The number of connected sets in Apollonian networks

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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.