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The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of Kf((T)) and the first four maximum values of Kf((U)), where (T) and (U) are the complements of a tree T and unicyclic graph U, respectively.