ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.05
https://doi.org/10.26493/2590-9770.1280.4da
(Also available at http://adam-journal.eu)
An alternate proof of the monotonicity of the
number of positive entries in nonnegative matrix
powers*
Slobodan Filipovski†
FAMNIT, University of Primorska, Glagoljaška 8, Koper, Slovenia
Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia
Received 6 December 2018, accepted 19 July 2020, published online 23. January 2021
Abstract
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) ≤ 3 or f(A) ≥ n2 − 2n + 2, then the
sequence (f(Ak))∞k=1 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.
Keywords: Digraphs, walks, monotonicity, adjacency matrix.
Math. Subj. Class.: 05C20, 05C81, 15B48
*We would like to thank the anonymous referee for helpful comments which have improved the paper.
†Supported by VEGA 1/0423/20.
E-mail address: slobodan.filipovski@famnit.upr.si (Slobodan Filipovski)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 4 (2021) #P1.05
https://doi.org/10.26493/2590-9770.1280.4da
(Dostopno tudi na http://adam-journal.eu)
Alternativni dokaz monotonosti števila
pozitivnih in negativnih matričnih potenc*
Slobodan Filipovski†
FAMNIT, University of Primorska, Glagoljaška 8, Koper, Slovenia
Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia
Prejeto 6. decembra 2018, sprejeto 19. julija 2020, objavljeno na spletu 23. januarja 2021
Povzetek
Naj bo A nenegativna realna matrika reda n in naj f(A) označuje število pozitivnih
elementov v A. Leta 2018 je Xie dokazal, da če je f(A) ≤ 3 ali f(A) ≥ n2−2n+2, potem
je zaporedje (f(Ak))∞k=1 monotono za pozitivna cela števila k. V tem kratkem članku
podamo alternativen dokaz tega rezultata, do katerega pridemo s preštevanjem sprehodov
v digrafu reda n.
Ključne besede: Digrafi, sprehodi, monotonost, matrika sosednosti.
Math. Subj. Class.: 05C20, 05C81, 15B48
*Rad bi se zahvalil anonimnemu recenzentu za koristne pripombe, ki so izboljšale članek.
†Podprt s strani VEGA 1/0423/20.
E-poštni naslov: slobodan.filipovski@famnit.upr.si (Slobodan Filipovski)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/