PreprintCC BY-NC-SA 4.0arXiv
On the Spectral Region of 4-Cycle Stochastic Matrices
Abstract
We study the spectrum of 4-cycle row-stochastic matrices. For real eigenvalues the spectral region is [-1,1]. For nonreal eigenvalues a+ib we derive necessary conditions in terms of the real and imaginary parts, including the inequality a+|b| <= 1 and the condition (b^2+a^2+a)^2+2a^2-b^2 >= 0. We also prove conversely that every point in the corresponding interior region occurs as an eigenvalue of a 4-cycle matrix. The proof is organized through a reformulation of the characteristic equation, an argument parametrization, a convex-analytic criterion, and explicit boundary constructions. Hence, the spectral region for the 4-cycle row-stochastic matrices is exactly and explicitly determined.
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BibTeX
@misc{vagenende2026spectral,
title = {{On the Spectral Region of 4-Cycle Stochastic Matrices}},
author = {Brando Vagenende and Brecht F. Verbeken and Andres Algaba and Marie-Anne Guerry},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2605.06743},
eprint = {2605.06743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.06743},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On the Spectral Region of 4-Cycle Stochastic Matrices
AU - Brando Vagenende
AU - Brecht F. Verbeken
AU - Andres Algaba
AU - Marie-Anne Guerry
PY - 2026
DA - 2026/06/09
PB - arXiv
DO - 10.48550/arxiv.2605.06743
UR - https://arxiv.org/abs/2605.06743
AB - We study the spectrum of 4-cycle row-stochastic matrices. For real eigenvalues the spectral region is [-1,1]. For nonreal eigenvalues a+ib we derive necessary conditions in terms of the real and imaginary parts, including the inequality a+|b| <= 1 and the condition (b^2+a^2+a)^2+2a^2-b^2 >= 0. We also prove conversely that every point in the corresponding interior region occurs as an eigenvalue of a 4-cycle matrix. The proof is organized through a reformulation of the characteristic equation, an argument parametrization, a convex-analytic criterion, and explicit boundary constructions. Hence, the spectral region for the 4-cycle row-stochastic matrices is exactly and explicitly determined.
ER -