Multi-g base index of primitive anti-symmetric sign pattern matrices

In [Z. Li, F. Hall, and C. Eschenbach, On the period and base of a sign pattern matrix, Lin. Alg. Appl., 212/213 (1994), pp. 101–120], extended the concepts of base and period from non-negative matrices to powerful sign pattern matrices. In [L. You, J. Shao, and H. Shan, Bounds on the bases of irreducible generalized sign pattern matrices, Lin. Alg. Appl., 427 (2007), pp. 285–300], extended the concept of base from powerful sign pattern matrices to non-powerful generalized sign pattern matrices. In [Q. Li and B. Liu, Multi-g base index of non-powerful generalized sign pattern matrices, Ln. Multilin. Alg. (to appear)], extended the concept of kth multi-g base index from non-negative matrices to non-powerful generalized sign pattern matrices. In this article, we mainly study the bounds on kth multi-g base index, extremal graphs for the generalized base index for primitive anti-symmetric sign pattern matrices.