Principal eigenvectors and principal ratios in hypergraph Turán problems
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917570494332928 |
|---|---|
| author | Cooper, Joshua Desai, Dheer Noal Sahay, Anurag |
| author_facet | Cooper, Joshua Desai, Dheer Noal Sahay, Anurag |
| contents | For a general class of hypergraph Turán problems with uniformity $r$, we investigate the principal eigenvector for the $p$-spectral radius (in the sense of Keevash--Lenz--Mubayi and Nikiforov) for the extremal graphs, showing in a strong sense that these eigenvectors have close to equal weight on each vertex (equivalently, showing that the principal ratio is close to $1$). We investigate the sharpness of our result; it is likely sharp for the Turán tetrahedron problem.
In the course of this latter discussion, we establish a lower bound on the $p$-spectral radius of an arbitrary $r$-graph in terms of the degrees of the graph. This builds on earlier work of Cardoso--Trevisan, Li--Zhou--Bu, Cioabă--Gregory, and Zhang.
The case $1 < p < r$ of our results leads to some subtleties connected to Nikiforov's notion of $k$-tightness, arising from the Perron-Frobenius theory for the $p$-spectral radius. We raise a conjecture about these issues, and provide some preliminary evidence for our conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Principal eigenvectors and principal ratios in hypergraph Turán problems Cooper, Joshua Desai, Dheer Noal Sahay, Anurag Combinatorics 05C65 (Primary), 05C50 (Secondary) For a general class of hypergraph Turán problems with uniformity $r$, we investigate the principal eigenvector for the $p$-spectral radius (in the sense of Keevash--Lenz--Mubayi and Nikiforov) for the extremal graphs, showing in a strong sense that these eigenvectors have close to equal weight on each vertex (equivalently, showing that the principal ratio is close to $1$). We investigate the sharpness of our result; it is likely sharp for the Turán tetrahedron problem. In the course of this latter discussion, we establish a lower bound on the $p$-spectral radius of an arbitrary $r$-graph in terms of the degrees of the graph. This builds on earlier work of Cardoso--Trevisan, Li--Zhou--Bu, Cioabă--Gregory, and Zhang. The case $1 < p < r$ of our results leads to some subtleties connected to Nikiforov's notion of $k$-tightness, arising from the Perron-Frobenius theory for the $p$-spectral radius. We raise a conjecture about these issues, and provide some preliminary evidence for our conjecture. |
| title | Principal eigenvectors and principal ratios in hypergraph Turán problems |
| topic | Combinatorics 05C65 (Primary), 05C50 (Secondary) |
| url | https://arxiv.org/abs/2401.10344 |