Iranian Journal of Mathematical Chemistry

Iranian Journal of Mathematical Chemistry

Extremal Results and Inequalities for the Diminished Sombor Index of Graphs

Document Type : Research Paper

Authors
Mohammad Hadi Akhbari 1
Fateme Movahedi 2
1 Department of Mathematics‎, ‎Es.C.‎, ‎Islamic Azad University‎, ‎Estahban‎, ‎Iran
2 Department of Mathematics‎, ‎Faculty of Sciences‎, ‎Golestan University‎, ‎Gorgan‎, ‎Iran
10.22052/ijmc.2026.258144.2101
Abstract
‎For any simple graph \(G \) having an edge set \(E\)‎, ‎the Diminished Sombor index (DSO) is formulated as‎:
‎DSO(G)‎= \sum_{uv \in E} \frac{\sqrt{\zeta_u^2‎ + ‎\zeta_v^2}}{\zeta_u‎ + ‎\zeta_v}‎,
‎where \(\zeta_u\) is the degree of the vertex u‎.
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.
Keywords
Diminished Sombor index
Mathematical chemistry
Degree-based topological index
Subjects
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Iranian Journal of Mathematical Chemistry
Volume 17, Issue 3
In Progress
Summer 2026
Pages 259-270