An inequality between the edge-Wiener index and the Wiener index of a graph

摘要

The Wiener index W(G) of a connected graph G is defined to be the sum ∑u, vd(u, v) of distances between all unordered pairs of vertices in G. Similarly, the edge-Wiener index We(G) of G is defined to be the sum ∑e, fd(e, f) of distances between all unordered pairs of edges in G, or equivalently, the Wiener index of the line graph L(G). Wu (2010) showed that We(G) ≥ W(G) for graphs of minimum degree 2, where equality holds only when G is a cycle. Similarly, in Knor et al. (2014), it was shown that We(G)≥δ2−14W(G) where δ denotes the minimum degree in G. In this paper, we extend/improve these two results by showing that We(G)≥δ24W(G) with equality satisfied only if G is a path on 3 vertices or a cycle. Besides this, we also consider the upper bound for We(G) as well as the ratio We(G)W(G). We show that among graphs G on n vertices We(G)W(G) attains its minimum for the star.