Minimum of Product of Wiener and Harary Indices

For a connected graph G, the Wiener index W and the Harary index H are defined as W = u,v d(u, v) and H = u,v 1/d(u, v), respectively.Recently, in MATCH 91 (2024) 287, the extremal value of the product W • H was studied and shown that W • H ≥ 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 •H-value.