Vol. 14, No. 4, 2021

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On the monotonicity of the number of positive entries in nonnegative five-element matrix powers

Matthew Rodriguez Galvan, Vadim Ponomarenko and John Salvadore Sabio

Vol. 14 (2021), No. 4, 703–721
Abstract

Let A be an m × m square matrix with nonnegative entries and let F(A) denote the number of positive entries in A. We consider the adjacency matrix A with a corresponding digraph with m vertices. F(A) corresponds to the number of directed edges in the corresponding digraph. We consider conditions on A to make the sequence {F(An)}n=1 monotonic. Monotonicity is known for F(A) 4 (except for three nonmonotonic cases) and F(A) m2 2m + 2; we extend this to F(A) = 5.

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Keywords
nonnegative matrix, power, monotonicity, directed graph, adjacency matrix
Mathematical Subject Classification
Primary: 05C20, 15B34, 15B48
Milestones
Received: 12 March 2021
Revised: 10 April 2021
Accepted: 28 April 2021
Published: 23 October 2021

Communicated by Ronald Gould
Authors
Matthew Rodriguez Galvan
Department of Mathematics and Statistics
San Diego State University
San Diego, CA
United States
Vadim Ponomarenko
Department of Mathematics and Statistics
San Diego State University
San Diego, CA
United States
John Salvadore Sabio
Department of Computer Science
San Diego State University
San Diego, CA
United States