On the Kirchhoff index of the complement of a bipartite graph

The Kirchhoff index of a graph G is defined as , where rij is the resistance distance between vi and vj. Let G be a simple bipartite graph, the complement graph of G. First, we get an expression for in terms of the numbers of closed walks of subdivision graph of G. Then, as an application, we determine the trees with the first and second maximum and minimum values of .