Extremal trees with respect to the first and second reformulated Zagreb index

Abstract

Let $G$ be a graph with edge set $E(G)$. The first and second reformulated Zagreb indices of $G$ are defined as $E M_1(G)=\sum_{e \in E(G)} \operatorname{deg}(e)^2$ and $E M_2(G)=\sum_{e \sim f} \operatorname{deg}(e) \operatorname{deg}(f)$,respectively, where $\operatorname{deg}(e)$ denotes the degree of the edge $e$, and $e \sim f$ means that the edges $e$ and $f$ are incident. In this paper, the extremal trees with respect to the first and second reformulated Zagreb indices are presented.