# The p-spectral radius of k-partite and k-chromatic uniform hypergraphs

## Abstract

The p-spectral radius of an r-uniform hypergraph G of order n is defined for every real number p≥1 asλ(p(G)=max|x1p+xnp=1r'{i1,...,ir}E(G)x i1xir. It generalizes several hypergraph parameters, including the Lagrangian, the spectral radius, and the number of edges. This paper presents solutions to several extremal problems about the p-spectral radius of k-partite and k-chromatic hypergraphs of order n. Two of the main results are: (I) Let k≥r≥2, and let G be a k-partite r-graph of order n. For every p>1,λ(p(G)<λ(p(Tkr(n)), unless G=Tkr(n), where Tkr(n) is the complete k-partite r-graph of order n, with parts of size n/k or n/k (II) Let k≥2, and let G be a k-chromatic 3-graph of order n. For every p≥1,λ(p(G)<λ(p(Qk3(n)), unless G=Qk3(n), where Qk3(n) is a complete k-chromatic 3-graph of order n, with classes of size n/k or n/k The latter statement generalizes a result of Mubayi and Talbot.

## Publication Title

Linear Algebra and Its Applications

## Recommended Citation

Kang, L., Nikiforov, V., & Yuan, X.
(2015). The p-spectral radius of k-partite and k-chromatic uniform hypergraphs.* Linear Algebra and Its Applications**, 478*, 81-107.
https://doi.org/10.1016/j.laa.2015.03.016