Abstract
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 \( W(G) =\sum_{\{u,v\} \subseteq G} d(u,v) \). Another index of graphs closely related to Wiener index is the Harary index, defined as \( H(G) =\sum_{\{u,v\} \subseteq G,u \neq v} 1/d(u, v) \). Recently, Gutman posed a the following conjecture:
For a positive integer \( n\geq5 \), let \( T_n \) be any \( n \)-vertex tree different from the star \( S_n \) and the path \( P_n \). Then
\( W(S_n)\cdot H(S_n)
