The hyper-Wiener index of graph operations

Let GG be a graph. The distance d(u,v)d(u,v) between the vertices uu and vv of the graph GG is equal to the length of a shortest path that connects uu and vv. The Wiener index W(G)W(G) is the sum of all distances between vertices of GG, whereas the hyper-Wiener index WW(G)WW(G) is defined as WW(G)=12W(G)+12∑{u,v}⊆V(G)d(u,v)2. In this paper the hyper-Wiener indices of the Cartesian product, composition, join and disjunction of graphs are computed. We apply some of our results to compute the hyper-Wiener index of C4C4 nanotubes, C4C4 nanotori and qq-multi-walled polyhex nanotori.