关于加权树的平均到达时间与肯尼系数
On average hitting time and Kemeny's constant for weighted trees
AI总结：
研究加权树中随机游走的平均到达时间与肯尼系数，证明在固定边权多重集下，极化路径最大化平均到达时间，星型图最小化，同时给出肯尼系数极值的特征。
摘要：
对于连通图G，平均到达时间α(G)和肯尼系数κ(G)是两个相似的量，均衡量随机游走在G上两个随机选择顶点之间的所需时间。我们证明，在所有边权形成固定多重集的加权树中，α在极化路径中最大，在唯一加权星型图中最小。我们还获得了此类加权树集合中κ最大化和最小化的特征。我们的证明基于α和κ的森林公式。
英文摘要：
For a connected graph $G$, the average hitting time $α(G)$ and the Kemeny's constant $κ(G)$ are two similar quantities, both measuring the time for the random walk on $G$ to travel between two randomly chosen vertices. We prove that, among all weighted trees whose edge weights form a fixed multiset, $α$ is maximized by a special type of ``polarized'' paths and is minimized by a unique weighted star graph. We also obtain a similar characterization of the $κ$-maximizing and $κ$-minimizing elements among such a collection of weighted trees. Our proofs are based on the forest formulas for $α$ and $κ$.
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