Diminished Sombor index and its relationship with topological indices

TL;DR

This paper introduces the Diminished Sombor index, explores its mathematical bounds and relationships with classical topological indices, and characterizes extremal graphs where these bounds are tight.

Contribution

It establishes sharp bounds and inequalities for the Diminished Sombor index in relation to classical topological indices, providing new insights into graph invariants.

Findings

Sharp bounds for DSO in terms of classical indices

Relationships and inequalities between DSO and other indices

Characterizations of extremal graphs achieving bounds

Abstract

In this paper, we investigate the Diminished Sombor index (DSO), a recently introduced degree-based topological index for a simple graph $G$, defined as \[ DSO(G) = \sum_{uv \in E} \frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}, \] 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, Albertson, Harmonic, Randi\'c, and geometric-arithmetic indices. The relationships and inequalities between DSO and these indices are analyzed thoroughly, with characterizations of extremal graphs achieving equality conditions.