کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6420183 | 1631785 | 2015 | 8 صفحه PDF | دانلود رایگان |
The Wiener index W(G) of a connected graph G is defined to be the sum âu, 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 âe, 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)â¥Î´2â14W(G) where δ denotes the minimum degree in G. In this paper, we extend/improve these two results by showing that We(G)â¥Î´24W(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.
Journal: Applied Mathematics and Computation - Volume 269, 15 October 2015, Pages 714-721