AI 中文总结
本文针对直径为2的连通图，通过推导有效电阻估计，证实Kim等人2026年关于Kemeny常数为O(n)的猜想，还建立其与补图Kemeny常数乘积的Nordhaus-Gaddum界，证实另一相关猜想。
AI 中文摘要
设G为阶数n、直径为2的连通图，对任意两个不同顶点u,v∈V(G)，本文证明u与v之间的有效电阻r_G(u,v)满足r_G(u,v)≤(3+√5)(1/d_G(u)+1/d_G(v))，其中d_G(u)表示G中顶点u的度数。利用该电阻估计，进一步证明G的Kemeny常数𝒦(G)满足𝒦(G)≤(3+√5)(n-1)，从而证实Kim等人（2026）提出的“所有直径为2的连通图的Kemeny常数为O(n)”的猜想。此外，本文还建立了Nordhaus-Gaddum界：当G及其补图$\u00af{G}$均连通时，𝒦(G)𝒦($\u00af{G}$)=O(n⁴)，证实了Kim等人（2026）提出的另一猜想。
英文摘要
Kemeny's constant for a connected graph $G$, denoted by $\mathcal{K}(G)$, is the expected time for a random walk to reach a randomly chosen vertex $u$, regardless of the choice of the initial vertex. Recently, Kim et al. (2026) proposed two conjectures on Kemeny's constant. The first conjecture asserts that if $G$ is a connected graph of order $n$ and diameter 2, then $\mathcal{K}(G) = O(n)$. The second conjecture asserts that if $G$ be a graph of order $n$, then $\min\{\mathcal{K}(G), \mathcal{K}(\overline{G})\} = O(n)$, and if both $G$ and $\overline{G}$ are connected, then $\mathcal{K}(G)\mathcal{K}(\overline{G}) = O(n^4)$, where $\overline{G}$ denotes the complement of $G$. In this paper, we confirm both conjectures. For the first conjecture, we prove that if $G$ is a connected graph of order $n$ and diameter 2, then \[ \mathcal{K}(G) \leq (3 + \sqrt{5})(n - 1). \] For the second conjecture, we prove that for any $n$-vertex graph $G$, \[ \min\{\mathcal{K}(G), \mathcal{K}(\overline{G})\} \leq (8 + 2\sqrt{5})n - (10 + 2\sqrt{5}). \] Moreover, if both $G$ and $\overline{G}$ are connected, then \[ \mathcal{K}(G)\mathcal{K}(\overline{G}) \leq \frac{3 + \sqrt{5}}{2}n^4. \] Our proof relies on effective estimates on resistance distances and spectral gaps of graphs.
