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Research Article | Open Access

Spectral radius of biased random walks on regular trees

He Song, Meng Liu( )
School of Mathematics and Statistics, Huaiyin Normal University, Huai'an 223300, China
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Abstract

For the biased random walk on d-regular trees, we obtain the spectral radius, first return probability and n-step transition probability, speed. We prove that the spectral radius depends continuously on the bias parameter and is strictly increasing with respect to it, while the speed remains strictly positive and decreases monotonically as the bias increases.

Keywords

biased random walkspectral radiuscontinuous
CLC number: Primary 05C81, 60G50, 60J10; Secondary 05C63, 05C80, 60C05

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AIMS Mathematics
Volume 11 Issue 2,
February 2026
Pages 4787-4804
DOI: 10.3934/math.2026195

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Song H, Liu M. Spectral radius of biased random walks on regular trees. AIMS Mathematics, 2026, 11(2): 4787-4804. https://doi.org/10.3934/math.2026195

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Received: 23 November 2025
Revised: 27 January 2026
Accepted: 03 February 2026
Published: 27 February 2026
©2026 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)