arxiv: 2605.06695 · v1 · submitted 2026-05-04 · 🧮 math.GM

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Sharp Bounds and Extremal Fuzzy Graphs for the Fuzzy Sombor Index

Jasem Hamoud

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Pith reviewed 2026-05-11 00:44 UTC · model grok-4.3

classification 🧮 math.GM

keywords fuzzy graphsSombor indextopological indicesregular fuzzy graphsfuzzy degreeextremal valuesboundsinequalities

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The pith

The fuzzy Sombor index attains its sharp maximum and minimum on regular fuzzy graphs and obeys inequalities with other fuzzy topological indices.

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

The paper extends the classical Sombor index to the setting of fuzzy graphs, where edges and vertices carry membership values between 0 and 1. It focuses on the general fuzzy Sombor index for exponent alpha greater than 1 and determines its largest and smallest possible values when the fuzzy graph is regular. The work also derives inequalities that relate this index to other established fuzzy topological indices. A reader would care because such bounds and relations give concrete ways to compare and control graded network structures that arise in uncertain or imprecise systems.

Core claim

For the general fuzzy Sombor index SO^μ_α defined by summing (μ(u,v) times the square root of μ_u squared plus μ_v squared) raised to alpha over all edges, the maximum and minimum values are characterized when the fuzzy graph is regular, and the index satisfies significant inequalities with other well-known fuzzy topological indices.

What carries the argument

The fuzzy Sombor index SO^μ_α, the sum over edges of the alpha-power of each edge membership multiplied by the Euclidean norm of the pair of endpoint fuzzy degrees, which aggregates the weighted degree contributions across the fuzzy edge set.

Load-bearing premise

The fuzzy graph is regular, so every vertex has exactly the same fuzzy degree, and the membership functions satisfy the usual fuzzy-graph axioms.

What would settle it

An explicit regular fuzzy graph on a small number of vertices whose computed fuzzy Sombor index value lies strictly outside the characterized maximum or minimum range.

Figures

Figures reproduced from arXiv: 2605.06695 by Jasem Hamoud.

**Figure 2.** Figure 2: demonstrate that the behavior of the fuzzy Sombor index across fundamental families of fuzzy graphs based on the value of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: compares four significant fuzzy topological indices, where the visualisation demonstrates the unique behaviour of each index [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

read the original abstract

The fuzzy Sombor index applies the classical Sombor index to fuzzy graphs, incorporating both edge membership values and fuzzy vertex degrees. For $\alpha>1$, the general fuzzy Sombor index it is defined as \[ \mathrm{SO}^{\mu}_{\alpha}(\Gamma)=\sum_{uv\in V(\Gamma)} \left( \mu(u,v)\, \sqrt{\mu_u^2+\mu_v^2} \right)^{\alpha}. \] This paper analyses extremal features of $\mathrm{SO}^{\mu}$ across different types of fuzzy graphs. We determine the maximum value (resp. minimum value) of $\mathrm{SO}^{\mu}$ characterise in regular fuzzy graph. We established significant inequality between the fuzzy Sombor index and other well-known fuzzy topological indices.

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

The paper defines a generalized alpha-powered fuzzy Sombor index and claims max/min values plus inequalities on regular fuzzy graphs, but the maximum result is probably impossible without an unstated fixed vertex count.

read the letter

The main takeaway is that this work extends the Sombor index into fuzzy graphs by raising the usual term to an alpha power greater than 1, then tries to pin down its largest and smallest values on regular fuzzy graphs while also relating it to other fuzzy indices. The definition itself is straightforward: sum over edges of (edge membership times square root of the sum of squared fuzzy degrees) to the alpha. That part is clear and follows the pattern of other fuzzy degree-based indices without adding extra assumptions beyond the standard fuzzy graph axioms on memberships in [0,1]. The inequalities section appears to be direct comparisons that fall out from the definitions, so those should be easy to verify and could serve as quick reference tools inside the subfield. Credit is due for keeping the setup explicit and for working with the usual regularity condition that every vertex has the same fuzzy degree. The soft spot sits in the extremal claims. The abstract states that the maximum and minimum of the index are determined and the attaining regular fuzzy graphs are characterized. Yet nothing in the statement fixes the number of vertices or imposes a global bound on total membership. The complete fuzzy graph on n vertices with every membership equal to 1 is regular, and its index value grows without limit as n increases because the number of edges increases while each term stays positive. Without an n parameter or equivalent constraint, no finite maximum exists and the characterization cannot hold as written. If the theorems inside actually restrict to fixed n, that needs to be stated up front; otherwise the central result has a gap. This paper is aimed at the small group already working on fuzzy topological indices. A reader who follows fuzzy Zagreb, Randic, or Sombor variants might find the inequalities handy for their own calculations, but the work is unlikely to interest anyone outside that niche. It deserves a serious referee because the claims are concrete and mathematical, so a reviewer can check the derivations and ask for the missing bound condition if needed. I would send it to peer review with the expectation that the extremal section gets clarified or corrected.

Referee Report

1 major / 1 minor

Summary. The manuscript defines the general fuzzy Sombor index SO^μ_α(Γ) = ∑_{uv} (μ(uv) √(μ_u² + μ_v²))^α for α > 1 on fuzzy graphs Γ. It claims to determine the maximum and minimum values of this index on regular fuzzy graphs, characterize the extremal graphs attaining these values, and establish inequalities relating the fuzzy Sombor index to other known fuzzy topological indices.

Significance. If the extremal characterizations were valid for graphs of fixed order n, the results would extend classical Sombor-index bounds to the fuzzy setting and provide concrete inequalities with other indices. The current formulation, however, does not restrict to fixed n, rendering the claimed maximum unattainable and the characterization impossible.

major comments (1)

[Abstract] Abstract (and the corresponding theorems): the claim that the maximum value of SO^μ is determined and the attaining regular fuzzy graphs are characterized does not include any restriction on the order n. The complete fuzzy graph K_n with all memberships equal to 1 is regular of degree n-1; the index is a sum of n(n-1)/2 nonnegative terms each of which grows with n, so SO^μ_α can be made arbitrarily large. No finite global maximum therefore exists, and the stated characterization cannot hold.

minor comments (1)

[Abstract] Abstract: the sentence 'We determine the maximum value (resp. minimum value) of SO^μ characterise in regular fuzzy graph' is grammatically incomplete and should be rephrased for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to restrict the extremal results to fuzzy graphs of fixed order n. We agree that this clarification is essential for the claims to be valid.

read point-by-point responses

Referee: [Abstract] Abstract (and the corresponding theorems): the claim that the maximum value of SO^μ is determined and the attaining regular fuzzy graphs are characterized does not include any restriction on the order n. The complete fuzzy graph K_n with all memberships equal to 1 is regular of degree n-1; the index is a sum of n(n-1)/2 nonnegative terms each of which grows with n, so SO^μ_α can be made arbitrarily large. No finite global maximum therefore exists, and the stated characterization cannot hold.

Authors: We agree with the referee's assessment. Without a fixed order n the index SO^μ_α is indeed unbounded above, as shown by the sequence of complete fuzzy graphs with all memberships equal to 1. This omission in the abstract and theorem statements was an oversight. In the revised manuscript we will explicitly restrict all extremal results to the class of regular fuzzy graphs of order n, reformulate the abstract and theorems accordingly, and provide the sharp bounds together with the characterization of the extremal graphs within this fixed-n setting. The revision will make the results precise and consistent with standard practice for topological indices. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived directly from explicit index definition and regularity axioms

full rationale

The paper defines SO^μ_α explicitly as a sum over edges and proceeds to bound it on regular fuzzy graphs using the standard membership axioms (μ(uv) ≤ min(σ(u),σ(v)), degrees equal) and elementary inequalities on nonnegative terms. No step equates a derived quantity to a fitted parameter, renames a known result, or reduces the central claim to a self-citation chain. The extremal characterization rests on the given formula and the regularity hypothesis without self-referential closure. The absence of any load-bearing self-citation or ansatz smuggling keeps the derivation self-contained against external graph-theoretic facts.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition of the fuzzy Sombor index and the regularity assumption for fuzzy graphs.

axioms (1)

domain assumption Fuzzy graphs have vertex and edge membership functions taking values in [0,1] that obey the standard axioms of fuzzy set theory.
Invoked in the definition of SO^μ_α and in the notion of regularity.

invented entities (1)

General fuzzy Sombor index SO^μ_α no independent evidence
purpose: To extend the classical Sombor index to fuzzy graphs by incorporating edge and vertex membership values raised to power alpha.
Newly defined in the paper; no independent evidence outside the definition is provided.

pith-pipeline@v0.9.0 · 5416 in / 1339 out tokens · 64244 ms · 2026-05-11T00:44:35.386974+00:00 · methodology

discussion (0)

Reference graph

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