Linear and Multilinear Algebra

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. extended the concepts of base and period from non-negative matrices to powerful sign pattern matrices. 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. 1. Introduction The sign of a real number a, denoted by sgn(a), is defined to be 1, À1 or 0, if a > 0, a < 0, or a ¼ 0 respectively. The sign pattern matrix of a real matrix A, denoted by sgn(A), is the (0, 1, À1)-matrix obtained from A by replacing each entry by its sign. The powers (especially the sign patterns of the powers) of square sign pattern matrices have been studied to some extent [4,5,9,10]. Notice that in the computation of (the signs of) the entries of the power A k , an ambiguous sign may arise when we add a positive sign to a negative sign, so a new symbol '#' has been introduced to denote the ambiguous sign in [5]. The set À ¼ {0, 1, À1, #} is called generalized sign set, where '#' denotes the ambiguous sign. We define addition and multiplication involving the symbol '#' as follows: for all a 2 À and for all b 2 À\{0},