Minimizing Kirchhoff index among graphs with a given vertex bipartiteness

The resistance distance between any two vertices of a graph G is defined as the effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index Kf(G) is the sum of the resistance distances between all the pairs of vertices in G. The vertex bipartiteness vb of a graph G is the minimum number of vertices whose deletion from G results in a bipartite graph. In this paper, we characterize the graph having the minimum Kf(G) values among graphs with a fixed number n   of vertices and fixed vertex bipartiteness, 1≤vb≤n−3.1≤vb≤n−3.