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arxiv: 2510.02008 · v2 · submitted 2025-10-02 · 🧮 math.CO

A Note on Conjectures of Gullerud, Johnson, and Mbirika

Robert Davis , Nayda Farnsworth This is my paper

Pith reviewed 2026-05-18 10:34 UTC · model grok-4.3

classification 🧮 math.CO
keywords conjecturesroot distributioncharacteristic polynomialspath graphstridiagonal matricesnonhomogeneous equationsFibonacci numbers
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The pith

Partial progress on two conjectures about root distributions, with counterexamples for a third and evidence against a fourth.

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

Gullerud, Johnson, and Mbirika studied roots of nonhomogeneous equations connected to characteristic polynomials of path graph adjacency matrices, noting that some evaluate to a Fibonacci number at the imaginary unit. This led to several conjectures on root distributions. The current paper makes partial progress toward proving two of those conjectures, finds an infinite family of polynomials that disproves a third conjecture, and supplies evidence that casts doubt on a fourth conjecture.

Core claim

We further explore their conjectures regarding the distribution of roots. We make partial progress towards establishing two conjectures, identify an infinite class of polynomials for which a third is false, and give evidence against a fourth.

What carries the argument

nonhomogeneous equations related to the characteristic polynomials of adjacency matrices for path graphs

If this is right

  • Two of the conjectures receive partial support from the analysis of specific cases.
  • An infinite class of polynomials serves as counterexamples to one conjecture.
  • Computational or analytic evidence weakens support for the remaining conjecture.

Where Pith is reading between the lines

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

  • Future work could seek to classify exactly for which polynomials each conjecture holds.
  • The methods used here might extend to other types of graphs or matrices.
  • Resolving the remaining conjectures could clarify connections to Fibonacci numbers and root symmetries.

Load-bearing premise

The definitions and partial results on the nonhomogeneous equations and characteristic polynomials from the 2023 paper are taken as given.

What would settle it

Explicit calculation of the roots for polynomials in the infinite counterexample class to check if they fail to satisfy the third conjecture as claimed.

Figures

Figures reproduced from arXiv: 2510.02008 by Nayda Farnsworth, Robert Davis.

Figure 1
Figure 1. Figure 1: Polynomial Degree vs. RMSE Error The behavior of the purely real and imaginary roots observed in our data both corroborates and challenges aspects of the authors’ claims. Their central assertion is that the roots of Fn lie on an ellipse. Our least￾squares fit, which yields minimal error, supports this claim. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Semi-major and semi-minor axis values The authors further contend in Conjecture 1.1 that the purely imaginary roots lie between −i and i, a claim consistent with our findings. However, their assertion that the maximum real part of a point in Rn grows unbounded as n increases is not supported by our data. On the contrary, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of the roots of fn(λ) = c for varying c ∈ Z>0 4 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

In 2023, Gullerud, Johnson, and Mbirika presented results on their study of certain tridiagonal real symmetric matrices. As part of their work, they studied the roots to nonhomogeneous equations related to characteristic polynomials of adjacency matrices for path graphs. They showed that a subset of these polynomials give a Fibonacci number when evaluated at the imaginary unit, leading them to make several intriguing conjectures. In this work, we further explore their conjectures regarding the distribution of roots. We make partial progress towards establishing two conjectures, identify an infinite class of polynomials for which a third is false, and give evidence against a fourth.

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 manuscript investigates conjectures from Gullerud, Johnson, and Mbirika (2023) on the roots of nonhomogeneous equations tied to characteristic polynomials of adjacency matrices for path graphs. It supplies case-by-case analytic arguments establishing partial results toward two conjectures, constructs an infinite parametric family of counterexamples that falsifies a third conjecture, and presents numerical root-distribution plots as evidence against a fourth.

Significance. If the arguments and constructions hold, the note provides concrete advances on these conjectures by delivering an explicit infinite counterexample family to one of them and independent analytic and numerical support for the others. The work builds directly on the 2023 definitions and partial results without circularity or new unverified assumptions, and the falsification step is a clear strength.

minor comments (3)
  1. The introduction would benefit from a concise restatement of all four conjectures (with their original numbering) so that the partial results and counterexamples can be read without constant reference to the 2023 paper.
  2. §3 (counterexample construction): the parametric choice of inhomogeneity is described, but an explicit statement of the parameter interval guaranteeing infinitely many distinct polynomials would make the infinitude claim immediate.
  3. Figure captions for the root plots should specify the range of degrees or parameter values examined and the total number of roots plotted, to allow readers to assess the scope of the numerical evidence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. The report correctly captures our partial analytic progress on two conjectures, the explicit infinite counterexample family disproving a third, and the numerical evidence against the fourth. We have no major comments to address point by point, as none were raised.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript takes the definitions, notation, and partial results on nonhomogeneous recurrences and characteristic polynomials as given from the external 2023 Gullerud-Johnson-Mbirika paper. It then constructs independent counterexamples via specialization of those recurrences to parametric forms, supplies case-by-case analytic arguments for two conjectures, and provides numerical root plots for the fourth. None of these steps reduce a claimed prediction or result to a quantity defined by its own fitted parameters, nor do they rely on self-citation chains or imported uniqueness theorems. The derivation chain remains self-contained against the external benchmarks and exhibits none of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of characteristic polynomials, roots, and Fibonacci numbers together with the specific setup from the 2023 paper; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math Standard algebraic properties of characteristic polynomials of adjacency matrices and evaluation at the imaginary unit
    Invoked when discussing the Fibonacci evaluation and root locations.

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Works this paper leans on

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