The maximum p-Spectral Radius of Hypergraphs with m Edges

2018/03/23 by Linyuan Lü, Lu, Linyuan
Computer Science · Mathematics · #05C35 #05C50 #05C65 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.1803.08653

openalex publication_date 2018/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

For r≥ 2 and p≥ 1, the p-spectral radius of an r-uniform hypergraph H=(V,E) on n vertices is defined to be ρ_p(H)=max_\bf x∈ ℝⁿ: ‖\bf x‖_p=1r ⋅ ∑_\i₁,i₂,…, i_r\∈ E(H) x_i1x_i2⋯ x_ir, where the maximum is taken over all \bf x∈ ℝⁿ with the p-norm equals 1. In this paper, we proved for any integer r≥ 2, and any real p≥ 1, and any r-uniform hypergraph H with m=s\choose r edges (for some real s≥ r-1), we have λ_p(H)≤ \fracrms^r/p. The equality holds if and only if s is an integer and H is the complete r-uniform hypergraph K^r_s with some possible isolated vertices added. Thus, we completely settled a conjecture of Nikiforov. In particular, we settled all the principal cases of the Frankl-Füredi's Conjecture on the Lagrangians of r-uniform hypergraphs for all r≥ 2.

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