ABSTRACT

Kemeny's constant $\kappa \left(G\right)$ of a connected graph G is a measure of the expected transit time for the random walk associated with G. In the current work, we consider the case when G is a tree and, in this setting, we provide lower and upper bounds for $\kappa \left(G\right)$ in terms of the order n and diameter δ of G by using two different techniques. The lower bound is given as Kemeny's constant of a particular caterpillar tree and, as a consequence, it is sharp. The upper bound is found via induction, by repeatedly removing pendent vertices from G. By considering a specific family of trees – the broom-stars – we show that the upper bound is asymptotically sharp.

中文翻译：

---

关于具有固定阶数和直径的树的 Kemeny 常数

摘要

凯梅尼常数$\kappa \left(G\right)$连通图G是与G相关的随机游走的预期传输时间的度量。在当前的工作中，我们考虑G是一棵树的情况，在这种情况下，我们为$\kappa \left(G\right)$通过使用两种不同的技术，根据G的阶数 n和直径δ 。下界作为特定毛虫树的 Kemeny 常数给出，因此，它是尖锐的。上限是通过归纳法找到的，通过从G中反复移除下垂的顶点。通过考虑一个特定的树族——扫帚星——我们证明了上限是渐近尖锐的。