Proof of a conjecture on the $ \epsilon $-spectral radius of trees

Jianping Li; Leshi Qiu; Jianbin Zhang; Jianping Li; Leshi Qiu; Jianbin Zhang

doi:10.3934/math.2023217

AIMS Mathematics

2023, Volume 8, Issue 2: 4363-4371. doi: 10.3934/math.2023217

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Research article

Proof of a conjecture on the $ \epsilon $-spectral radius of trees

1.
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510090, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received: 06 October 2022 Revised: 13 November 2022 Accepted: 16 November 2022 Published: 05 December 2022
MSC : 05C50

The $ \epsilon $-spectral radius of a connected graph is the largest eigenvalue of its eccentricity matrix. In this paper, we identify the unique $ n $-vertex tree with diameter $ 4 $ and matching number $ 5 $ that minimizes the $ \epsilon $-spectral radius, and thus resolve a conjecture proposed in [W. Wei, S. Li, L. Zhang, Characterizing the extremal graphs with respect to the eccentricity spectral radius, and beyond, Discrete Math. 345 (2022) 112686].
- eccentricity matrix,
- spectral radius,
- matching number
Citation: Jianping Li, Leshi Qiu, Jianbin Zhang. Proof of a conjecture on the $ \epsilon $-spectral radius of trees[J]. AIMS Mathematics, 2023, 8(2): 4363-4371. doi: 10.3934/math.2023217

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Abstract

The $ \epsilon $-spectral radius of a connected graph is the largest eigenvalue of its eccentricity matrix. In this paper, we identify the unique $ n $-vertex tree with diameter $ 4 $ and matching number $ 5 $ that minimizes the $ \epsilon $-spectral radius, and thus resolve a conjecture proposed in [W. Wei, S. Li, L. Zhang, Characterizing the extremal graphs with respect to the eccentricity spectral radius, and beyond, Discrete Math. 345 (2022) 112686].

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