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)over-bar, Kf(G) + Kf((G)over-bar) <= n(3) - n/6 + n Sigma(n-1)(k=1)1/n-4sin(2)k pi/2n, with equality if and only if G or (G)over-bar is the path graph P-n. In this paper, by employing com- binatorial and electrical techniques, we show that the conjecture is true except for a complementary pair of small graphs on five vertices. (C) 2018 Elsevier Inc. All rights reserved.