A Class of Minimally Spectrally Arbitrary Sign Patterns

Abstract

A spectrally arbitrary pattern $A$ is a sign pattern of order $n$ such that every monic real polynomial of degree $n$ can be achieved as the characteristic polynomial of a matrix with sign pattern $A$. A sign pattern $A$ is minimally spectrally arbitrary if it is spectrally arbitrary but is not spectrally arbitrary if any nonzero entry (or entries) of $A$ is replaced by zero. In this paper, we introduce some new sign patterns which are minimally spectrally arbitrary for all orders $n\ge 7$.