A sharp lower bound for the Wiener index of a graph

Given a simple connected undirected graph G, the Wiener index W(G) of G is defined as half the sum of the distances over all pairs of vertices of G. In practice, G corresponds to what is known as the molecular graph of an organic compound. We obtain a sharp lower bound for W(G) of an arbitrary graph in terms of the order, size and diameter of G.

💡 Research Summary

The paper addresses a long‑standing gap in the study of the Wiener index W(G), a distance‑based topological invariant widely used in chemical graph theory. While many works have derived upper bounds or exact formulas for specific families of graphs (trees, cycles, complete graphs, etc.), a sharp universal lower bound expressed solely in terms of elementary graph parameters has been missing. The authors fill this void by proving a new inequality that relates W(G) to the order n, size m, and diameter D of any simple connected undirected graph G.

The main result (Theorem 1) states that for every such graph
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