For a simple graph \(\Gamma \) and a real number \(\lambda \) , the general reduced second Zagreb index is defined by the formula \(\begin{aligned} GRM_\lambda (\Gamma )=\sum _{ab\in E(\Gamma )}[(\deg _{\Gamma }(a)+\lambda )(\deg _{\Gamma }(b)+\lambda )]\,. \end{aligned}\) 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.
