Degree Variance and the Fuzzy Sigma Index in Fuzzy Graphs

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 $${\sigma }^{\ast }(\Gamma )=\frac{1}{n}\sum _{v\in V(\Gamma )}{\left(d_{\Gamma }(v)-\frac{2\mathrm{ew}}{n}\right)}^2,$$ 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 ${\sigma }^{\ast }(\Gamma )$ under standard fuzzy graph operations. This work provides a foundation for further study of variance-based topological indices in fuzzy graph theory.

11 pages, 4 figures