Some Turán-type results for the signless Laplacian spectral radius
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910060373868544 |
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| author | Zheng, Jian Li, Yongtao Fan, Yi-Zheng |
| author_facet | Zheng, Jian Li, Yongtao Fan, Yi-Zheng |
| contents | Half a century ago, Bollobás and Erdős [Bull. London Math. Soc. 5 (1973)] proved that every $n$-vertex graph $G$ with $e(G)\ge (1- \frac{1}{k} + \varepsilon )\frac{n^2}{2}$ edges contains a blowup $K_{k+1}[t]$ with $t=Ω_{k,\varepsilon}(\log n)$. A well-known theorem of Nikiforov [Combin. Probab. Comput. 18 (3) (2009)] asserts that if $G$ is an $n$-vertex graph with adjacency spectral radius $λ(G)\ge (1- \frac{1}{k} + \varepsilon)n$, then $G$ contains a blowup $K_{k+1}[t]$ with $t=Ω_{k,\varepsilon}(\log n)$. This gives a spectral version of the Bollobás--Erdős theorem. In this paper, we systematically explore variants of Nikiforov's result in terms of the signless Laplacian spectral radius, extending the supersaturation, blowup of cliques and the stability results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02263 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some Turán-type results for the signless Laplacian spectral radius Zheng, Jian Li, Yongtao Fan, Yi-Zheng Combinatorics 15A42, 05C35, 05C50 Half a century ago, Bollobás and Erdős [Bull. London Math. Soc. 5 (1973)] proved that every $n$-vertex graph $G$ with $e(G)\ge (1- \frac{1}{k} + \varepsilon )\frac{n^2}{2}$ edges contains a blowup $K_{k+1}[t]$ with $t=Ω_{k,\varepsilon}(\log n)$. A well-known theorem of Nikiforov [Combin. Probab. Comput. 18 (3) (2009)] asserts that if $G$ is an $n$-vertex graph with adjacency spectral radius $λ(G)\ge (1- \frac{1}{k} + \varepsilon)n$, then $G$ contains a blowup $K_{k+1}[t]$ with $t=Ω_{k,\varepsilon}(\log n)$. This gives a spectral version of the Bollobás--Erdős theorem. In this paper, we systematically explore variants of Nikiforov's result in terms of the signless Laplacian spectral radius, extending the supersaturation, blowup of cliques and the stability results. |
| title | Some Turán-type results for the signless Laplacian spectral radius |
| topic | Combinatorics 15A42, 05C35, 05C50 |
| url | https://arxiv.org/abs/2507.02263 |