Nordhaus-Gaddum-type inequality for the hyper-Wiener index of graphs when decomposing into three parts

Let k≥2 be an integer, a k-decomposition(^G1, ^G2,⋯,^Gk) of a graph G is a partition of its edge set to form k spanning subgraphs ^G1,^G2,.,^Gk. The hyper-Wiener index WW is one of the recently conceived distance-based graph invariants (Randi 1993 [15]): WW=WW(G):=12W(G)+12^W2(G), where W is the Wiener index (Wiener 1947 [18]) and ^W2 is the sum of squares of distance of all pairs of vertices in G. In this paper, we investigate the Nordhaus-Gaddum-type inequality of a 3-decomposition of ^Kn for the hyper-Wiener index: 7n2≤WW(^G1)+WW(^G2)+WW( ^G3)≤2n+24+n2+4(n-1). The corresponding extremal graphs are characterized.