2024/01/18 by Cooper, Joshua, Desai, Dheer Noal, Sahay, Anurag
#05C50 (Secondary) #05C65 (Primary) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2401.10344
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.