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An alternate proof of the monotonicity of the number of positive entries in nonnegative matrix powers. (English) Zbl 1454.05044

Summary: Let \(A\) be a nonnegative real matrix of order \(n\) and \(f(A)\) denote the number of positive entries in \(A\). In 2018, Xie proved that if \(f(A) \leq 3\) or \(f(A) \geq n^2 - 2n + 2\), then the sequence \((f(A^k))_{k = 1}^{\infty}\) is monotone for positive integers \(k\). In this note we give an alternate proof of this result by counting walks in a digraph of order \(n\).

MSC:

05C20 Directed graphs (digraphs), tournaments
05C81 Random walks on graphs
15B48 Positive matrices and their generalizations; cones of matrices
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References:

[1] R. B. Bapat and T. E. S. Raghavan,Nonnegative matrices and applications, volume 64 ofEncyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1997, doi:10.1017/CBO9780511529979. · Zbl 0879.15015
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[5] Q. Xie, Monotonicity of the number of positive entries in nonnegative matrix powers,J. Inequal. Appl.(2018), Paper No. 255, 5, doi:10.1186/s13660-018-1833-5
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