Square nearly nonpositive sign pattern matrices

Abstract

A sign pattern matrix A is called square nearly nonpositive if all entries but one of A 2 are nonpositive. We characterize the irreducible sign pattern matrices that are square nearly nonpositive. Further we determine the maximum (resp. minimum) number of negative entries that can occur in A 2 when A is irreducible square nearly nonpositive (SNNP), and then we characterize the sign patterns that achieve this maximum (resp. minimum) number. Finally, we discuss some spectral properties of the sign patterns which are square nonpositive or square nearly nonpositive.