Abstract
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.