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

Let INSn,pINSn,p be the set of n×nn×n irreducible non-powerful (generalized) sign pattern matrices with period pp, and let A∈INSn,pA∈INSn,p. In this paper, we introduce a new parameter called the index of maximum ambiguous density of AA. Furthermore, the generalized index of maximum ambiguous density of AA, 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∈INSn,pA∈INSn,p. Finally, the index and the generalized index of maximum ambiguous density for irreducible non-powerful zero-symmetric sign pattern matrices are discussed.