On the period and base of a sign pattern matrix
Let \(Q_ n\) denote the set of all \(n \times n\) sign pattern matrices (each entry is \(+\), \(-\), or \(0)\). A pattern \(A \in Q_ n\) is powerful provided all powers \(A^ 1\), \(A^ 2\), \(A^ 3, \dots\) are unambiguously defined (no two walks \(a_{i_ 0 i_ 1} a_{i_ 1 i_ 2} \dots a_{i_{k - 1} i_ k}\) of length \(k\) from \(i_ 0\) to \(i_ k\) have opposite sign). For a powerful pattern \(A\), if \(A^ l = A^{l + p}\) with \(l\) and \(p\) minimal, then \(l\) is called the base of \(A\) and \(p\) the period of \(A\). Irreducible powerful sign pattern matrices are characterized, and the period and base of powerful patterns are investigated. The paper contains a good review of the basic concepts and several previously known results. It concludes by producing families of examples of powerful patterns by using directed graphs known as negative \(C_ r\)- cockades studied in \textit{J. R. Lundgren} and \textit{J. S. Maybee} [J. Comb. Inf. Syst. Sci., 16, No. 1, 29-36 (1991; Zbl 0764.05032)].
scientific article; zbMATH DE number 720471
scientific article