New classes of spectrally arbitrary ray patterns (Q632461)

A ray (sign) pattern of order \(n\) is said to be spectrally arbitrary if each monic polynomial of degree \(n\) with coefficient from \(\mathbb C\) is the characteristic polynomial of some matrix in the ray (sign) pattern class. In 2000, \textit{J. H. Drew, C. R. Johnson, D. D. Olesky} and \textit{P. van den Driessche} [Linear Algebra Appl. 308, No. 1-3, 121--137 (2000; Zbl 0957.15012)] developed the nilpotent-Jacobian method for the classification of spectrally arbitrary sign patterns. The generalization of this method for ray pattern was done by \textit{J. J. McDonald} and \textit{J. Stuart} [Linear Algebra Appl. 429, No. 4, 727--734 (2008; Zbl 1143.15007)]. In 2004, \textit{T. Britz, J. J. McDonald, D. D. Olesky} and \textit{P. van den Driessche} [SIAM J. Matrix Anal. Appl. 26, No.~1, 257--271 (2004; Zbl 1082.15016)] showed that every \(n\times n\) irreducible spectrally arbitrary sign pattern must have at least \(2n -1\) non-zero entries and they provided families of sign patterns that have exactly \(2n\) non-zero entries. In this article the author has given a class of \(n \times n\) spectrally arbitrary ray patterns with exactly \(3n\) non-zero entries for \(n \geq 6\).