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arxiv: 2605.17847 · v1 · pith:MQG3YPKRnew · submitted 2026-05-18 · 🧮 math.RA · math.CO

On Zero-Divisor Graph of the Ring frac{mathbb{F}_p[u, v]}{langle u²,\, v², \, uv-vurangle}

N. Annamalai This is my paper

Pith reviewed 2026-05-20 00:54 UTC · model grok-4.3

classification 🧮 math.RA math.CO
keywords zero-divisor graphcommutative ringsgraph invariantstopological indicesspectral graph theoryadjacency matrixLaplacian matrix
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The pith

The zero-divisor graph of the commutative ring F_p + uF_p + vF_p + uvF_p has its graph-theoretic properties and matrix spectra determined.

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

This paper investigates the zero-divisor graph Γ(R) of the ring R = F_p[u, v] / for an odd prime p. Vertices in the graph are the non-zero elements that multiply to zero with some other non-zero element, and edges connect pairs whose product is zero. The work computes the clique number, chromatic number, vertex connectivity, edge connectivity, diameter, and girth of this graph. Additionally, certain topological indices are calculated, and the eigenvalues, energy, and spectral radius are found for the adjacency matrix, Laplacian matrix, and eccentricity matrix.

Core claim

The central discovery is that for this particular ring R of characteristic p with two nilpotent generators that commute, the associated zero-divisor graph Γ(R) admits complete determination of its basic invariants including clique and chromatic numbers, connectivities, diameter and girth, as well as the full spectral information from its standard matrices.

What carries the argument

The zero-divisor graph Γ(R) with vertices the non-zero zero-divisors and an edge between two if their product is the zero element in R.

Load-bearing premise

The ring is exactly the four-dimensional vector space over F_p with the specified multiplication rules that make it commutative and non-chain.

What would settle it

Explicit construction of the graph for p=3 by listing all 81 elements, identifying the zero-divisors, and calculating its clique number or the eigenvalues of the adjacency matrix to match or contradict the paper's results.

Figures

Figures reproduced from arXiv: 2605.17847 by N. Annamalai.

Figure 1
Figure 1. Figure 1: Zero-divisor graph of R = Fp + uFp + vFp + uvFp As u 2 = 0, v2 = 0 and uv = vu, every vertices of Auv is adjacent to all vertices of Au, Av, Au+v, Au+uv, Av+uv and Au+v+uv, every vertices of Au is adjacent to all vertices of Au+uv, every vertices of Av is adjacent to all vertices of Av+uv, Also, any two distinct vertices of Au are adjacent, any two distinct vertices of Av are adjacent, and any two distinct… view at source ↗
Figure 2
Figure 2. Figure 2: Zero-divisor graph of R = F3 + uF3 + vF3 + uvF3 The number of vertices is 26 and the number of edges is 67. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

In this article, we study the zero-divisor graph of the commutative non-chain ring with identity $ \mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p,$ where \(u^2 = 0\), \(v^2 = 0\), \(uv = vu\), and \(p\) is an odd prime. We determine several graph-theoretic properties of the associated zero-divisor graph \(\Gamma(R)\), including clique number, chromatic number, vertex connectivity, edge connectivity, diameter, and girth. In addition, we compute certain topological indices of \(\Gamma(R)\). Furthermore, we obtain the eigenvalues, energy, and spectral radius of the adjacency matrix, the Laplacian matrix and the Eccentricity matrix of \(\Gamma(R)\).

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.

Referee Report

0 major / 2 minor

Summary. The paper studies the zero-divisor graph Γ(R) of the commutative ring R = F_p[u, v]/⟨u², v², uv-vu⟩ for odd prime p. It determines the clique number, chromatic number, vertex connectivity, edge connectivity, diameter, and girth of Γ(R). It also computes certain topological indices and the eigenvalues, energy, and spectral radius of the adjacency, Laplacian, and eccentricity matrices of Γ(R).

Significance. If the explicit computations hold, the manuscript contributes concrete data on zero-divisor graphs for a parameterized family of finite commutative rings of order p⁴. The inclusion of spectral properties across three matrices (adjacency, Laplacian, eccentricity) provides a multifaceted view that can aid comparison with other known examples in the literature.

minor comments (2)
  1. The abstract would benefit from stating the order of R or the number of vertices in Γ(R) to contextualize the scale of the computations performed.
  2. Notation for the ideal generators is slightly inconsistent between the title (uv-vu) and the abstract (uv=vu); a uniform presentation would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments were provided in the report, so we have no individual points to rebut. We will incorporate any minor suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper starts from the explicit ring presentation R = F_p[u,v]/<u²,v²,uv-vu> (p odd prime) and constructs the zero-divisor graph Γ(R) by enumerating non-zero zero-divisors and their products under the given relations. All claimed results—clique/chromatic numbers, connectivity, diameter, girth, topological indices, and spectra/energy of adjacency/Laplacian/eccentricity matrices—are obtained by direct, finite computation on this explicitly described graph. No fitted parameters, self-referential definitions, load-bearing self-citations, or imported uniqueness theorems appear in the derivation chain; each invariant follows from standard graph theory applied to the concrete vertex/edge set.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the explicit definition of the ring and the standard construction of its zero-divisor graph; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption R equals F_p + u F_p + v F_p + uv F_p with u² = 0, v² = 0, uv = vu, and p odd prime.
    This fixes the algebraic object whose zero-divisors generate the graph under study.

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Reference graph

Works this paper leans on

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