arxiv: 2604.26603 · v2 · submitted 2026-04-29 · 🧮 math.CO · math.AC

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· Lean Theorem

On main eigenvalues of zero-divisor graphs of reduced rings

R. Barabde, Sakshi Jain, Y. M. Borse

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Pith reviewed 2026-05-11 01:49 UTC · model grok-4.3

classification 🧮 math.CO math.AC

keywords zero-divisor graphsreduced ringsmain eigenvaluesspectral graph theoryequitable partitionsadjacency spectrumbipartite subgraphsinfinite families

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The pith

Zero-divisor graphs of reduced rings form infinite families of simple connected graphs with exactly s main eigenvalues for any positive integer s.

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

The long-standing problem in spectral graph theory is to characterize or construct graphs with a prescribed number of main eigenvalues, where only a few infinite families of simple connected examples are known for s greater than or equal to 2. This paper shows that zero-divisor graphs of reduced rings supply such families, with the number of main eigenvalues fixed at exactly s for any chosen positive integer s. The same holds for certain induced bipartite subgraphs of these graphs. The construction works because the adjacency spectra of these algebraic graphs can be computed explicitly through equitable partitions, which separate the main eigenvalues from the rest. A sympathetic reader sees this as a systematic algebraic source of examples that bypasses the scarcity of known constructions.

Core claim

We prove that the zero-divisor graphs of reduced rings provide an infinite family of simple connected graphs with exactly s main eigenvalues, and that certain induced bipartite subgraphs also have exactly s main eigenvalues for any positive integer s.

What carries the argument

Equitable partitions of the zero-divisor graph of a reduced ring, which yield an explicit block-diagonal form of the adjacency matrix and thereby isolate the main eigenvalues (those with eigenvectors not orthogonal to the all-ones vector).

If this is right

For every positive integer s there exist infinitely many simple connected graphs whose adjacency spectrum contains exactly s main eigenvalues.
Certain induced bipartite subgraphs extracted from these zero-divisor graphs likewise possess exactly s main eigenvalues.
The explicit equitable-partition description of the spectrum gives a concrete way to control the main-eigenvalue count by varying the underlying reduced ring.
These families enlarge the known supply of graphs with prescribed main-eigenvalue multiplicity beyond the few previously recorded constructions.

Where Pith is reading between the lines

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

The same equitable-partition technique could be applied to zero-divisor graphs of other ring classes to test whether the main-eigenvalue count remains controllable.
One could examine whether the induced bipartite subgraphs inherit additional spectral properties, such as integrality, that are not claimed in the paper.
The construction supplies concrete test cases for conjectures about the possible values of the main-eigenvalue multiplicity in arbitrary graphs.

Load-bearing premise

That the zero-divisor graphs of reduced rings are always simple and connected, and that the equitable partitions of their adjacency matrices always produce exactly s main eigenvalues for arbitrary s.

What would settle it

A reduced ring whose zero-divisor graph has a number of main eigenvalues different from the count predicted by its equitable partition, or a reduced ring whose zero-divisor graph is disconnected.

Figures

Figures reproduced from arXiv: 2604.26603 by R. Barabde, Sakshi Jain, Y. M. Borse.

**Figure 5.1.** Figure 5.1: Zero-divisor graph Γ(Z 4 2 ) and the bipartite subgraph Γ ′ (Z 4 2 ). 6 Conclusion In this paper, we present two infinite families of graphs arising from algebraic structures that have exactly s main eigenvalues for any given positive integer s. In particular, we show that the zero-divisor graph of the ring Rn := Fm ×Fm ×· · ·×Fm (with n terms) has exactly n −1 main eigenvalues. Moreover, we prove that … view at source ↗

read the original abstract

The problem of characterizing graphs with a prescribed number of main eigenvalues is a long-standing problem in spectral graph theory. Although some constructions are known, only a few produce infinite families of simple connected graphs with exactly $s \ge 2$ main eigenvalues. Zero-divisor graphs form a well-structured class of algebraic graphs whose spectra can be described explicitly using equitable partitions, making them a convenient setting to study main eigenvalues. In this paper, we prove that the zero-divisor graphs of reduced rings provide an infinite family of simple connected graphs with exactly $s$ main eigenvalues, and that certain induced bipartite subgraphs also have exactly $s$ main eigenvalues for any positive integer $s$.

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

Zero-divisor graphs of reduced rings give explicit infinite families with exactly s main eigenvalues via equitable partitions on support patterns.

read the letter

The main thing here is that zero-divisor graphs of finite products of fields produce infinite families of simple connected graphs with exactly s main eigenvalues for any positive integer s, and the same holds for certain induced bipartite subgraphs on two support classes. The construction tunes the number of field factors to control s directly. They partition the vertices by the zero-support patterns, which turns out to be equitable. The quotient matrix eigenvalues become the main ones because the lifted eigenvectors have nonzero inner product with the all-ones vector. The remaining eigenvalues sit in the orthogonal complement to the partition subspace and stay orthogonal to the all-ones vector under the adjacency action, shown by direct computation. The reduced ring condition keeps the graph simple with no loops. The same partition works on the restricted bipartite subgraphs. This is cleaner than earlier limited constructions because the ring structure supplies the partitions and the spectrum control without extra fitting. The arguments look reproducible from the partition details and do not rely on finiteness or characteristic bounds beyond the reduced hypothesis. A small soft spot is confirming connectedness for every choice of fields and verifying the inner-product signs hold uniformly for arbitrary s, though the direct calculations appear to cover it. No circularity or invented entities show up. This is useful for algebraic graph theorists studying main eigenvalues or looking for concrete families to test conjectures. It deserves peer review because the construction is grounded in standard ring and graph tools and the claims are checkable.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that the zero-divisor graphs of reduced rings (finite products of fields with varying numbers of factors) form an infinite family of simple connected graphs with exactly s main eigenvalues for any positive integer s. It further shows that certain induced bipartite subgraphs, obtained by restricting to two complementary support classes, also have exactly s main eigenvalues. The argument proceeds by defining equitable partitions on the vertex sets according to support patterns, computing the quotient matrices explicitly, verifying that the quotient eigenvalues are main (via non-zero inner product of lifted eigenvectors with the all-ones vector), and showing that the remaining eigenvalues (from the orthogonal complement) are orthogonal to the all-ones vector by direct adjacency action.

Significance. If correct, the result supplies an explicit algebraic construction of infinite families of simple connected graphs with a prescribed number of main eigenvalues, addressing a long-standing problem in spectral graph theory where only a few such families were previously known. The use of equitable partitions on zero-divisor graphs of reduced rings yields direct, parameter-free spectral control without additional boundedness or characteristic assumptions, and the same partition technique applies uniformly to the induced bipartite subgraphs. This algebraic-graph-theoretic approach is a clear strength.

minor comments (3)

The definition of 'main eigenvalue' (eigenvalue whose eigenvector has non-zero inner product with the all-ones vector) is standard but should be recalled explicitly in §2 before the equitable-partition arguments begin.
In the construction of the reduced ring as a product of k fields, the dependence of s on k (or on the number of factors) should be stated as a lemma or proposition with an explicit formula, even if it is immediate from the quotient matrix.
The proof that the graphs are simple (no loops or multiple edges) follows immediately from the reduced hypothesis, but a one-sentence reminder in the opening paragraph of §3 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript, the accurate summary of our results, and the positive recommendation to accept. The referee's assessment correctly identifies the key contributions regarding infinite families of graphs with a prescribed number of main eigenvalues arising from zero-divisor graphs of reduced rings and their induced bipartite subgraphs.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation relies on explicit constructions of reduced rings as finite products of fields, equitable partitions of the zero-divisor graph vertices by support patterns, and direct computation of the quotient matrices from the adjacency rules. Main eigenvalues are identified by verifying that the lifted eigenvectors from the quotient have nonzero inner product with the all-ones vector, while the orthogonal complement is shown by direct adjacency action to be orthogonal to the all-ones vector. The identical partition argument is applied to the induced bipartite subgraphs on complementary support classes. These steps use only the algebraic definition of reduced rings, the standard definition of main eigenvalues, and linear-algebraic verification internal to each graph; no parameter is fitted to the target count s, no uniqueness theorem is imported from prior self-work, and no ansatz or renaming of a known pattern is smuggled in. The central claim therefore follows from the ring-theoretic input without reduction to the output.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or nonstandard axioms are mentioned. The work relies on standard properties of reduced rings and graph spectra.

axioms (1)

domain assumption Zero-divisor graphs of reduced rings are simple connected graphs whose spectra admit equitable partitions
Invoked to describe the spectra explicitly and control the number of main eigenvalues.

pith-pipeline@v0.9.0 · 5411 in / 1209 out tokens · 33767 ms · 2026-05-11T01:49:11.863876+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean phi_golden_ratio, Jcost definitions and recurrence identities echoes
generalized Fibonacci sequence {F_{α,k}} ... F_{α,k}=F_{α,k-1}+(α-1)F_{α,k-2} ... γ_{m,k}=F_{m,k+1}/F_{m,k} ... Vandermonde matrix V ... det(V)=∏(γ_{m,l}-γ_{m,r}) ... rank(W(P))=n-1
IndisputableMonolith/Constants.lean phi_golden_ratio echoes
for m=2 ... φ=(1+√5)/2 ... eigenvalues of P[2,n] are main ... exactly n-1 main eigenvalues

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring.J. Algebra, 217 (1999), 434–447

work page 1999
[2]

Beck, Coloring of commutative rings.J

I. Beck, Coloring of commutative rings.J. Algebra, 116 (1988), 208–226

work page 1988
[3]

Cvetković, P

D. Cvetković, P. Rowlinson and S. K. Simić,An Introduction to the Theory of Graph Spectra, Cambridge Univ. Press, Cambridge, 2010

work page 2010
[4]

Cvetkovič, The main part of the spectrum, divisors, and switching of graphs.Publ

D. Cvetkovič, The main part of the spectrum, divisors, and switching of graphs.Publ. Inst. Math. (Beograd), 23 (1978), 37–41

work page 1978
[5]

Z. Du, F. Liu, S. Liu and Z. Qin, Graphs withn−1main eigenvalues.Discrete Math., 344(7) (2021), 112397. 13

work page 2021
[6]

Z. Du, L. You and H. Liu, Almost controllable graphs and beyond.Discrete Math., 347(1) (2024), 113743

work page 2024
[7]

França, A

F. França, A. Brondani and D. Jaume, Graphs with exactly three main eigenvalues.Discrete Appl. Math., 378 (2026), 49–64

work page 2026
[8]

Godsil, Controllable subsets in graphs.Ann

C. Godsil, Controllable subsets in graphs.Ann. Comb., 16 (2012), 733–744

work page 2012
[9]

Ghebleh, S

M. Ghebleh, S. Al-Yakoob, A. Kanso and D. Stevanović, Graphs having two main eigenvalues and arbitrarily many distinct vertex degrees.Appl. Math. Comput., 495 (2025), 129311

work page 2025
[10]

E. M. Hagos, Some results on graph spectra.Linear Algebra Appl., 356 (2002), 103–111

work page 2002
[11]

Harary and A

F. Harary and A. J. Schwenk, The spectral approach to determining the number of walks in a graph.Pacific J. Math., 80(2) (1979), 443–449

work page 1979
[12]

Hou and H

Y. Hou and H. Zhou, Trees with exactly two main eigenvalues.J. Nat. Sci. Hunan Norm. Univ., 28 (2005)

work page 2005
[13]

Hou and F

Y. Hou and F. Tian, Unicyclic graphs with exactly two main eigenvalues.Appl. Math. Lett., 19(11) (2006), 1143–1147

work page 2006
[14]

Y. Hou, Z. Tang and W. C. Shiu, Some results on graphs with exactly two main eigenvalues. Appl. Math. Lett., 25(10) (2012), 1274–1278

work page 2012
[15]

Huang, Q

X. Huang, Q. Huang and L. Lu, Construction of graphs with exactlykmain eigenvalues.Linear Algebra Appl., 486 (2015), 204–218

work page 2015
[16]

Khare,Matrix Analysis and Entrywise Positivity Preservers, Cambridge Univ

A. Khare,Matrix Analysis and Entrywise Positivity Preservers, Cambridge Univ. Press, 2022

work page 2022
[17]

G. S. Kadu, G. Sonawane and Y. M. Borse, Reciprocal eigenvalue properties using the zeta and Möbius functions.Linear Algebra Appl., 694 (2024), 186–205

work page 2024
[18]

Koshy,Fibonacci and Lucas Numbers with Applications, Wiley, New York, 2001

T. Koshy,Fibonacci and Lucas Numbers with Applications, Wiley, New York, 2001

work page 2001
[19]

J. D. LaGrange, Spectra of Boolean graphs and certain matrices of binomial coefficients.Int. Electron. J. Algebra, 9 (2011), 78–84

work page 2011
[20]

J. D. LaGrange, Eigenvalues of Boolean graphs and Pascal-type matrices.Int. Electron. J. Al- gebra, 13 (2013), 109–119

work page 2013
[21]

J. D. LaGrange, A combinatorial development of Fibonacci numbers in graph spectra.Linear Algebra Appl., 438(11) (2013), 4335–4347

work page 2013
[22]

J. D. LaGrange, Boolean rings and reciprocal eigenvalue properties.Linear Algebra Appl., 436(7) (2012), 1863–1871

work page 2012
[23]

Rowlinson, The main eigenvalues of a graph: a survey.Appl

P. Rowlinson, The main eigenvalues of a graph: a survey.Appl. Anal. Discrete Math., 1 (2007), 445–471

work page 2007
[24]

Sonawane, G

G. Sonawane, G. S. Kadu and Y. M. Borse, Spectra of zero-divisor graphs of finite reduced rings. J. Algebra Appl., 24(3) (2025), 2550082

work page 2025
[25]

Sonawane, Pascal-type matrices and eigenvalues of zero-divisor graphs of reduced rings.Linear Multilinear Algebra, (2025), 1–15

G. Sonawane, Pascal-type matrices and eigenvalues of zero-divisor graphs of reduced rings.Linear Multilinear Algebra, (2025), 1–15. 14

work page 2025