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The maximal tree with respect to the exponential of the second Zagreb index

The second Zagreb index is $M_2(G)=\sum_{uv\in E(G)}d_{G}(u)d_{G}(v)$. It was found to occur in certain approximate expressions of the total $π$-electron energy of alternant hydrocarbons and used by various researchers in their QSPR and QSAR studies. Recently the exponential of a vertex-degree-based topological index was introduced. It is known that among all trees with $n$ vertices, the exponential of the second Zagreb index $e^{M_2}$ attains its minimum value in the path $P_n$. In this paper, we show that $e^{M_2}$ attains its maximum value in the balanced double star with $n$ vertices and solve an open problem proposed by Cruz and Rada [R. Cruz, J. Rada, The path and the star as extremal values of vertex-degree-based topological indices among trees, MATCH Commun. Math. Comput. Chem. 82 (3) (2019) 715-732].