Bases of primitive nonpowerful sign patterns

For a square primitive nonpowerful sign pattern A, the base of A, denoted by l(A), is the least positive integer l such that every entry of Al is #. For a square sign pattern matrix A with order n, the associated digraph of A, denoted by D(A), has vertex set V={1,2,…,n} and arc set . The associated signed digraph of A, denoted by S(A), is obtained from D(A) by assigning sign of aij to arc (i,j) for all i and j. In this paper, we consider the base set of the primitive nonpowerful sign pattern matrices. For a square primitive nonpowerful sign pattern A with order n and base at least , some properties about the cycles in S(A) are obtained, and a bound on the base is given. Some sign pattern matrices with given bases are characterized and some “gaps” in the base set are shown as well.