Bases of Primitive Nonpowerful Zero-Symmetric Sign Pattern Matrices Without Nonzero Diagonal Entry

Abstract

It is well known that the properties about the power sequences of different classes of sign pattern matrices may be very different. In this paper, we consider the base of primitive nonpowerful zero-symmetric square sign pattern matrices without nonzero diagonal entry. The base set is shown to be \(\{2, 3, \ldots, 2n – 1\}\); the extremal sign pattern matrices with base \(2n – 1\) are characterized. As well, for the sign patterns with order \(3\), the sign patterns with bases \(3\), \(4\), \(5\) are characterized, respectively.