A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching Vertices

The Wiener polarity index of a graph G , usually denoted by W p G , is defined as the number of unordered pairs of those vertices of G that are at distance 3. A vertex of a tree with degree at least 3 is called a branching vertex. A segment of a tree T is a nontrivial path S whose end-vertices have degrees different from 2 in T and every other vertex (if exists) of S has degree 2 in T . In this note, the best possible sharp lower bounds on the Wiener polarity index W p are derived for the trees of fixed order and with a given number of branching vertices or segments, and all the trees attaining this lower bound are characterized.