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) = ₔₕ ₄ dᵤ²+dᵥ²dᵤ+dᵥ, \ 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.