Combinatorics and Discrete Probability Seminar
Anurag Sahay
Purdue University
Principal eigenvectors and principal ratios in hypergraph Turán problems
Abstract: For a general class of (uniform) hypergraph Turán problems, we will discuss the principal eigenvector for the 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 will also comment on the sharpness of our result -- it is likely sharp for the Turán tetrahedron problem. In the course of this latter discussion, we shall establish a lower bound on the p-spectral radius of an arbitrary uniform hypergraph in terms of its degrees. 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. If time permits, we will raise a conjecture about these issues and provide some preliminary evidence for the conjecture.
Joint work with Joshua Cooper and Dheer Noal Desai.
Monday February 12, 2024 at 3:00 PM in 1227 SEO