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

Abstract

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

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