Line graphs with the largest eigenvalue multiplicity
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910758247333888 |
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| author | Zhen, Wenhao Wong, Dein Xu, Songnian |
| author_facet | Zhen, Wenhao Wong, Dein Xu, Songnian |
| contents | For a connected graph $G$, we denote by $L(G)$, $m_{G}(λ)$, $c(G)$ and $p(G)$ the line graph of $G$, the eigenvalue multiplicity of $λ$ in $G$, the cyclomatic number and the number of pendant vertices in $G$, respectively. In 2023, Yang et al. \cite{WL LT} proved that $m_{L(T)}(λ)\leq p(T)-1$ for any tree $T$ with $p(T)\geq 3$, and characterized all trees $T$ with $m_{L(T)}(λ) = p(T)-1$. In 2024, Chang et al. \cite{-1 LG} proved that, if $G$ is not a cycle, then $m_{L(G)}(λ)\leq 2c(G)+p(G)-1$, and characterized all graphs $G$ with $m_{L(G)}(-1) = 2c(G)+p(G)-1$. The remaining ploblem is to characterize all graphs $G$ with $m_{L(G)}(λ)= 2c(G)+p(G)-1$ for an arbitrary eigenvalue $λ$ of $L(G)$. In this paper, we give this problem a complete solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14835 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Line graphs with the largest eigenvalue multiplicity Zhen, Wenhao Wong, Dein Xu, Songnian Spectral Theory 05C50 For a connected graph $G$, we denote by $L(G)$, $m_{G}(λ)$, $c(G)$ and $p(G)$ the line graph of $G$, the eigenvalue multiplicity of $λ$ in $G$, the cyclomatic number and the number of pendant vertices in $G$, respectively. In 2023, Yang et al. \cite{WL LT} proved that $m_{L(T)}(λ)\leq p(T)-1$ for any tree $T$ with $p(T)\geq 3$, and characterized all trees $T$ with $m_{L(T)}(λ) = p(T)-1$. In 2024, Chang et al. \cite{-1 LG} proved that, if $G$ is not a cycle, then $m_{L(G)}(λ)\leq 2c(G)+p(G)-1$, and characterized all graphs $G$ with $m_{L(G)}(-1) = 2c(G)+p(G)-1$. The remaining ploblem is to characterize all graphs $G$ with $m_{L(G)}(λ)= 2c(G)+p(G)-1$ for an arbitrary eigenvalue $λ$ of $L(G)$. In this paper, we give this problem a complete solution. |
| title | Line graphs with the largest eigenvalue multiplicity |
| topic | Spectral Theory 05C50 |
| url | https://arxiv.org/abs/2411.14835 |