Degree Variance and the Fuzzy Sigma Index in Fuzzy Graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918446920368128 |
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| author | Abdullah, Duaa |
| author_facet | Abdullah, Duaa |
| contents | 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 $$ σ^*(Γ) = \frac{1}{n} \sum_{v \in V(Γ)} \left( d_Γ(v) - \frac{2\,\mathrm{ew}}{n}\right)^2, $$ where $d_Γ(v)$ denotes the fuzzy degree of a vertex $v$, and $\mathrm{ew}$ represents the fuzzy size of the fuzzy graph $Γ=(V,ν, μ)$. We establish several fundamental properties of this topological index. In particular, we derive sharp lower and upper bounds. Analyze the behavior of $σ^*(Γ)$ under standard fuzzy graph operations. This work provides a foundation for further study of variance-based topological indices in fuzzy graph theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_13113 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Degree Variance and the Fuzzy Sigma Index in Fuzzy Graphs Abdullah, Duaa General Mathematics 05C72, 05C07, 05C09, 05C35, 05C76 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 $$ σ^*(Γ) = \frac{1}{n} \sum_{v \in V(Γ)} \left( d_Γ(v) - \frac{2\,\mathrm{ew}}{n}\right)^2, $$ where $d_Γ(v)$ denotes the fuzzy degree of a vertex $v$, and $\mathrm{ew}$ represents the fuzzy size of the fuzzy graph $Γ=(V,ν, μ)$. We establish several fundamental properties of this topological index. In particular, we derive sharp lower and upper bounds. Analyze the behavior of $σ^*(Γ)$ under standard fuzzy graph operations. This work provides a foundation for further study of variance-based topological indices in fuzzy graph theory. |
| title | Degree Variance and the Fuzzy Sigma Index in Fuzzy Graphs |
| topic | General Mathematics 05C72, 05C07, 05C09, 05C35, 05C76 |
| url | https://arxiv.org/abs/2604.13113 |