The index of maximum ambiguous density for irreducible non-powerful sign pattern matrices

Abstract

Let I N S n , p be the set of n × n irreducible non-powerful (generalized) sign pattern matrices with period p , and let A ∈ I N S n , p . In this paper, we introduce a new parameter called the index of maximum ambiguous density of A . Furthermore, the generalized index of maximum ambiguous density of A , which generalizes the concept of the index of maximum ambiguous density, is introduced. Moreover, some bounds on these indices are obtained, and we exhibit a system of gaps in the set of the index of maximum ambiguous density for A ∈ I N S n , p . Finally, the index and the generalized index of maximum ambiguous density for irreducible non-powerful zero-symmetric sign pattern matrices are discussed.