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

Let INS_n,p be the set of n×n irreducible non-powerful (generalized) sign pattern matrices with period p, and let AINS_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 AINS_n,p. Finally, the index and the generalized index of maximum ambiguous density for irreducible non-powerful zero-symmetric sign pattern matrices are discussed.