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arxiv: 2605.22160 · v1 · pith:3N5PO3W5new · submitted 2026-05-21 · 🧮 math.RA

Minimum second neighborhood degree energy of commuting graphs of finite rings

Payal Tak , Jutirekha Dutta , Rajat Kanti Nath This is my paper

Pith reviewed 2026-05-22 02:29 UTC · model grok-4.3

classification 🧮 math.RA
keywords commuting graphfinite ringminimum second neighborhood degree spectrumgraph energyMSN-integralnon-commutativeprime power
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The pith

Commuting graphs of non-commutative rings of orders p squared to p to the fifth and mixed prime powers have their minimum second neighborhood degree spectra and energies computed.

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

This paper sets out to compute the minimum second neighborhood degree spectrum and the associated energy for the commuting graphs of non-commutative rings whose orders are p to the power two, three, four or five, or p to the power two or three times q. These orders allow a complete listing of the rings so the graphs can be analyzed directly. If these computations are accurate they provide a concrete link between the algebraic property of commutativity in rings and a numerical graph invariant. The results also confirm that the graphs are integral with respect to this spectrum measure but not hyperintegral and resolve two conjectures on related energy concepts.

Core claim

We compute the minimum second neighborhood degree spectrum and energy of commuting graphs of certain finite non-commutative rings of order p to the power two, three, four, five, two q and three q where p and q are primes. We show that the commuting graphs of these rings are MSN-integral but not MSN-hyperintegral. The techniques used prove two conjectures from prior work on common neighbourhood energy of commuting graphs of finite groups and of rings.

What carries the argument

The minimum second neighborhood degree energy of the commuting graph which is the sum of the absolute values of the eigenvalues in the minimum second neighborhood degree spectrum of the graph whose edges connect commuting elements of the ring.

If this is right

  • The spectrum and energy are determined explicitly for each such ring.
  • The commuting graphs are MSN-integral.
  • The commuting graphs are not MSN-hyperintegral.
  • Two conjectures on neighbourhood energies of commuting graphs are proved.

Where Pith is reading between the lines

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

  • The case-by-case analysis for these orders may extend to other small orders of rings.
  • The results on integrality could guide investigations into why certain graph spectra from algebraic structures have integer eigenvalues.
  • The proof of the conjectures demonstrates the utility of the minimum second neighborhood degree approach for similar problems.

Load-bearing premise

All non-commutative rings of the stated orders can be exhaustively listed and their commuting graphs constructed for individual analysis.

What would settle it

A mismatch between the paper's computed minimum second neighborhood degree energy and an independent calculation for any one non-commutative ring of order p cubed would indicate an error in the general result for that order.

read the original abstract

In this paper, we compute minimum second neighborhood degree spectrum and energy of commuting graphs of certain finite non-commutative rings. In particular, we consider non-commutative rings of order $p^2, p^3, p^4, p^5, p^2q$ and $p^3q$, where $p$ and $q$ are primes. We shall also show that the commuting graphs of these rings are MSN-integral but not MSN-hyperintegral. Finally, employing the techniques used in this paper, we prove Conjecture 3 of [Nath, R. K., Fasfous, W. N. T., Das, K. C. and Shang, Y. Common neighbourhood energy of commuting graphs of finite groups, {\em Symmetry} {\bf 13}(9), Article No. 1651, 2021.] and Conjecture 3.12 of [W. N. T. Fasfous and Nath, R. K. Common neighborhood spectrum and energy of commuting graphs of finite rings, \emph{ Palestine J. Math.} \textbf{13}(1), 66--76, 2024.]. We conclude this paper with two open problems.

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 / 3 minor

Summary. The paper computes the minimum second neighborhood degree spectra and energies of the commuting graphs of all non-commutative rings of orders p², p³, p⁴, p⁵, p²q and p³q (p, q primes). It establishes that these graphs are MSN-integral but not MSN-hyperintegral and applies the same case-analysis techniques to prove Conjecture 3 of Nath et al. (2021) and Conjecture 3.12 of Fasfous and Nath (2024). The manuscript concludes with two open problems.

Significance. If the exhaustive enumerations are complete, the explicit spectra and energies supply concrete, verifiable data for a standard family of algebraic graphs whose ring classifications are already known in the literature. Resolving the two cited conjectures is a clear contribution to the study of common-neighborhood energies on commuting graphs of rings and groups.

minor comments (3)
  1. [Introduction] The definition of 'minimum second neighborhood degree' and the associated spectrum/energy should be recalled or referenced at the beginning of the main text rather than assumed from prior papers.
  2. [Section 3] A summary table listing the MSN-energy for each ring order (or each isomorphism type) would make the case-by-case results easier to survey.
  3. [Section 5] In the proofs that settle the two conjectures, an explicit sentence stating how the enumerated cases cover the general statement of each conjecture would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of the explicit computations for the minimum second neighborhood degree spectra and energies, and the recommendation for minor revision. We are pleased that the resolution of the two cited conjectures is viewed as a clear contribution.

Circularity Check

0 steps flagged

No significant circularity; explicit computations on classified rings

full rationale

The paper's central results consist of case-by-case computations of minimum second-neighborhood degree spectra and energies for commuting graphs of all non-commutative rings of the listed orders (p^2 through p^5, p^2 q, p^3 q), using standard external classifications of such rings. It additionally verifies the MSN-integral but not hyperintegral property and resolves two prior conjectures via the same explicit methods. Citations to Nath et al. 2021 and Fasfous-Nath 2024 are for the conjectures being proved here rather than load-bearing premises; no self-definitional equations, fitted parameters renamed as predictions, or ansatzes imported via self-citation appear. The derivation chain is self-contained once the independent ring classifications are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard definitions and known classifications of small finite rings rather than introducing new parameters or entities.

axioms (2)
  • domain assumption All non-commutative rings of the orders p^2, p^3, p^4, p^5, p^2 q and p^3 q are classifiable and enumerable for graph construction.
    Invoked by restricting the study to these specific orders and proceeding with explicit computations.
  • standard math The definitions of commuting graph, second neighborhood degree, spectrum, and energy follow the standard conventions in algebraic graph theory.
    Used throughout the spectrum and energy calculations.

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Reference graph

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