University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 17 3 2026 09 01 Extremal Results and Inequalities for the Diminished Sombor Index of Graphs 259 270 115729 10.22052/ijmc.2026.258144.2101 EN Mohammad Hadi Akhbari Department of Mathematics‎, ‎Es.C.‎, ‎Islamic Azad University‎, ‎Estahban‎, ‎Iran 0000-0002-2669-4966 Fateme Movahedi Department of Mathematics‎, ‎Faculty of Sciences‎, ‎Golestan University‎, ‎Gorgan‎, ‎Iran 0000-0001-7863-7915 Journal Article 2025 12 16 ‎For any simple graph \(G \) having an edge set \(E\)‎, ‎the Diminished Sombor index (DSO) is formulated as‎: <br>‎DSO(G)‎= \sum_{uv \in E} \frac{\sqrt{\zeta_u^2‎ + ‎\zeta_v^2}}{\zeta_u‎ + ‎\zeta_v}‎,<br>‎where \(\zeta_u\) is the degree of the vertex u‎.<br>The present paper focuses on the DSO index of simple graphs‎, ‎where precise bounds are derived based on fundamental structural parameters‎, ‎including order‎, ‎size‎, ‎maximum degree‎, ‎and minimum degree‎. ‎We also establish sharp inequalities relating DSO to a collection of standard topological indices‎, ‎namely the Randi\'c‎, ‎Geometric-Arithmetic‎, ‎Zagreb‎, ‎Sombor‎, ‎and Harmonic indices‎, ‎as well as inverse degree-related indices‎. ‎For each bound‎, ‎extremal graphs attaining equality are fully characterized‎. ‎Additionally‎, ‎we analyze the behavior of DSO for specific graph families‎, ‎including trees and regular graphs‎. Diminished Sombor index mathematical chemistry Degree-based topological index https://ijmc.kashanu.ac.ir/article_115729_c5c2a991d92b4f9244c12c38238fbbb6.pdf