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

Abstract

The Wiener polarity index of a graph $G$ , usually denoted by $W_p\left(G\right)$ , 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.