Degree Variance and the Fuzzy Sigma Index in Fuzzy Graphs

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 $${\sigma }^{\ast }(\mathrm{\Gamma })=\frac{1}{n}\sum _{v\in V(\mathrm{\Gamma })}{(d_{\mathrm{\Gamma }}(v)-\frac{2\mathrm{e}\mathrm{w}}{n})}^2,$$ where $d_{\mathrm{\Gamma }}(v)$ denotes the fuzzy degree of a vertex $v$, and $\mathrm{e}\mathrm{w}$ represents the fuzzy size of the fuzzy graph $\mathrm{\Gamma }=(V,\nu ,\mu )$. We establish several fundamental properties of this topological index. In particular, we derive sharp lower and upper bounds. Analyze the behavior of ${\sigma }^{\ast }(\mathrm{\Gamma })$ under standard fuzzy graph operations. This work provides a foundation for further study of variance-based topological indices in fuzzy graph theory.