The exponential of the second Zagreb index of a graph G with n vertices is defined as

$$\begin{aligned} e^{{\mathcal {M}}_{2}}\left( G\right) =\sum _{1\le i\le j\le n-1}m_{i,j}\left( G\right) e^{ij}, \end{aligned}$$

where \(m_{i,j}\) 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 \(e^{M_{2}}\). In this paper we show that the balanced double star tree has maximum value of \(e^{{\mathcal {M}}_{2}}\).

中文翻译：

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平衡双星具有最大指数第二萨格勒布指数

具有n个顶点的图G的第二Zagreb指数的指数定义为

$$ \ begin {aligned} e ^ {{\ mathcal {M}} _ {2}} \ left（G \ right）= \ sum _ {1 \ le i \ le j \ le n-1} m_ {i ，j} \ left（G \ right）e ^ {ij}，\ end {aligned} $$

其中\（m_ {i，j} \）是连接度为i和j的顶点的边数。众所周知，在所有具有n个顶点的树中，路径的最小值为\（e ^ {M_ {2}} \）。在本文中，我们证明平衡双星树的最大值为\（e ^ {{\数学{M}} _ {2}} \）。