The base of a primitive, nonpowerful sign pattern with exactly nonzero diagonal entries

In [J.Y. Shao, L.H. You, H.Y. Shan, Bound on the base of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 2–3, 285–300], Shao, You and Shan extended the concept of the base from powerful sign pattern matrices to nonpowerful (generalized) sign pattern matrices. It is well known that the properties of the power sequences of different classes of sign pattern matrices may be very different. In this paper, we consider the base set of the primitive nonpowerful square sign pattern matrices of order n with exactly d (with d ≥ 1 ) nonzero diagonal entries. The base set is shown to be { 2 , 3 , … , 3 n − d − 1 } . The extremal sign pattern matrices with both the least number n + d nonzero entries and the maximum base 3 n − d − 1 are characterized.