发表机构
Okegawa City Okegawa West Junior High School; University of New Mexico; University of Novi Sad（尾久川市尾久川西初中; 新墨西哥大学; 诺维萨德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过基尔霍夫指标与不对称项分解击中时间指标，推导其上下界，并证明星图在固定阶数树中唯一最小化该指标。
AI 中文摘要
我们研究了击中时间指标，这是一个通过有限连通图上简单随机游走的期望击中时间定义的图不变量，并探讨了它与基尔霍夫指标的关系。利用通勤时间恒等式和Tetali公式，我们将击中时间指标分解为一个基尔霍夫指标项和一个非负的不对称项，后者以度加权有效电阻和的形式表达。我们推导了不对称项的下界和上界及其相等条件，并基于度加权电阻和的范围获得了击中时间指标的下界。对于树，我们获得了不对称项的下界和上界，其中上界以维纳指标表达，并刻画了下界中的相等情形。特别地，我们证明了在所有固定阶数的树中，星图唯一地最小化击中时间指标。最后，我们将该分解应用于完全二部图、路径以及通过连接完全图得到的图，并恢复了若干已知的击中时间公式。
英文摘要
We study the hitting time index, a graph invariant defined in terms of expected hitting times of simple random walks on finite connected graphs, through its relation with the Kirchhoff index. Using the commute time identity and Tetali's formula, we decompose the hitting time index into a Kirchhoff-index term and a nonnegative asymmetry term expressed by degree-weighted effective resistance sums. We derive lower and upper bounds for the asymmetry term, together with their equality conditions, and obtain a lower bound for the hitting time index in terms of the range of the degree-weighted resistance sums. For trees, we obtain lower and upper bounds for the asymmetry term, with the upper bound expressed in terms of the Wiener index, and characterize the equality case in the lower bound. In particular, we show that among all trees of fixed order, the hitting time index is minimized uniquely by the star graph. Finally, we apply the decomposition to complete bipartite graphs, paths, and graphs obtained by conjoining complete graphs, and recover several known hitting-time formulas.
Comments23 pages
