Extremal Results for the General Reduced Second Zagreb Index and Sombor-Type Indices on Trees

Abstract

We investigate extremal properties of vertex-degree-based graph invariants on trees, focusing primarily on the general reduced second Zagreb index $GRM_{-2}$ and the fifth Sombor-type index $SO_5$. We provide sharp lower and upper bounds for these indices in terms of the maximum degree and order of trees. Furthermore, we characterize the extremal structures attaining equality. These results provide complete solutions to several open problems recently posed in the literature concerning extremal behavior of $GRM_{-2}$ and $SO_5$ on trees.