Further results on the lower bound on reduced Zagreb index of trees
This work provides incremental improvements in graph theory, specifically for researchers studying topological indices in chemical graph theory.
The paper extends and corrects previous results on the lower bound of the general reduced second Zagreb index for trees, specifically addressing minimal values for certain parameters and characterizing extremal trees, with a focus on molecular trees for specific degree constraints.
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 $Î$. 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 $Î= 3$ and $Î= 4$ in $λ= -2$, and characterize the extremal trees.