Abstract
For a connected graph \( G \), the Wiener index \( W \) and the Harary index \( H \) are defined as \( W = \sum_{u,v} d(u,v) \) and \( H= \sum_{u,v} 1/d(u,v) \), respectively. Recently, in MATCH91 (2024) 287, the extremal value of the product \( W \cdot H \) was studied and shown that \( W \!\cdot H\! \geq \binom{n}{2} \), with equality for the complete graph. We now extend this result to all graphs of order \( n \) and size \( m \), and characterize the respective species with minimum \( W \!\cdot\! H \)-value.
