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

The p-spectral radius of a uniform hypergraph G of order n is de…ned for every real number p 1 as (p) (G) = max jx1j + + jxnj = 1 r! X fi1;:::;irg2E(G) xi1 xir : It generalizes several hypergraph parameters, including the Lagrangian, the spectral radius, and the number of edges. The 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; (p) (G) < (p) (T r k (n)) ; unless G = T r k (n) ; where T r k (n) is the complete k-partite r-graph of order n; with parts of size bn=kc or dn=ke. (II) Let k 2; and let G be a k-chromatic 3-graph of order n: For every p 1; (p) (G) < (p) Qk (n) ; unless G = Qk (n) ; where Q 3 k (n) is a complete k-chromatic 3-graph of order n; with classes of size bn=kc or dn=ke. The latter statement generalizes a result of Mubayi and Talbot.