An improved spectral lower bound of treewidth
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909167828074496 |
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| author | Gima, Tatsuya Hanaka, Tesshu Noro, Kohei Ono, Hirotaka Otachi, Yota |
| author_facet | Gima, Tatsuya Hanaka, Tesshu Noro, Kohei Ono, Hirotaka Otachi, Yota |
| contents | We show that for every $n$-vertex graph with at least one edge, its treewidth is greater than or equal to $n λ_{2} / (Δ+ λ_{2}) - 1$, where $Δ$ and $λ_{2}$ are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [Inf. Process. Lett., 2003] and the subsequent one by the authors of the present paper [IEICE Trans. Inf. Syst., 2024]. The new lower bound is almost tight in the sense that there is an infinite family of graphs such that the lower bound is only $1$ less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08520 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An improved spectral lower bound of treewidth Gima, Tatsuya Hanaka, Tesshu Noro, Kohei Ono, Hirotaka Otachi, Yota Combinatorics Data Structures and Algorithms We show that for every $n$-vertex graph with at least one edge, its treewidth is greater than or equal to $n λ_{2} / (Δ+ λ_{2}) - 1$, where $Δ$ and $λ_{2}$ are the maximum degree and the second smallest Laplacian eigenvalue of the graph, respectively. This lower bound improves the one by Chandran and Subramanian [Inf. Process. Lett., 2003] and the subsequent one by the authors of the present paper [IEICE Trans. Inf. Syst., 2024]. The new lower bound is almost tight in the sense that there is an infinite family of graphs such that the lower bound is only $1$ less than the treewidth for each graph in the family. Additionally, using similar techniques, we also present a lower bound of treewidth in terms of the largest and the second smallest Laplacian eigenvalues. |
| title | An improved spectral lower bound of treewidth |
| topic | Combinatorics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2404.08520 |