  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 $\pi$-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].
