On Extremal Graphs of Weighted Szeged Index
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An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index (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 sz(G). Finding a tree with the minimal 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 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_ n/2 n/2 attains the maximal sz(G) . We believe that the K_ n/2 n/2 is a connected graph of given order that attains the maximum sz(G).
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