Improved Lower Bounds on the General Reduced Second Zagreb Index of Trees and Unicyclic Graphs
For a simple graph $\Gamma$ and a real number $\lambda$, the general reduced second Zagreb index is defined by the formula $$GRM_\lambda(\Gamma)=\sum_{ab\in E(\Gamma)}[(\deg_{\Gamma}(a)+\lambda)(\deg_{\Gamma}(b)+\lambda)]\,.$$ A sharp lower bound for $GRM_\lambda$ 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 $GRM_\lambda$ of trees and unicyclic graphs of given order.
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