A Note on Conjectures of Gullerud, Johnson, and Mbirika

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.