Potentially nilpotent and spectrally arbitrary even cycle sign patterns

An n x n sign pattern P-n is potentially nilpotent if there is a real matrix having sign pattern P-n and characteristic polynomial x(n). 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 pattem Wn. 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 'Kn and characteristic polynomial x(n) + Sigma(n)(i=1) (-1)(i) mu(i)x(n-i) for any real mu(i). Some results and a conjecture on minimality of these spectrally arbitrary sign patterus are given. (c) 2006 Elsevier Inc. All rights reserved.