Algebraic connectivity of the second power of a graph
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866929405080633344 |
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| author | Afshari, B. |
| author_facet | Afshari, B. |
| contents | Denote the Laplacian of a graph $G$ by $L(G)$ and its second smallest Laplacian eigenvalue by $λ_2(G)$. If $G$ is a graph on $n\ge 2$ vertices, then it is shown that the second smallest eigenvalue of $L(G) + \frac{1}{n} L(\overline{G^2})$ is at least 1, where $\overline{G^2}$ is the complement of the second power of $ G $. As a corollary of this result, it is shown that \begin{itemize}
\item $ n \, λ_2(G) \ge λ_2(G^2), $
\item $ λ_2(G) \ge 1-\frac{|D_G|}{n}, $
\item $ λ_2(G) + λ_2(\Gb) \ge 1, $ \end{itemize} where $|D_G|$ is the number of vertices of eccentricity at least 3 in $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_04568 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Algebraic connectivity of the second power of a graph Afshari, B. Combinatorics 05C50, 15A18 Denote the Laplacian of a graph $G$ by $L(G)$ and its second smallest Laplacian eigenvalue by $λ_2(G)$. If $G$ is a graph on $n\ge 2$ vertices, then it is shown that the second smallest eigenvalue of $L(G) + \frac{1}{n} L(\overline{G^2})$ is at least 1, where $\overline{G^2}$ is the complement of the second power of $ G $. As a corollary of this result, it is shown that \begin{itemize} \item $ n \, λ_2(G) \ge λ_2(G^2), $ \item $ λ_2(G) \ge 1-\frac{|D_G|}{n}, $ \item $ λ_2(G) + λ_2(\Gb) \ge 1, $ \end{itemize} where $|D_G|$ is the number of vertices of eccentricity at least 3 in $G$. |
| title | Algebraic connectivity of the second power of a graph |
| topic | Combinatorics 05C50, 15A18 |
| url | https://arxiv.org/abs/2109.04568 |