We study Nordhaus-Gaddum problems for Kemeny's constant K(G) of a
connected graph G. We prove bounds on
min{K(G),K(G)} and the product
K(G)K(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−Ω(n), then
min{K(G),K(G)} is at most O(n)