Solution to a conjecture on a Nordhaus-Gaddum type result for the Kirchhoff index

Let G be a connected graph. The resistance distance between any two vertices of G is defined as the net effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index of G, denoted by Kf(G), is the sum of resistance distances between all pairs of vertices in G. In [28], it was conjectured that for a connected n-vertex graph G with a connected complement G¯,Kf(G)+Kf(G¯)≤n3−n6+n∑k=1n−11n−4sin2kπ2n,with equality if and only if G or G¯ is the path graph Pn. In this paper, by employing combinatorial and electrical techniques, we show that the conjecture is true except for a complementary pair of small graphs on five vertices.