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arxiv: 2604.03101 · v1 · submitted 2026-04-03 · 🧮 math.CO · cs.DM· math.AC

Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings

Bilal Ahmad Rather This is my paper

Pith reviewed 2026-05-13 18:24 UTC · model grok-4.3

classification 🧮 math.CO cs.DMmath.AC
keywords zero-divisor graphtruncated polynomial ringA_alpha matrixadjacency spectrumLaplacian spectrumdistance spectrumsignless Laplacian
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The pith

For the zero-divisor graph of the ring Z_p[x]/(x^c), the A_α-matrix spectrum is determined in closed form and the Laplacian and distance eigenvalues are all integers.

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

The paper computes the eigenvalues of the A_α-matrix on the zero-divisor graph Γ(R) for R equal to Z_p[x] modulo x to the c. Special cases recover the full adjacency spectrum and the signless Laplacian spectrum in explicit formulas that depend only on p and c. It further shows that every Laplacian eigenvalue and every distance eigenvalue of these graphs is an integer. A reader would care because the result supplies complete algebraic expressions for the main spectral invariants of a concrete family of graphs that arise naturally from algebra.

Core claim

For R = Z_p[x]/(x^c) the zero-divisor graph Γ(R) has vertex set the nonzero zero-divisors and edges between distinct vertices whose product is zero; the eigenvalues of its A_α-matrix are given by explicit algebraic expressions in p and c, which immediately yield the adjacency spectrum and signless Laplacian spectrum, while the Laplacian spectrum and the distance spectrum consist entirely of integer values.

What carries the argument

The A_α-matrix of Γ(R), defined as a convex combination of the adjacency matrix and the diagonal degree matrix, whose characteristic polynomial is computed directly from the ring structure.

If this is right

  • The adjacency eigenvalues of Γ(R) are given by a short explicit list depending on p and c.
  • The signless Laplacian eigenvalues of Γ(R) are likewise given by an explicit list.
  • Every eigenvalue of the ordinary Laplacian matrix of Γ(R) is an integer.
  • Every eigenvalue of the distance matrix of Γ(R) is an integer.

Where Pith is reading between the lines

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

  • The same ring family may admit similar closed-form spectra for other matrices such as the Seidel matrix or normalized Laplacian.
  • The integrality result suggests these graphs belong to the class of integral graphs, which could be used to test conjectures about integral distance-regular graphs.

Load-bearing premise

The ring must be exactly Z_p[x] modulo x^c with the zero-divisor graph using only nonzero zero-divisors and edges precisely when the product is zero.

What would settle it

For the concrete case p=2 and c=2, compute the characteristic polynomial of the A_α-matrix of the resulting graph by hand or machine and check whether its roots match the closed-form expressions given in the paper.

Figures

Figures reproduced from arXiv: 2604.03101 by Bilal Ahmad Rather.

Figure 1
Figure 1. Figure 1: zero-divisor graph Γ(R) for R ∼= F2[x]/(x 6 ). The following result gives the Aα spectra of Zp[x]/⟨x 2b ⟩. Theorem 2.2. Let p be a prime and a = 2b with b ∈ N, and let Γ(R) be the zero-divisor graph of R = Zp[x]/⟨x 2b ⟩ ∼= Fp[x]/(x 2b ). Then the Aα–spectrum of Γ(R) is as follows: For each i = 1, . . . , 2b − 1, there is an eigenvalue λi =    α [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Zero–divisor graph Γ(R) for R ∼= Z2[x]/⟨x 5 ⟩ [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
read the original abstract

Let $R$ be a commutative ring with identity and let $Z^{\ast}(R)$ denote the set of nonzero zero-divisors of $R$. The \emph{zero-divisor graph} $ \varGamma(R)$ is the simple graph with vertex set $V( \varGamma(R))=Z^{\ast}(R)$, where two distinct vertices$x,y\in Z^{\ast}(R)$ are adjacent if and only if $xy=0$ in $R$. In this paper we investigate the zero-divisor graph of the truncated polynomial ring $R=\mathbb{Z}_{p}[x]/\langle x^{c}\rangle,$ for $c\in\mathbb{N}.$ We determine the spectrum of the $A_{\alpha}$-matrix associated with $ \varGamma(R)$, and, as special cases, explicitly obtain both the adjacency spectrum and the signless Laplacian spectrum of $ \varGamma(R)$. Furthermore, we prove that the Laplacian eigenvalues, as well as the distance eigenvalues, of these graphs are all integers.

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 truncated polynomial ring R = ℤ_p[x]/(x^c) for prime p and natural number c. It determines the spectrum of the A_α-matrix associated to Γ(R), obtains the adjacency spectrum and signless Laplacian spectrum explicitly as special cases, and proves that the Laplacian eigenvalues and the distance eigenvalues are all integers.

Significance. The central contribution is an equitable partition of the vertex set into valuation level sets V_k (k=1 to c-1) that reduces every claimed spectrum to the eigenvalues of an explicit (c-1)×(c-1) quotient matrix together with internal eigenvalues on each block. The quotient matrices have integer entries and the induced subgraphs on the blocks are empty or complete, so the internal eigenvalues are integers by inspection; the paper supplies closed-form expressions for the quotient eigenvalues. This yields fully explicit, parameter-free spectral formulas and integrality results for a concrete infinite family of graphs arising from commutative algebra.

minor comments (2)
  1. [Theorem 3.1] In the statement of the main theorems, explicitly record the dependence on the prime p and the exponent c (e.g., “for every prime p and every integer c ≥ 2”).
  2. [Section 2] The definition of the A_α-matrix is standard, but a one-sentence reminder of the formula A_α = αD + (1-α)A would improve readability for readers outside spectral graph theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, detailed assessment of the significance of the equitable partition approach, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the spectra of the A_α-matrix, adjacency matrix, signless Laplacian, Laplacian, and distance matrix of Γ(R) for R = ℤ_p[x]/(x^c) by partitioning the vertex set into level sets V_k (k=1 to c-1) according to the valuation of the polynomials. This yields an equitable partition whose quotient matrix is explicitly constructed with integer entries from the ring multiplication rules; the eigenvalues of the quotient are solved directly, while the internal eigenvalues on each V_k follow from the induced subgraphs being complete or empty (hence integer spectra) and constant distances within blocks. All steps are obtained from the graph definition and the explicit structure of R without any fitted parameters, self-referential equations, or load-bearing self-citations. The results are therefore independent computations rather than reductions to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions from ring theory and spectral graph theory with no free parameters fitted to data, no invented entities, and only domain-standard axioms.

axioms (2)
  • domain assumption R is a commutative ring with identity
    Invoked in the definition of Z^*(R) and the zero-divisor graph Γ(R) as stated in the abstract.
  • domain assumption The zero-divisor graph is simple with vertices Z^*(R) and edges when xy=0
    Standard construction used to define the adjacency relation for spectral computations.

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Reference graph

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