Although connectivity is a basic concept in graph theory, the enumeration of connected subgraphs of a graph have only recently received attention. The topic of this paper is the average order of a connected induced subgraph of a graph. This generalizes, to graphs in general, the average order of a subtree of a tree. For various infinite families of graphs, we investigate the asymptotic behavior of the proportion of vertices in an induced connected subgraph of average order. For ladders and circular ladders, an explicit closed formula is derived for the average order of a connected induced subgraph in terms of the classic Pell numbers. These formulas imply that, asymptotically, 3/4 of the vertices of a ladder or circular ladder, on average, are present in a connected induced subgraph. Results on such infinite families motivate an assortment of open problems.

中文翻译：

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图的连通顶点集的平均大小——显式公式和开放问题

尽管连通性是图论中的一个基本概念，但图的连通子图的枚举直到最近才受到关注。本文的主题是图的一个连通诱导子图的平均阶数。这概括了一般的图形，树的子树的平均顺序。对于各种无限的图族，我们研究了平均阶的诱导连通子图中顶点比例的渐近行为。对于梯形和圆形梯形，根据经典佩尔数导出连通诱导子图的平均阶数的显式闭合公式。这些公式意味着，渐近地，阶梯或圆形阶梯的顶点的 3/4 平均存在于连通的诱导子图中。