Primitive non-powerful sign pattern matrices with base 2

Abstract

A sign pattern matrix M with zero trace is primitive non-powerful if for some positive integer k, M k = J #. The base l(M) of the primitive non-powerful matrix M is the smallest integer k. By considering the signed digraph S whose adjacent matrix is the primitive non-powerful matrix M, we will show that if l(M) = 2, the minimum number of non-zero entries of M is 5n − 8 or 5n − 7 depending on whether n is even or odd.