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

The p-spectral radius of an r-uniform hypergraph G of order n   is defined for every real number p≥1p≥1 asλ(p)(G)=max|x1|p+⋯+|xn|p=1⁡r!∑{i1,…,ir}∈E(G)xi1⋯xir. 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≥2k≥r≥2, and let G be a k-partite r-graph of order n  . For every p>1p>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⌋⌊n/k⌋ or ⌈n/k⌉⌈n/k⌉.(II) Let k≥2k≥2, and let G be a k-chromatic 3-graph of order n  . For every p≥1p≥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⌋⌊n/k⌋ or ⌈n/k⌉⌈n/k⌉.The latter statement generalizes a result of Mubayi and Talbot.