A Note on Conjectures of Gullerud, Johnson, and Mbirika
October 02, 2025
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.
1 Introduction and background↩︎
An \(n \times n\) matrix \(A = (a_{i,j})\) is called tridiagonal if \(a_{i,j} = 0\) whenever \(|i - j| \geq 2\). Tridiagonal matrices appear in many practical applications (e.g., [1]–[3]) but are of great interest in their own right (e.g., [4]–[6]). In this note, we consider a particular class of symmetric tridiagonal matrices and study properties of characteristic polynomials.
Recall that for an undirected graph \(G = (V,E)\) with vertex set \(V = \{v_1,\dots,v_n\}\), the adjacency matrix for \(G\) is \(A(G) = (a_{ij}) \in \mathbb{R}^{n \times n}\) where \[a_{ij} = \begin{cases} 1 & \text{ if } v_iv_j \in E, \\ 0 & \text{ otherwise}. \end{cases}\] Note that, since \(G\) is undirected, \(a_{ij} = a_{ji}\). The characteristic polynomial of \(G\) is therefore the characteristic polynomial of \(A(G)\), that is, \[f_G(\lambda) = \det(A(G) - \lambda I_n).\]
In [7], the authors investigated the characteristic polynomial of \(A(P_n)\), where \(P_n\) is the path graph on \(n\) vertices. Namely, they investigated the solutions to the equation \(f_{P_n}(\lambda) = F_{n+1}\), where \(F_n\) is the \(n^{th}\) Fibonacci number (using the convention \(F_0 = 0\) and \(F_1 = 1\)). Note the closed formula \[f_n(\lambda) = \sum_{k=0}^{\lfloor \frac{n}{2}\rfloor} (-1)^{n+k} \binom{n-k}{k}\lambda^{n-2k}\] for the characteristic polynomial of the path on \(n\) vertices, which can be deduced from [8]. Among the authors’ conjectures are the following.
Conjecture 1 ([7]). Let \(R_n\) be the set of roots of \(f_n(\lambda) = F_{n+1}\).
For each \(n\), the points in \(R_n\) lie on an ellipse.
The maximum real part of a point in \(R_n\) is unbounded as \(n\) increases.
For all \(n\), the imaginary parts of points in \(R_n\) lie within \([-1,1]\).
If \(n\) is even, then \(R_n\) contains exactly two distinct real roots.
If \(n\) is odd, then \(R_n\) contains exactly one real root, which is negative.
In Section 2.1, we provide the results of a least squares analysis for the roots of \(f_n\) and show how the roots trend towards lying on an ellipse. Although this leaves Conjecture 1 (1) open, in Section 3 we draw connections with work of Oyengo [9] related to Chebyshev polynomials to show that the points of \(R_n\) are contained inside one particular ellipse \(E\) for all \(n\). This shows that (2) is in fact false for an infinite class of polynomials and (3) is true for that same subclass. In Section 4, we make progress towards (4) and (5) of Conjecture 1 by showing that Further, when \(n \equiv 0 \pmod{4}\), we show that the only two purely imaginary roots of \(f_{4k}(\lambda) = F_{4k+1}\) are \(\pm \boldsymbol{i}\).
2 A least squares approach↩︎
2.1 Computational Framework↩︎
To analyze whether the solutions to \(f_n(\lambda) = F_{n+1}\) fall on an ellipse, we first gathered data using SageMath [10]. Our overarching approach, together with accompanying pseudocode, is outlined as follows:
We begin by computing the roots of \(\mathcal{F}_n(\lambda)\), the \(n\)th-degree polynomial obtained by subtracting the \((n+1)\)th Fibonacci number from the characteristic polynomial of the adjacency matrix of the path graph on \(n\) vertices. We then separate these roots into their real and imaginary parts, which are stored in two lists corresponding, respectively, to the \(x\)-axis and \(y\)-axis for plotting.
Below, we use \(\odot\) to indicate the Hadamard product of matrices. The pseudocode below computes approximations of the coefficients of the best-fit ellipse for the roots. A detailed explanation of the procedure follows.
To analyze the geometric distribution of the roots of \(\mathcal{F}_n(\lambda)\), we model them using the general equation of an ellipse, \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \text{ equivalently, } Ax^2 + B y^2 = 1,\] where \(A = \frac{1}{a^2}\) and \(B = \frac{1}{b^2}\).
Let \(\{\rho_k\}_{k=1}^m\) denote the roots of \(\mathcal{F}_n(\lambda)\), and write \(\rho_k = \Re(\rho_k) + i\,\Im(\rho_k)\), so that the \(k^{th}\) root corresponds to the point \((x_k, y_k) = (\Re(\rho_k), \Im(\rho_k))\). We then form the overdetermined linear system \[\begin{bmatrix} (\Re(\rho_1))^2 & (\Im(\rho_1))^2 \\ (\Re(\rho_2))^2 & (\Im(\rho_2))^2 \\ \vdots & \vdots \\ (\Re(\rho_m))^2 & (\Im(\rho_m))^2 \end{bmatrix} \begin{bmatrix} \tilde{A} \\ \tilde{B} \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}.\]
To approximate the root distribution geometrically, we first compute the best-fit coefficients \(\tilde{A}\) and \(\tilde{B}\) by solving the system in the least squares sense, as
implemented in Algorithm 2 using our function least_squares_solver() [11]. This function
solves the overdetermined system via the normal equations and then converts \((\tilde{A}, \tilde{B})\) into the corresponding ellipse parameters \((\tilde{a}, \tilde{b})\) according to \[\tilde{a} = \frac{1}{\sqrt{\tilde{A}}}, \quad \tilde{b} = \frac{1}{\sqrt{\tilde{B}}}.\] The resulting approximated ellipse, \[\frac{x^2}{\tilde{a}^2} + \frac{y^2}{\tilde{b}^2} = 1,\] provides
a geometric representation of the root distribution.
Finally, we fit the ellipse to the root data and quantitatively evaluate the model by computing the root-mean-square error (RMSE). The RMSE is computed as \[\mathrm{RMSE} = \sqrt{\frac{1}{m} \sum_{k=1}^{m} \left( \frac{\Re(\rho_k)^2}{\tilde{a}^2} + \frac{\Im(\rho_k)^2}{\tilde{b}^2} - 1 \right)^2 }.\] This metric provides a quantitative measure of the extent to which the roots conform to an elliptical pattern.
2.2 Empirical Results↩︎
Our findings are summarized in Figure 3. As the polynomial degree increases, the error approaches zero.
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 \(\mathcal{F}_n\) lie on an ellipse. Our least-squares fit, which yields minimal error, supports this claim.
The authors further contend in Conjecture 1 that the purely imaginary roots lie between \(-\boldsymbol{i}\) and \(\boldsymbol{i}\), a claim consistent with our findings. However, their assertion that the maximum real part of a point in \(R_n\) grows unbounded as \(n\) increases is not supported by our data. On the contrary, Figure 4 suggests that the real parts of the points in \(R_n\) stabilize at approximately \(\sqrt{5}\) as \(n \to \infty\). This observation leads us to conjecture that, for all \(n\), the real parts of points in \(R_n\) lie within \([-\sqrt{5}, \sqrt{5}]\).
Moreover, the authors’ attention to the case \(f_n(\lambda) = F_{n+1}\) makes one wonder if these properties are unique to \(F_{n+1}\). In contrast, our experiments indicate that the elliptic structure persists even when the constant is varied, independent of its connection to the Fibonacci sequence. Preliminary evidence further suggests that the eccentricity of the ellipse depends on the chosen constant. This is illustrated in Figure 5.
Figure 5: Visualization of the roots of \(f_n(\lambda) = c\) for varying \(c \in \mathbb{Z}_{> 0}\). a — \(f_{13}(\lambda) = 3\), b — \(f_{13}(\lambda) = F_{14}\), c — \(f_{13}(\lambda) = 1000\)
3 A connection to Chebyshev polynomials↩︎
Chebyshev polynomials constitute a classical sequence of orthogonal polynomials with fundamental applications in numerical analysis, approximation theory, and various areas of applied mathematics (see, e.g., [12]–[14]). There exist two commonly studied types of Chebyshev polynomials.
Definition 2 (Type 1 Chebyshev polynomials). The Chebyshev polynomials of the first kind, denoted \(T_n(x)\), are defined recursively by \[T_0(x) = 1, \quad T_1(x) = x, \quad T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x), \quad n \ge 1.\]
These polynomials play a central role in function approximation, particularly in minimizing the maximum error in polynomial interpolation, a problem known as the Chebyshev approximation problem.
Definition 3 (Type 2 Chebyshev polynomials). The Chebyshev polynomials of the second kind, denoted \(U_n(x)\), are defined recursively by \[U_0(x) = 1, \quad U_1(x) = 2x, \quad U_{n+1}(x) = 2x U_n(x) - U_{n-1}(x), \quad n \ge 1.\]
While they arise in similar approximation contexts, Type 2 polynomials are especially prevalent in the analysis of differential equations and in spectral methods. Gullerud, Johnson, and Mbirika observed that \(f_n\) and \(U_n\) are related via \[\label{connectionfandu} f_n(x) = U_n\!\left(-\frac{x}{2}\right),\tag{1}\] a fact we will make use of. Consequently, the roots of \(f_n\) are precisely scaled roots of \(U_n\). Hence, if the distribution of the roots of \(U_n\!\left(-\frac{x}{2}\right) - F_{n+1}\) can be associated with a conic form, the same holds for \(f_n(\lambda) - F_{n+1}\).
While there is existing literature on conic distributions of the Chebyshev polynomials of the first kind [15], to the best of our knowledge, no direct correspondence has been established between the roots of \(T_n\) and those of \(U_n\). Furthermore, we found no prior work indicating that the roots of the Chebyshev polynomials of the second kind exhibit a direct correspondence with ellipses, as is observed for the first kind. That said, Oyengo was able to establish in his dissertation the following relationship between the roots of \(U_n\) and ellipses.
Proposition 4 ([9]). Let \(\lambda, \kappa \in \mathbb{R}\) be non-zero with \(\lambda^2 - \kappa^2 = \alpha^2\). Then the roots of \[U_n\Bigl(\frac{x}{\alpha}\Bigr) - U_n\Bigl(\frac{\boldsymbol{i}\kappa}{\alpha}\Bigr)\] lie inside, but very close to, the ellipse \(E(\lambda, \kappa)\).
Our objective is to identify \(\alpha\) and \(\kappa\) such that \[U_n\Bigl(-\frac{x}{2}\Bigr) - F_{n+1} = U_n\Bigl(\frac{x}{\alpha}\Bigr) - U_n\Bigl(\frac{\boldsymbol{i}\kappa}{\alpha}\Bigr).\] Furthermore, we aim to set \(\lambda = \pm \sqrt{5}\) and \(\kappa = \pm 1\), consistent with our conjectured semi-major and semi-minor axes of \(\sqrt{5}\) and \(1\), respectively. We will demonstrate that such a choice is indeed possible. First, we establish a lemma we will need.
Lemma 5. For all \(n \geq 1\), \(U_n\left(-\frac{\boldsymbol{i}}{2}\right) = \boldsymbol{i}^{3n}F_{n+1}\).
Proof. We proceed by strong induction on \(n\). First observe that when \(n = 0\), \[U_0\Big(-\frac{\boldsymbol{i}}{2}\Big) = 1 = \boldsymbol{i}^{3\cdot 0} F_1,\] and when \(n = 1\), \[U_1\Big(-\frac{\boldsymbol{i}}{2}\Big) = 2\Big(-\frac{\boldsymbol{i}}{2}\Big) = -\boldsymbol{i}= \boldsymbol{i}^{3\cdot 1} F_2.\]
For the inductive step, we use the definition of \(U_n\) and the Fibonacci recurrence to see \[\begin{align} U_{k+1}\Big(-\frac{\boldsymbol{i}}{2}\Big) &= 2\Big(-\frac{\boldsymbol{i}}{2}\Big) U_k\Big(-\frac{\boldsymbol{i}}{2}\Big) - U_{k-1}\Big(-\frac{\boldsymbol{i}}{2}\Big) \\ &= -\boldsymbol{i}\cdot \boldsymbol{i}^{3k} F_{k+1} - \boldsymbol{i}^{3(k-1)} F_k \\ &= \boldsymbol{i}^{3(k+1)} F_{k+1} + \boldsymbol{i}^{3(k+1)} F_k \\ &= \boldsymbol{i}^{3(k+1)} (F_{k+1} + F_k) \\ &= \boldsymbol{i}^{3(k+1)} F_{k+2}. \end{align}\] Hence, \(U_n\left(-\frac{\boldsymbol{i}}{2}\right) = \boldsymbol{i}^{3n}F_{n+1}\) is true for all \(n \ge 0\). ◻
Theorem 6. Let \(\mathcal{F}_n(x) = f_n(x) - F_{n+1}.\) If \(n \equiv 0 \pmod{4}\), then the roots of \(\mathcal{F}_n(x)\), denoted \(\{\rho_k\}_{k=1}^n\) with \(\rho_k = \Re(\rho_k) + \boldsymbol{i}\,\Im(\rho_k)\), are bounded as \[|\Im(\rho_k)| \leq 1 \quad \text{and} \quad |\Re(\rho_k)| \leq \sqrt{5}\] for all \(1 \leq k \leq n\).
Proof. Consider \(\lambda = \sqrt{5}, \alpha = -2\), and \(\kappa = 1\). These choices satisfy \[\lambda^2 - \kappa^2 = \alpha^2,\] so by Proposition 4, the roots of \[U_n\!\left(-\frac{x}{2}\right) - U_n\!\left(-\frac{\boldsymbol{i}}{2}\right)\] lie within the ellipse \(E(\sqrt{5}, 1)\). It remains to show that \[U_n\!\left(-\frac{x}{2}\right) - U_n\!\left(-\frac{\boldsymbol{i}}{2}\right) = f_n(x) - F_{n+1}.\] The equality \(U_n\!\left(-\frac{x}{2}\right) = f_n(x)\) follows directly from 1 , so it suffices to verify that \(U_n\!\left(-\frac{\boldsymbol{i}}{2}\right) = F_{n+1}\). Notice that when \(n \equiv 0 \pmod{4}\), we have \(\boldsymbol{i}^{3n} = 1\). Then, by Lemma 5, it follows immediately that \[U_n\Big(-\frac{\boldsymbol{i}}{2}\Big) = F_{n+1}.\] Consequently, \[U_n\!\left(-\frac{x}{2}\right) - U_n\!\left(-\frac{\boldsymbol{i}}{2}\right) = f_n(x) - F_{n+1},\] and hence the roots of \(\mathcal{F} = f_n(x) - F_{n+1}\) are bounded by the ellipse \(E(\sqrt{5},1)\). ◻
The above theorem assures us that the roots of \(\mathcal{F}_n\) are bounded by an ellipse when \(n\) is divisible by \(4\). On the other hand, for general \(n\), we have verified that the roots lie near to \(E(\sqrt{5},1)\) with a decreasing error on the order of \(10^{-4}\) for \(n \leq 1000\). These findings agree with the axis values conjectured from computational evidence demonstrated in Figure 4, and they prove part (3) of Conjecture 1 while disproving part (2) of Conjecture 1 when restricting to those polynomials with \(n \equiv 0 \pmod{4}\). Future work may extend our approach, or apply a new approach, to generalize these results to all \(n \in \mathbb{Z}_{> 0}\).
4 Root counts for \(f_n(\lambda) = c\)↩︎
A sequence related to Fibonacci numbers is the sequence of Pell numbers. The Pell numbers are the integers \(P_0,P_1,P_2,\dots\) defined recursively by \(P_0 = 0\), \(P_1 = 1\), and \(P_n = 2P_{n-1} + P_{n-2}\) for all \(n \geq 2\). These numbers allow us to make progress towards proving Conjecture 1 (4) and (5).
Theorem 7. When \(n\) is odd, the equation \(f_n(\lambda) = c\) has exactly one real solution for all constants \(c\) satisfying \(|c| \geq P_{n+1}\). When \(n\) is even, \(f_n(\lambda) = c\) has no real solutions when \(c \leq -P_{n+1}\) and exactly two real solutions when \(c \geq P_{n+1}\).
Proof. First note that, \(f_n\) has a negative leading coefficient when \(n\) is odd. Further, since \(\deg(f_n)\) is odd when \(n\) is odd, \(f_n(\lambda)\) will always have at least one real solution. When \(n\) is even, \(f_n\) is monic and \(\deg(f_n)\) is even. Additionally, it is known [16] the solutions to \(f_n(\lambda) = 0\) are exactly the values \[r_s = 2\cos\left(\frac{s\pi}{n+1}\right)\] for \(s = 1,\dots,n\). Hence, to establish our claims, suffices to show that \(|f_n(\lambda)| < P_{n+1}\) for \(\min(r_s) \leq \lambda \leq \max(r_s)\), that is, \[r_n = -2\cos\left(\frac{\pi}{n+1}\right) \leq \lambda \leq 2\cos\left(\frac{\pi}{n+1}\right) = r_1.\] Importantly, \(|r_1| = |r_n| < 2\). Additionally, since \(f_n\) is even or odd, it is enough for us to restrict ourselves to \(\lambda \geq 0\).
We proceed by induction. When \(n=1\), \(f_1(\lambda) = -\lambda\). This has only one root, \(0\), and so \(|f_1(\lambda)| = 0 = P_0\) in the range \([r_1,r_1] = \{0\}\).
When \(n=2\), \(f_2(\lambda) = \lambda^2 - 1\), which satisfies \[|f_2(\lambda)| \leq 1 = P_1\] in the range \([r_2,r_1] = [-1,1]\).
Now consider \(n > 2\). We apply the triangle inequality to the recurrence \[f_n(\lambda) = -\lambda f_{n-1}(\lambda) - f_{n-2}(\lambda)\] and apply the inductive hypothesis to obtain \[|f_n(\lambda)| = |\lambda||f_{n-1}(\lambda)| + |f_{n-2}(\lambda)| < 2P_{n-2} + P_{n-3} = P_{n-1},\] as claimed. ◻
4.1 Uniqueness of imaginary roots for \(\mathbf{n \equiv 0}\) (mod 4).↩︎
In [7], it is established that for every positive integer \(k\), \(\lambda = \pm \boldsymbol{i}\) is a root of \(f_{4k}(\lambda) = F_{4k+1}\). We show that these are the only purely imaginary roots.
Theorem 8. For every positive integer \(k\), \(f_{4k}(a\boldsymbol{i}) = F_{4k+1}\) if and only if \(a = 1\).
Proof. Consider \(a \in \mathbb{R}\), \(M \in \mathbb{Z}_{> 0}\). If \(a=1\), then \(f_{4k}(\boldsymbol{i}) = F_{4k+1}\) by [7]. So, suppose \(f_{4k}(a\boldsymbol{i}) = F_{4k+1}\), that is, \[f_{4k}(a\boldsymbol{i}) = \sum_{j = 0}^{2k} (-1)^{4k + j}{4k - j \choose j}((a\boldsymbol{i})^2)^{2k-j} = F_{4k + 1}.\] By elementary algebra and the well-known identity \(F_{4k+1} = \sum_{j = 0}^{2k}{4k-j \choose j}\) (see, e.g., [17]), this is equivalent to writing \[\label{eq:32equation1} \sum_{j = 0}^{2k} (a^2)^{2k-j}{4k - j \choose j} = F_{4k + 1}.\tag{2}\] If \(|a| < 1\), then \((a^2)^{2k-j}{4k - j \choose j} < {4k - j \choose j}\), in which case 2 is false, a contradiction. We obtain a similar contradiction if \(|a| > 1\). Thus, \(|a| = 1\). Since \(a\) is real, we obtain \(a = \pm 1\). ◻
5 Future Direction↩︎
Our experiments indicate that the conic patterns observed for the characteristic polynomials of adjacency matrices of path graphs are not unique to this family. In particular, cycle graphs display similar conic behavior, likely reflecting the close relationship between their polynomials. Indeed, the characteristic polynomials of cycle graphs can be expressed recursively in terms of those of path graphs (see, e.g., [8]):
\[f_n^{\mathrm{cyc}}(\lambda) = f_n(\lambda) - f_{n-2}(\lambda) - 2.\]
Beyond cycles, the roots of polynomials associated with star graphs and complete bipartite graphs also display related patterns. These observations suggest that methods developed for identifying the conic structure in polynomials related to path graphs may be adaptable to other graph families. Consequently, a promising direction for future research is to formalize a general framework for detecting conic structures across diverse classes of graphs, potentially uncovering broader principles governing the spectral behavior of graph polynomials.
6 Acknowledgements↩︎
This work was completed in part for the course MATH 310E: Combinatorial Problem Solving, Extended Study. The authors thank Colgate University for approving the program, providing us with the opportunity to visit Institute of Science Tokyo and the Graduate School of Information Science and Technology at Osaka University to present and further our work there. We in particular want to thank Tamás Kálmán, Akiyoshi Tsuchiya, and Akihiro Higashitani for helping arrange the visit, Nick Moore for his insights on connections between combinatorics and numerical analysis, and the other students in the course for their support, questions, and feedback.