Further results on the lower bound on reduced Zagreb index of trees

TL;DR

This paper refines and extends lower bound results for the general reduced second Zagreb index in trees, providing new bounds and characterizations for specific maximum degrees and parameter values.

Contribution

It corrects and extends previous equality results for the minimal GRM_or trees, and introduces two approaches for molecular trees with specific degrees.

Findings

Corrected and extended bounds for GRM_or trees with or rom -1.

Determined minimal GRM_or molecular trees with or rom 3 and 4 at rom -2.

Characterized extremal trees achieving these bounds.

Abstract

For a graph $G$, the general reduced second Zagreb index is defined as $$GRM_\lambda (G) = \sum_{uv \in E} (deg(u) + \lambda) (deg(v) + \lambda),$$ where $\lambda$ 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 $GRM_\lambda$ for $\lambda \geq -1$ among trees with $n$ vertices and a maximal degree $\Delta$. 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 $\Delta = 3$ and $\Delta = 4$ in $\lambda = -2$, and characterize the extremal trees.