Potentially nilpotent and spectrally arbitrary even cycle sign patterns

An n × n sign pattern Sn is potentially nilpotent if there is a real matrix having sign pattern Sn and characteristic polynomial xn. A new family of sign patterns Cn with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern Cn. These nilpotent matrices are used together with a Jacobian argument to show that Cn is spectrally arbitrary, i.e., there is a real matrix having sign pattern Cn and characteristic polynomial for any real μi. Some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given.