For a graph $G$, the general reduced second Zagreb index is defined as $$GRM_λ(G) = \sum_{uv \in E} (deg(u) + λ) (deg(v) + λ),$$ where $λ$ is an arbitrary real number and $deg (v)$ is the degree of the vertex $v$. In this paper, we extend and correct the equality results from [N. Dehgardia, S. Klav\v zar, {\it Improved lower bounds on the general reduced second Zagreb index of trees}, preprint (2023)] regarding the minimal value of $GRM_λ$ for $λ\geq -1$ among trees with $n$ vertices and a maximal degree $Δ$. Furthermore, we complement these results with two distinct approaches to determine the minimum value of the general reduced second Zagreb index for molecular trees with $Δ= 3$ and $Δ= 4$ in $λ= -2$, and characterize the extremal trees.

翻译：对于图$G$，广义约化第二Zagreb指数定义为 $$GRM_λ(G) = \sum_{uv \in E} (deg(u) + λ) (deg(v) + λ),$$ 其中$λ$为任意实数，$deg(v)$表示顶点$v$的度数。本文扩展并修正了文献[N. Dehgardia, S. Klav\v zar, {\it 树的一般约化第二Zagreb指数的改进下界}, 预印本 (2023)]中关于在$λ\geq -1$条件下、顶点数为$n$且最大度为$Δ$的树族中$GRM_λ$最小值的等式结论。此外，我们通过两种不同方法补充了当$λ= -2$时、分子树在$Δ= 3$和$Δ= 4$情形下广义约化第二Zagreb指数最小值的确定结果，并刻画了极值树的结构特征。