On two conjectures concerning Kemeny's constant of graphs

Abstract

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.