  An extension of the well-known Szeged index was introduced recently, named as
weighted Szeged index ($\textrm{sz}(G)$). This paper is devoted to
characterizing the extremal trees and graphs of this new topological invariant.
In particular, we proved that the star is a tree having the maximal
$\textrm{sz}(G)$. Finding a tree with the minimal $\textrm{sz}(G)$ is not an
easy task to be done. Here, we present the minimal trees up to 25 vertices
obtained by computer and describe the regularities which retain in them. Our
preliminary computer tests suggest that a tree with the minimal
$\textrm{sz}(G)$ is also the connected graph of the given order that attains
the minimal weighted Szeged index. Additionally, it is proven that among the
bipartite connected graphs the complete balanced bipartite graph
$K_{\left\lfloor n/2\right\rfloor\left\lceil n/2 \right\rceil}$ attains the
maximal $\textrm{sz}(G)$\,. We believe that the $K_{\left\lfloor
n/2\right\rfloor\left\lceil n/2 \right\rceil}$ is a connected graph of given
order that attains the maximum $\textrm{sz}(G)$.
