The kth lower bases of primitive non-powerful signed digraphs

In [J. Shao, L. You, H. Shan, Bound on the bases of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 285-300], the authors extended the concept of the base from powerful sign pattern matrices to non-powerful irreducible sign pattern matrices. Recently, the kth local bases and the kth upper bases, which are generalizations of the bases, of primitive non-powerful signed digraphs were introduced. In this paper, we introduce a new parameter called the kth lower bases of primitive non-powerful signed digraphs and obtain some bounds for it. For some cases, the bounds we obtain are best possible and the extremal signed digraphs are characterized, respectively. Moreover, we show that there exist “gaps” in the kth lower bases set of primitive non-powerful signed digraphs.