A Nordhaus--Gaddum problem for the spectral gap of a graph
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929343747325952 |
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| author | Kim, Sooyeong Madras, Neal |
| author_facet | Kim, Sooyeong Madras, Neal |
| contents | Let $G$ be a graph on $n$ vertices, with complement $\overline{G}$. The spectral gap of the transition probability matrix of a random walk on $G$ is used to estimate how fast the random walk becomes stationary. We prove that the larger spectral gap of $G$ and $\overline{G}$ is $Ω(1/n)$. Moreover, if all degrees are $Ω(n)$ and $n-Ω(n)$, then the larger spectral gap of $G$ and $\overline{G}$ is $Θ(1)$. We also show that if the maximum degree is $n-O(1)$ or if $G$ is a join of two graphs, then the spectral gap of $G$ is $Ω(1/n)$. Finally, we provide a family of connected graphs with connected complements such that the larger spectral gap of $G$ and $\overline{G}$ is $O(1/n^{3/4})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15167 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Nordhaus--Gaddum problem for the spectral gap of a graph Kim, Sooyeong Madras, Neal Combinatorics Probability 60J10, 05C81 Let $G$ be a graph on $n$ vertices, with complement $\overline{G}$. The spectral gap of the transition probability matrix of a random walk on $G$ is used to estimate how fast the random walk becomes stationary. We prove that the larger spectral gap of $G$ and $\overline{G}$ is $Ω(1/n)$. Moreover, if all degrees are $Ω(n)$ and $n-Ω(n)$, then the larger spectral gap of $G$ and $\overline{G}$ is $Θ(1)$. We also show that if the maximum degree is $n-O(1)$ or if $G$ is a join of two graphs, then the spectral gap of $G$ is $Ω(1/n)$. Finally, we provide a family of connected graphs with connected complements such that the larger spectral gap of $G$ and $\overline{G}$ is $O(1/n^{3/4})$. |
| title | A Nordhaus--Gaddum problem for the spectral gap of a graph |
| topic | Combinatorics Probability 60J10, 05C81 |
| url | https://arxiv.org/abs/2404.15167 |