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 >= 1 as [GRAPHICS] 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, lambda((p)) (G) < lambda((p)) (T-k(gamma) (n)), unless G = T-k(gamma) (n), where T-k(gamma) (n) is the complete k-partite gamma-graph of order n, with parts of size left perpendicularn/kright perpendicular or inverted left perpendicularn/kinverted right perpendicular. (II) Let k >= 2, and let G be a k-chromatic 3-graph of order n. For every p >= 1, lambda((p)) (G) < lambda((p)) (Q(k)(3) (n)), unless G = Q(k)(3) (n), where Q(k)(3) (n) is a complete k-chromatic 3-graph of order n, with classes of size left perpendicularn/kright perpendicular or inverted left perpendicularn/kinverted right perpendicular The latter statement generalizes a result of Mubayi and Talbot. (C) 2015 Elsevier Inc. All rights reserved.