The balanced double star has maximum exponential second Zagreb index

The exponential of the second Zagreb index of a graph G with n vertices is defined as $$\begin{aligned} e^{{\mathcal {M}}_{2}}\left( G\right) =\sum _{1\le i\le j\le n-1}m_{i,j}\left( G\right) e^{ij}, \end{aligned}$$ e M 2 G = ∑ 1 ≤ i ≤ j ≤ n - 1 m i , j G e ij , where $$m_{i,j}$$ m i , j is the number of edges joining vertices of degree i and j. It is well known that among all trees with n vertices, the path has minimum value of $$e^{M_{2}}$$ e M 2 . In this paper we show that the balanced double star tree has maximum value of $$e^{{\mathcal {M}}_{2}}$$ e M 2 .