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

TL;DR

This paper identifies the tree structure that maximizes the exponential of the second Zagreb index, specifically showing that the balanced double star achieves this maximum among all trees.

Contribution

It proves that the balanced double star maximizes the exponential of the second Zagreb index among all trees, solving an open problem in the field.

Findings

The exponential of the second Zagreb index is minimized by the path $P_n$.

The exponential of the second Zagreb index is maximized by the balanced double star.

The paper resolves an open problem regarding extremal trees for this index.

Abstract

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].