Sharp Bounds and Extremal Fuzzy Graphs for the Fuzzy Sombor Index

TL;DR

This paper investigates the extremal properties of the fuzzy Sombor index in fuzzy graphs, establishing bounds and relationships with other fuzzy topological indices.

Contribution

It introduces bounds and extremal characterizations for the fuzzy Sombor index, extending classical graph invariants to fuzzy graph contexts.

Findings

Maximum and minimum fuzzy Sombor index values characterized for regular fuzzy graphs.

Significant inequalities established between fuzzy Sombor index and other fuzzy topological indices.

Abstract

The fuzzy Sombor index applies the classical Sombor index to fuzzy graphs, incorporating both edge membership values and fuzzy vertex degrees. For $\alpha>1$, the general fuzzy Sombor index it is defined as \[ \mathrm{SO}^{\mu}_{\alpha}(\Gamma)=\sum_{uv\in V(\Gamma)} \left( \mu(u,v)\, \sqrt{\mu_u^2+\mu_v^2} \right)^{\alpha}. \] This paper analyses extremal features of $\mathrm{SO}^{\mu}$ across different types of fuzzy graphs. We determine the maximum value (resp. minimum value) of $\mathrm{SO}^{\mu}$ characterise in regular fuzzy graph. We established significant inequality between the fuzzy Sombor index and other well-known fuzzy topological indices.