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Extremal spectral radius of nonregular graphs with a fixed odd maximum degree
Abstract
For integers n≥3 and 2≤Δ≤ n−1, let λ1(n,Δ) be the maximum adjacency spectral radius among all connected nonregular graphs of order n and maximum degree Δ. Liu conjectured that for each fixed integer Δ≥3, limn→∞n2(Δ−λ1(n,Δ))= (Δ−1)π2/4,&if $Δ$ is odd; (Δ−2)π2/2,&if $Δ$ is even. He proved the case Δ=3 and Δ=4. We prove the conjecture when Δ≥5 is odd.
Related
- Asymptotic maximum spectral radius of nonregular graphs
- Nonregular graphs of odd maximum degree with maximum spectral radius
- Degree sequences of spectrally extremal connected nonregular graphs with prescribed maximum degree: the dense case
- Extremal Graphs with Prescribed Parameters for the Spectral Radius of Weighted Adjacency Matrices with Property P∗
- The maximum spectral radius of connected graphs with prescribed order and size
BibTeX
@misc{huang2026extremal,
title = {{Extremal spectral radius of nonregular graphs with a fixed odd maximum degree}},
author = {Zejun Huang and Chenxi Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26044},
eprint = {2609.26044},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26044},
}
RIS
TY - UNPB
TI - Extremal spectral radius of nonregular graphs with a fixed odd maximum degree
AU - Zejun Huang
AU - Chenxi Yang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26044
UR - https://arxiv.org/abs/2609.26044
AB - For integers n≥3 and 2≤Δ≤ n−1, let λ_1(n,Δ) be the maximum adjacency spectral radius among all connected nonregular graphs of order n and maximum degree Δ. Liu conjectured that for each fixed integer Δ≥3, lim_n→∞n^2(Δ−λ_1(n,Δ))= (Δ−1)π^2/4,&if $Δ$ is odd; (Δ−2)π^2/2,&if $Δ$ is even. He proved the case Δ=3 and Δ=4. We prove the conjecture when Δ≥5 is odd.
ER -