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

Abstract

The p-spectral radius of a uniform hypergraph G of order n is defined for every real number p ≥ 1 as λ (p) (G) = maxIt 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,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,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.

Computing the p-Spectral Radii of Uniform Hypergraphs with Applications

“…. , i r } is an edge of G and s(e) is the weight of e. Definition 2.2 (p-spectral radius [32,29]). When p ≥ 1, the p-spectral radius of G, denoted by λ (p) (G), is defined as…”

“…When p is even and equals the order of this hypergraph, the p-spectral radius becomes the largest Heigenvalue of the adjacency tensor of G. Therefore, the p-spectral radius is connected with the (adjacency) spectral radius of a hypergraph [27,39,42]. Additionally, Kang et al provided solutions to several pspectral radius related extremal problems in [29]. Nikiforov in [50] did a comprehensive study and obtained many theoretical conclusions about p-spectral radius.…”

Extremal Problems for the $p$-Spectral Radius of Graphs

“…All required facts about the p-spectral radius of graphs are given below. Additional reference material can be found in [9], [19], and [20].…”

“…It should be noted that extremal problems for λ (p) of hypergraphs have been studied in [9], [10], [19], and [20], but 2-graphs are better understood, so it is worthwhile to delve into deeper extremal theory. Another line has been investigated in [18], where the emphasis is on hereditary properties.…”

Ordering of some uniform supertrees with larger spectral radii

“…They also generalized some basic spectral results from graphs to hypergraphs. The (adjacency) spectrum of uniform hypergraphs were further studied in [9,4,8,17,15].…”