Degree Variance and the Fuzzy Sigma Index in Fuzzy Graphs

TL;DR

This paper extends the classical degree variance measure to fuzzy graphs, introducing the fuzzy sigma index and analyzing its properties, bounds, and behavior under operations.

Contribution

It defines the fuzzy sigma index, establishes its fundamental properties, bounds, and behavior, providing a foundation for variance-based indices in fuzzy graph theory.

Findings

Derived sharp bounds for the fuzzy sigma index.

Analyzed the index's behavior under fuzzy graph operations.

Established fundamental properties of the fuzzy sigma index.

Abstract

The sigma index of a graph, defined as the population variance of its degree sequence, is a fundamental measure of structural irregularity. In this paper, we introduce and systematically investigate its natural extension to fuzzy graphs, termed the fuzzy sigma index $$ \sigma^*(\Gamma) = \frac{1}{n} \sum_{v \in V(\Gamma)} \left( d_\Gamma(v) - \frac{2\,\mathrm{ew}}{n}\right)^2, $$ where $d_\Gamma(v)$ denotes the fuzzy degree of a vertex $v$, and $\mathrm{ew}$ represents the fuzzy size of the fuzzy graph $\Gamma=(V,\nu, \mu)$. We establish several fundamental properties of this topological index. In particular, we derive sharp lower and upper bounds. Analyze the behavior of $\sigma^*(\Gamma)$ under standard fuzzy graph operations. This work provides a foundation for further study of variance-based topological indices in fuzzy graph theory.