Trees with minimum weighted Szeged index

The weighted Szeged index is a recent extension of the well-known Szeged index. Trees are conjectured to achieve the minimum weighted Szeged index among all graphs with a given number of vertices. In this paper, we present new tools to analyze and characterize trees with minimum weighted Szeged index. We exhibit the best trees with up to 130 vertices and use this information, together with our formal results, to propose certain conjectures on the structure of such minimal trees. In particular, we prove that they have maximum degrees at most ten, and conjecture that the right bound is six. (A recent paper of Atanasov, Furtula, and Škrekovski proves a bound of 16.) We hope these conjectures and initial results will motivate further developments on this interesting topological index.