Bounds on the kth multi-g base index of nearly reducible sign pattern matrices

Li et al. [On the period and base of a sign pattern matrix, Linear Algebra Appl. 212/213 (1994) 101-120.] extended the concepts of the base and period from nonnegative matrices to powerful sign pattern matrices. Then, Shao and You [Bound on the basis of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 285-300.] extended the concepts of the base from powerful sign pattern matrices to non-powerful irreducible sign pattern matrices. In this paper we mainly study the kth multi-g base index for non-powerful primitive nearly reducible sign pattern matrices. We obtain sharp upper bounds, together with a complete characterization of the equality cases of the kth multi-g base index for primitive nearly reducible generalized sign pattern matrices. We also show that there exist “gaps” in the kth multi-g base index set of the classes of such matrices.