A potentially stable and potentially nilpotent tree sign pattern that is not spectrally arbitrary

Abstract

Abstract It has been shown that a 4 x 4 tree sign pattern is spectrally arbitrary if and only if it is potentially stable and potentially nilpotent in [M. Arav, F. Hall, Z. Li, K. Kaphle, N. Manzagol, spectrally arbitrary tree sign patterns of order 4, Electron. J. Linear Algebra, 20 (2010), 180-197], where the authors asked whether the above important theorem can be extended to the 5 x 5 case. In this paper, we negatively answer this question by constructing a 5x5 potentially stable and potentially nilpotent tree sign pattern which is shown to be not spectrally arbitrary.