A note on the minimum Wiener polarity index of trees with a given number of vertices and segments or branching vertices (Q2657432)

Summary: 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.