For a graph $G$, the mean subtree order of $G$ is the average order of a subtree of $G$. In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph $G$ and every pair of distinct vertices $u$ and $v$ of $G$, the addition of the edge between $u$ and $v$ increases the mean subtree order. In fact, we show that the addition of a single edge between a pair of nonadjacent vertices in a graph of order $n$ can decrease the mean subtree order by as much as $n/3$ asymptotically. We propose the weaker conjecture that for every connected graph $G$ which is not complete, there exists a pair of nonadjacent vertices $u$ and $v$, such that the addition of the edge between $u$ and $v$ increases the mean subtree order. We prove this conjecture in the special case that $G$ is a tree.

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关于加边下图的平均子树顺序

对于图 $G$，$G$ 的平均子树顺序是 $G$ 子树的平均顺序。在这篇笔记中，我们提供了 Chin、Gordon、MacPhee 和 Vincent 的最近猜想的反例，即对于每个连通图 $G$ 以及 $G$ 的每对不同顶点 $u$ 和 $v$，添加$u$ 和 $v$ 之间的边增加了平均子树顺序。事实上，我们表明，在 $n$ 阶图中的一对不相邻顶点之间添加一条边可以使平均子树阶数渐近地减少 $n/3$。我们提出了较弱的猜想，对于每个不完整的连通图 $G$，都存在一对不相邻的顶点 $u$ 和 $v$，使得 $u$ 和 $v$ 之间的边的增加增加了平均子树顺序。我们在 $G$ 是一棵树的特殊情况下证明了这个猜想。