arxiv: 2604.13113 · v1 · submitted 2026-04-12 · 🧮 math.GM

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Degree Variance and the Fuzzy Sigma Index in Fuzzy Graphs

Duaa Abdullah

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classification 🧮 math.GM

keywords fuzzy sigma indexdegree variancefuzzy graphstopological indexsharp boundsgraph operationsfuzzy degree

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The pith

The fuzzy sigma index extends degree variance to fuzzy graphs and yields sharp bounds plus operational rules.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper takes the classical sigma index, the population variance of vertex degrees, and adapts it to fuzzy graphs in which edges carry membership grades between zero and one. The new index is defined as one over n times the sum over vertices of the squared deviation of each fuzzy degree from twice the fuzzy size divided by n. The authors establish sharp lower and upper bounds on its value and determine how the index changes when fuzzy graphs are combined or altered by the usual operations. A reader would care because fuzzy graphs represent networks with partial or uncertain connections, and a simple variance measure supplies a concrete numerical gauge of how uneven those connections are.

Core claim

We introduce the fuzzy sigma index σ*(Γ) = (1/n) ∑_{v ∈ V(Γ)} (d_Γ(v) - 2 ew / n)^2, where d_Γ(v) is the fuzzy degree of vertex v and ew is the fuzzy size of the fuzzy graph Γ. We derive sharp lower and upper bounds for σ*(Γ) and analyze the behavior of the index under standard fuzzy graph operations, thereby providing a foundation for variance-based topological indices in fuzzy graph theory.

What carries the argument

The fuzzy sigma index σ*(Γ), which is the population variance of the sequence of fuzzy degrees with mean equal to twice the fuzzy size divided by the number of vertices.

If this is right

The index attains its minimum precisely when every vertex has the same fuzzy degree.
Greater spread among the fuzzy degrees produces strictly larger values of the index.
The index transforms in a determined way under each standard fuzzy-graph operation such as union or join.
The bounds supply immediate numerical limits on the irregularity of any given fuzzy graph.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The index could be computed for families such as fuzzy paths or fuzzy complete graphs to produce explicit formulas that are unavailable for crisp graphs.
Direct numerical comparison of the fuzzy sigma index against other fuzzy irregularity measures would show which quantity best separates regular from irregular fuzzy graphs.
The same variance construction might be tested on fuzzy graphs whose edge memberships lie outside the unit interval to see whether the bounds continue to hold.

Load-bearing premise

The expression 2 ew / n correctly serves as the mean fuzzy degree for the variance calculation in any fuzzy graph.

What would settle it

A concrete fuzzy graph on which the claimed sharp lower or upper bound for σ*(Γ) is violated.

Figures

Figures reproduced from arXiv: 2604.13113 by Duaa Abdullah.

**Figure 1.** Figure 1: Fuzzy graph with edge thickness is proportional to µ(u, v). The fuzzy degrees are d(v1) = 0.8+0.3 = 1.1, d(v2) = 0.8+0.6 = 1.4 and d(v3) = 0.3+0.6 = 0.9. The fuzzy size is ew(Γ) = 0.8 + 0.3 + 0.6 = 1.7. Thus, for n = 3 we obtain λ = 17/15. Then, σ(Γ) = 19/450. 2.1. Behavior under Fuzzy Operations. Let Γ1 = (V1, ν1, µ1) and Γ2 = (V2, ν2, µ2) be two fuzzy graphs with |V1| = n1 and |V2| = n2, fuzzy sizes ew1(… view at source ↗

**Figure 2.** Figure 2: Fuzzy composition Γ = Γ1[Γ2] graph. As a result, the irregularity of Γ1[Γ2] is primarily determined by the irregularity of the outer graph Γ1. When n2 is high, σ ∗ (Γ1[Γ2]) ⩾ σ ∗ (Γ1). Theorem 2.1. For the Cartesian product Γ = Γ1□Γ2, the fuzzy sigma index is additive (10) σ ∗ (Γ) = σ(Γ1) + σ(Γ2). Proof. For any vertex (u, v) in Γ, its degree satisfies dΓ(u, v) = dΓ1 (u) + dΓ2 (v). Since the degree is for… view at source ↗

**Figure 3.** Figure 3: Fuzzy complement graph. 3. Extremal Characterization of Fuzzy Sigma Index We have now totally solved the second outstanding problem: characterise the fuzzy graphs that obtain the minimum and maximum values of σ ∗ (Γ). Theorem 3.1 had provided the fuzzy-regular graph. Theorem 3.1. Let Γ = (V, ν, µ) be a fuzzy graph. Then, σ ∗ (Γ) = 0 if and only if G is fuzzyregular, that is, there exists a constant r ≥ 0 … view at source ↗

**Figure 4.** Figure 4: A fuzzy 4-regular graph on 6 vertices with σ ∗ (Γ) = 0. 4. Conclusion This paper introduced and systematically analyzed the fuzzy sigma index, defined as the population variance of the fuzzy degree sequence of a fuzzy graph Γ. We established several fundamental properties of this index. In particular, we proved that σ ∗ (Γ) = 0 if and only if Γ is fuzzy-regular. We further showed that the maximum value of… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A direct but unremarkable extension of the sigma index to fuzzy graphs, with a likely gap in the claimed sharp upper bound.

read the letter

This paper defines the fuzzy sigma index as the variance of the fuzzy degrees around their mean, which is fixed at 2ew/n by the handshaking lemma. It then derives bounds and checks behavior under standard operations like union or join on fuzzy graphs. That is the core contribution: a straightforward translation of the classical degree-variance measure into the fuzzy setting using the usual definitions of fuzzy degree and fuzzy size. The definition itself is consistent and introduces no extra parameters or circular steps. The lower bound of zero on fuzzy-regular graphs is immediate and sharp. The paper does the mechanical work of writing out the properties in this new context, which is the sort of incremental cataloging that sometimes appears in this subfield. The soft spot is the upper bound. Treating the degrees as arbitrary reals that sum to 2ew gives a clean algebraic maximum, but fuzzy graphs constrain edge memberships to [0,1], so not every degree sequence is realizable. If the extremal case (one vertex taking as much weight as possible, others minimal) cannot actually occur, then the stated upper bound is not attained and therefore not sharp. The abstract and the stress-test note both suggest the derivation skips this realizability check. Readers who work specifically on topological indices for fuzzy graphs will find a clean definition and a few formulas to cite if they need the variance version. Most other graph theorists will see it as a minor extension that follows an established template without new machinery or applications. The thinking is clear on the definitional side and the claims are stated plainly, so the paper is coherent on its own terms. It is worth sending to peer review so a referee can verify the proofs and test whether the upper bound holds for actual fuzzy graphs with valid edge weights.

Referee Report

2 major / 2 minor

Summary. The paper defines the fuzzy sigma index σ*(Γ) = (1/n) ∑ (d_Γ(v) - 2ew/n)^2 as the population variance of the fuzzy degree sequence in a fuzzy graph Γ, where the mean is fixed by the handshaking lemma. It claims to establish fundamental properties of this index, derive sharp lower and upper bounds, and analyze its behavior under standard fuzzy graph operations such as union, join, and complement.

Significance. If the claimed sharp bounds are attained by realizable fuzzy graphs and the operational properties hold, the work supplies a natural variance-based topological index for measuring irregularity in fuzzy graphs, extending the classical sigma index in a parameter-free manner directly from fuzzy degree and size definitions. This could serve as a foundation for further study of variance-type indices in fuzzy graph theory.

major comments (2)

[Bounds derivation] The derivation of the sharp upper bound (presumably in the section following the definition) treats the fuzzy degrees as real numbers summing to 2ew without additional constraints. This does not automatically guarantee that the extremal sequence (one vertex absorbing as much weight as possible, others at minimum) is realizable by edge memberships μ(e) ∈ [0,1]; the bound may therefore not be attained or sharp for all ew.
[Abstract and main results] The abstract states that sharp lower and upper bounds are derived and that behavior under operations is analyzed, yet the provided statement contains no explicit proofs, extremal examples, or verification that the upper bound is realized by a concrete fuzzy graph. The manuscript must supply these to support the central claims.

minor comments (2)

[Definition] Notation for fuzzy size is given as ew; confirm consistency with standard fuzzy-graph literature (often denoted |μ| or m̃) and add a brief reminder of the handshaking lemma in fuzzy setting.
[Abstract] The abstract mentions 'standard fuzzy graph operations' without listing them; an explicit enumeration (union, join, etc.) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for your careful reading and constructive comments. We appreciate the feedback on the sharpness of the bounds and the need for explicit verification. We address each major comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

Referee: [Bounds derivation] The derivation of the sharp upper bound (presumably in the section following the definition) treats the fuzzy degrees as real numbers summing to 2ew without additional constraints. This does not automatically guarantee that the extremal sequence (one vertex absorbing as much weight as possible, others at minimum) is realizable by edge memberships μ(e) ∈ [0,1]; the bound may therefore not be attained or sharp for all ew.

Authors: We agree that the realizability constraint imposed by μ(e) ∈ [0,1] must be verified separately from the purely numerical bound on real numbers summing to 2ew. In the revised version we will add an explicit subsection containing constructions of fuzzy graphs (with concrete vertex sets and edge memberships) that attain the proposed upper bound for representative values of ew, thereby confirming sharpness under the fuzzy-graph constraints. revision: yes
Referee: [Abstract and main results] The abstract states that sharp lower and upper bounds are derived and that behavior under operations is analyzed, yet the provided statement contains no explicit proofs, extremal examples, or verification that the upper bound is realized by a concrete fuzzy graph. The manuscript must supply these to support the central claims.

Authors: We acknowledge that the current manuscript version presents the statements of the bounds and operational properties without sufficient accompanying proofs or examples. We will expand the relevant sections to include complete, self-contained proofs of both the lower and upper bounds, supply concrete extremal fuzzy graphs realizing each bound, and provide detailed calculations verifying the behavior under union, join, and complement operations. revision: yes

Circularity Check

0 steps flagged

No circularity: fuzzy sigma index defined directly as degree variance; bounds follow from variance properties and fuzzy graph definitions

full rationale

The central object is the explicit definition σ*(Γ) = (1/n) ∑ (d_Γ(v) − 2ew/n)^2, which is the population variance of the fuzzy degrees with mean fixed by the handshaking lemma (∑ d(v) = 2ew). This is a direct application of the variance formula to quantities already defined in fuzzy graph theory; no parameter fitting, self-referential equations, or renaming of prior results occurs. The lower bound of 0 is attained precisely when all degrees are equal (fuzzy-regular graphs), which is immediate from the non-negativity of variance and does not constitute a derivation that reduces to the input by construction. Upper bounds, if derived, would rest on extremal properties of real numbers summing to a fixed total or on fuzzy-graph realizability constraints; neither reduces the claimed result to a tautology or self-citation. No load-bearing self-citations, uniqueness theorems, or smuggled ansatzes are present in the abstract or stated claims. The derivation chain is therefore self-contained against the standard definitions of fuzzy degree and size.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the new definition of the fuzzy sigma index together with standard background assumptions from fuzzy graph theory; no free parameters are introduced and the only invented entity is the index itself.

axioms (1)

domain assumption Fuzzy graphs are equipped with a vertex set V, a vertex membership function ν, and an edge membership function μ; fuzzy degree d_Γ(v) and fuzzy size ew are defined in the usual way from these functions.
The definition of σ* directly invokes these standard fuzzy-graph quantities.

invented entities (1)

fuzzy sigma index σ*(Γ) no independent evidence
purpose: To serve as a variance-based topological index measuring degree irregularity in fuzzy graphs
Newly defined quantity whose properties are then investigated.

pith-pipeline@v0.9.0 · 5452 in / 1280 out tokens · 99617 ms · 2026-05-10T16:07:10.876624+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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