The Hyper-Wiener Index of Trees of Order n with Diameter d

The hyper-Wiener index is a kind of extension of the Wiener index, used for predicting physicochemical properties of organic compounds. The hyper-Wiener index WW(G) is defined as WW(G)=1/2∑u,v∈VGdGu,v+dG2u,v with the summation going over all pairs of vertices in G, and dGu,v denotes the distance of the two vertices u and v in the graph G. In this paper, we obtain the second-minimum hyper-Wiener indices among all the trees with n vertices and diameter d and characterize the corresponding extremal graphs