Abstract
An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index (wSz(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 wSz(G). Finding a tree with the minimal wSz(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 wSz(G) is, at the same time, 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-[n/2] [n/2] attains the maximal wSz(G). We believe that the K([n/2] [n/2] )is a connected graph of given order that attains the maximum wSz(G).