Diminished Sombor index and its relationship with topological indices

In this paper, we investigate the Diminished Sombor index (DSO), a recently introduced degree-based topological index for a simple graph G, defined as \[ DSO(G) = ∑uv ∈ E {√d_u^2+d_v^2}{d_u+d_v}, \] where dᵤ denotes the degree of a vertex u ∈ V. We establish several sharp bounds for this index in terms of classical topological indices such as the Zagreb, Albertson, Harmonic, Randić, and geometric-arithmetic indices. The relationships and inequalities between DSO and these indices are analyzed thoroughly, with characterizations of extremal graphs achieving equality conditions.