A note on chemical trees with minimum Wiener polarity index

The Wiener polarity index (usually denoted by W-p) of a graph G is defined as the number of unordered pairs of the vertices of G which are at distance 3. Denote by CTn the family of all n-vertex chemical trees. In a recent paper, Ashrafi and Ghalavand [1] determined the first three minimum W-p values of n-vertex chemical trees for n > 7 and characterized the chemical trees attaining the first two minimum W-p values among all the members of CTn for n >= 4. In this note, the chemical trees with the third minimum W-p value are characterized from the graph family CTn for n >= 7, and the chemical trees from the family CTn, n >= 4, with the first two minimum W-p values are also obtained in an alternative but shorter way. (C) 2018 Elsevier Inc. All rights reserved.