Nonregular graphs with a given maximum degree attaining maximum spectral radius
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909405456367616 |
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| author | Huang, Zejun Liu, Jiahui Yang, Chenxi |
| author_facet | Huang, Zejun Liu, Jiahui Yang, Chenxi |
| contents | Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $Δ$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(Δ,\ldots,Δ,δ)$ with $δ<Δ$. For $Δ=3$ and $Δ=4$, Liu (2024) confirmed this conjecture by characterizing the structure of such graphs. Liu also proposed a modified version of the conjecture for fixed $Δ$ and sufficiently large $n$, stating that the above $δ=Δ-1$ if $Δ$ and $n$ are both odd, $δ=1$ if $Δ$ is odd and $n$ is even, and $δ=Δ-2$ if $Δ$ is even. For the cases where $Δ=n-2$ with $n\ge 5$, and $Δ=n-3$ with $n\ge 59$, we fully characterize the structure of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17371 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonregular graphs with a given maximum degree attaining maximum spectral radius Huang, Zejun Liu, Jiahui Yang, Chenxi Combinatorics Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $Δ$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(Δ,\ldots,Δ,δ)$ with $δ<Δ$. For $Δ=3$ and $Δ=4$, Liu (2024) confirmed this conjecture by characterizing the structure of such graphs. Liu also proposed a modified version of the conjecture for fixed $Δ$ and sufficiently large $n$, stating that the above $δ=Δ-1$ if $Δ$ and $n$ are both odd, $δ=1$ if $Δ$ is odd and $n$ is even, and $δ=Δ-2$ if $Δ$ is even. For the cases where $Δ=n-2$ with $n\ge 5$, and $Δ=n-3$ with $n\ge 59$, we fully characterize the structure of $G$. |
| title | Nonregular graphs with a given maximum degree attaining maximum spectral radius |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.17371 |