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We study Nordhaus-Gaddum problems for Kemeny's constant $\mathcal{K}(G)$ of aconnected graph $G$. We prove bounds on$\min\{\mathcal{K}(G),\mathcal{K}(\overline{G})\}$ and the product$\mathcal{K}(G)\mathcal{K}(\overline{G})$ for various families of graphs. Inparticular, we show that if the maximum degree of a graph $G$ on $n$ verticesis $n-O(1)$ or $n-\Omega(n)$, then$\min\{\mathcal{K}(G),\mathcal{K}(\overline{G})\}$ is at most $O(n)$.

Bounds on Kemeny's constant of a graph and the Nordhaus-Gaddum problem