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A Note on Conjectures of Gullerud, Johnson, and Mbirika

Robert Davis, Nayda Farnsworth

TL;DR

The paper investigates the root distribution of $f_n(λ)=F_{n+1}$, where $f_n$ is the characteristic polynomial of the path-graph adjacency matrix, revealing an elliptic pattern in the complex roots via least-squares fits and connections to Chebyshev polynomials. The authors prove, for $n$ divisible by $4$, that all roots are bounded by the ellipse $E(\sqrt{5},1)$ and that the purely imaginary roots are exactly $\pm i$, substantiating part of the conjectures and disproving another for this infinite subfamily. They provide a Pell-number-based criterion for real-root counts of $f_n(λ)=c$ and establish exact imaginary-root uniqueness in the $n\equiv 0\pmod{4}$ case, while also suggesting broader applicability of the conic-pattern framework to other graph families. The work offers partial progress toward the conjectures, introduces a robust numerical-analytic approach, and opens directions for formalizing a general framework for conic root patterns in graph polynomials with potential extensions to cycles and other graphs.

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.

A Note on Conjectures of Gullerud, Johnson, and Mbirika

TL;DR

The paper investigates the root distribution of , where is the characteristic polynomial of the path-graph adjacency matrix, revealing an elliptic pattern in the complex roots via least-squares fits and connections to Chebyshev polynomials. The authors prove, for divisible by , that all roots are bounded by the ellipse and that the purely imaginary roots are exactly , substantiating part of the conjectures and disproving another for this infinite subfamily. They provide a Pell-number-based criterion for real-root counts of and establish exact imaginary-root uniqueness in the case, while also suggesting broader applicability of the conic-pattern framework to other graph families. The work offers partial progress toward the conjectures, introduces a robust numerical-analytic approach, and opens directions for formalizing a general framework for conic root patterns in graph polynomials with potential extensions to cycles and other graphs.

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.

Paper Structure

This paper contains 9 sections, 5 theorems, 30 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction and background
  2. A least squares approach
  3. Computational Framework
  4. Empirical Results
  5. A connection to Chebyshev polynomials
  6. Root counts for $f_n(\lambda) = c$
  7. Uniqueness of imaginary roots for $\mathbf{n \equiv 0}$ (mod 4).
  8. Future Direction
  9. Acknowledgements

Key Result

Proposition 3.3

Let $\lambda, \kappa \in \mathbb{R}$ be non-zero with $\lambda^2 - \kappa^2 = \alpha^2$. Then the roots of lie inside, but very close to, the ellipse $E(\lambda, \kappa)$.

Figures (3)

  • Figure 1: Polynomial Degree vs. RMSE Error
  • Figure 2: Semi-major and semi-minor axis values
  • Figure 3: Visualization of the roots of $f_n(\lambda) = c$ for varying $c \in \mathbb{Z}_{> 0}$

Theorems & Definitions (12)

  • Conjecture 1.1: Tridiagonal
  • Definition 3.1: Type 1 Chebyshev polynomials
  • Definition 3.2: Type 2 Chebyshev polynomials
  • Proposition 3.3: Oyengo
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 4.1
  • proof
  • ...and 2 more