Diminished Sombor index and its relationship with topological indices

Abstract

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