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

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