j.amc.2024.128883
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    References listed on IDEAS

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    1. Debarun Ghosh & Ervin Győri & Addisu Paulos & Nika Salia & Oscar Zamora, 2020. "The maximum Wiener index of maximal planar graphs," Journal of Combinatorial Optimization, Springer, vol. 40(4), pages 1121-1135, November.
    2. Yang, Yu & Liu, Hongbo & Wang, Hua & Fu, Hongsun, 2015. "Subtrees of spiro and polyphenyl hexagonal chains," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 547-560.
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