A vertex subset in a graph that induces a connected subgraph is referred to as a connected set. Counting the number of connected sets in a graph is generally a #P-complete problem. In our recent work [Graphs Combin. (2024)], a linear recursive algorithm was designed to count in any Apollonian network. In this paper we extend our research by establishing a tight upper bound on in Apollonian networks with an order of , 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 in Apollonian networks with a given order.

中文翻译：

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阿波罗网络中连通集的数量


图中导出连通子图的顶点子集称为连通集。计算图中连通集的数量通常是一个#P完全问题。在我们最近的工作中[Graphs Combin。 (2024)]，线性递归算法被设计用于在任何阿波罗网络中计数。在本文中，我们通过在阿波罗网络中建立一个严格的上限（阶数为 ）以及达到该上限的图的特征来扩展我们的研究。我们的方法主要利用线性规划技术。此外，我们提出了关于给定阶数的阿波罗网络下界的猜想。