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

Let k≥2k≥2 be an integer, a kk-decomposition(G1,G2,⋯,Gk)(G1,G2,⋯,Gk) of a graph GG is a partition of its edge set to form kk spanning subgraphs G1,G2,…,GkG1,G2,…,Gk. The hyper-Wiener index WWWW is one of the recently conceived distance-based graph invariants (Randi 1993 [15]): WW=WW(G):=12W(G)+12W2(G), where WW is the Wiener index (Wiener 1947 [18]) and W2W2 is the sum of squares of distance of all pairs of vertices in GG. In this paper, we investigate the Nordhaus–Gaddum-type inequality of a 33-decomposition of KnKn for the hyper-Wiener index: 7n2≤WW(G1)+WW(G2)+WW(G3)≤2n+24+n2+4(n−1). The corresponding extremal graphs are characterized.