Improved Lower Bounds on the General Reduced Second Zagreb Index of Trees and Unicyclic Graphs

For a simple graph $\mathrm{\Gamma }$ and a real number $\lambda$, the general reduced second Zagreb index is defined by the formula $$GR{M}_{\lambda }(\mathrm{\Gamma })=\sum _{ab\in E(\mathrm{\Gamma })}[({\mathrm{°}}_{\mathrm{\Gamma }}(a)+\lambda )({\mathrm{°}}_{\mathrm{\Gamma }}(b)+\lambda )]\mathrm{.}$$ A sharp lower bound for $GR{M}_{\lambda }$ over all trees of given order and maximum degree under the condition that $\lambda \ge -\frac{1}{2}$ is established. A parallel result is proved for unicyclic graphs under the condition $\lambda \ge -\frac{1}{2}$. The corresponding minimal trees and unicyclic graphs are identified. These findings improve upon the lower bounds previously established by Buyantogtokh, Horoldagva, and Das concerning $GR{M}_{\lambda }$ of trees and unicyclic graphs of given order.