The balanced double star has maximum exponential second Zagreb index

Abstract

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 $e^{{\mathcal {M}}_{2}}$.