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

Rodriguez Galvan, Matthew; Ponomarenko, Vadim; Salvadore Sabio, John

doi:10.2140/involve.2021.14.703

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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

DOI: 10.2140/involve.2021.14.703

Abstract

Let $A$ be an $m \times m$ 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 ${F (A^{n})}_{n = 1}^{\infty}$ monotonic. Monotonicity is known for $F (A) \leq 4$ (except for three nonmonotonic cases) and $F (A) \geq m^{2} - 2 m + 2$ ; we extend this to $F (A) = 5$ .

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

Vol. 14, No. 4, 2021