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Primitive non-powerful sign pattern matrices with base 2. (English) Zbl 1223.05104

Summary: 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.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C65 Hypergraphs
15A09 Theory of matrix inversion and generalized inverses
15B48 Positive matrices and their generalizations; cones of matrices
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References:

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