Potentially nilpotent and spectrally arbitrary even cycle sign patterns

An n × n sign pattern S_n is potentially nilpotent if there is a real matrix having sign pattern S_n and characteristic polynomial xⁿ. A new family of sign patterns C_n 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 C_n. These nilpotent matrices are used together with a Jacobian argument to show that C_n is spectrally arbitrary, i.e., there is a real matrix having sign pattern C_n and characteristic polynomial for any real μ_i. Some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given.