The signless Laplacian spectral Turán problems for hypergraphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917204847493120 |
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| author | Lu, Yongchun Wu, Jiadong Kang, Liying |
| author_facet | Lu, Yongchun Wu, Jiadong Kang, Liying |
| contents | Let $\mathcal{H}=(V, E)$ be an $r$-uniform hypergraph on $n$ vertices. The signless Laplacian spectral radius of $\mathcal{H}$ is defined as the maximum modulus of the eigenvalues of the tensor $\mathcal{Q}(\mathcal{H})=\mathcal{D}(\mathcal{H})+\mathcal{A}(\mathcal{H})$, where $\mathcal{D}(\mathcal{H})$ and $\mathcal{A}(\mathcal{H})$ are the degree diagonal tensor and the adjacency tensor of $\mathcal{H}$, respectively. In this paper, we establish a general theorem that extends the spectral Turán result of Keevash, Lenz and Mubayi [SIAM J. Discrete Math., 28 (4) (2014)]
to the setting of signless Laplacian spectral Turán problems. We prove that if a family $\mathcal{F}$ of $r$-uniform hypergraphs is degree-stable with respect to a family $\mathcal{H}_n$ of $r$-uniform hypergraphs and its extremal constructions satisfy certain natural assumptions, then the signless Laplacian spectral Turán problem for $\mathcal{F}$ can be effectively reduced to the corresponding problem restricted to the family $\mathcal{H}_n$. As a concrete application, we completely determine the extremal hypergraph that maximizes the signless Laplacian spectral radius among all Fano plane-free $3$-uniform hypergraphs, showing that the unique extremal hypergraph is the balanced complete bipartite $3$-uniform hypergraph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08595 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The signless Laplacian spectral Turán problems for hypergraphs Lu, Yongchun Wu, Jiadong Kang, Liying Combinatorics 05C50, 05C35 Let $\mathcal{H}=(V, E)$ be an $r$-uniform hypergraph on $n$ vertices. The signless Laplacian spectral radius of $\mathcal{H}$ is defined as the maximum modulus of the eigenvalues of the tensor $\mathcal{Q}(\mathcal{H})=\mathcal{D}(\mathcal{H})+\mathcal{A}(\mathcal{H})$, where $\mathcal{D}(\mathcal{H})$ and $\mathcal{A}(\mathcal{H})$ are the degree diagonal tensor and the adjacency tensor of $\mathcal{H}$, respectively. In this paper, we establish a general theorem that extends the spectral Turán result of Keevash, Lenz and Mubayi [SIAM J. Discrete Math., 28 (4) (2014)] to the setting of signless Laplacian spectral Turán problems. We prove that if a family $\mathcal{F}$ of $r$-uniform hypergraphs is degree-stable with respect to a family $\mathcal{H}_n$ of $r$-uniform hypergraphs and its extremal constructions satisfy certain natural assumptions, then the signless Laplacian spectral Turán problem for $\mathcal{F}$ can be effectively reduced to the corresponding problem restricted to the family $\mathcal{H}_n$. As a concrete application, we completely determine the extremal hypergraph that maximizes the signless Laplacian spectral radius among all Fano plane-free $3$-uniform hypergraphs, showing that the unique extremal hypergraph is the balanced complete bipartite $3$-uniform hypergraph. |
| title | The signless Laplacian spectral Turán problems for hypergraphs |
| topic | Combinatorics 05C50, 05C35 |
| url | https://arxiv.org/abs/2601.08595 |