Note on the Product of Wiener and Harary Indices

For a simple graph G, we use d(u, v) to denote the distance between two vertices u, v in G. The Wiener index is defined as the sum of distances between all unordered pairs of vertices in a graph. In other word, given a connected graph G, the Wiener index W(G) of G is W(G) = Sigma(d(u, v))({u,v}subset of G). Another index of graphs closely related to Wiener index is the Harary index, defined as H(G) = Sigma({u,v}subset of G,u not equal v) 1/d(u, v). Recently, Gutman posed a the following conjecture: For a positive integer n >= 5, let Tn be any n-vertex tree different from the star S-n and the path P-n. Then W(S-n) center dot H(S-n) < W(T-n) center dot H(T-n) < W(P-n) center dot H(Pn). In this paper, we confirm the lower bound of the conjecture and disproof the upper bound of it.