[Paper Review] The $α$-normal labeling method for computing the $p$-spectral radii of uniform hypergraphs

Lele Liu, Linyuan Lü|arXiv (Cornell University)|Mar 16, 2018

graph theory and CDMA systems8 references3 citations

TL;DR

This paper extends the α-normal labeling method—previously used for spectral radii (p = r) in uniform hypergraphs—to compute p-spectral radii for p ≠ r. By introducing a generalized α-normal labeling framework with weighted incidence matrices, the authors derive exact formulas for λ^(p)(G) in terms of α and hypergraph structure, enabling new monotonicity, convexity, and extremal results for p-spectral radii across various hypergraph classes.

ABSTRACT

Let $G$ be an $r$-uniform hypergraph of order $n$. For each $p\geq 1$, the $p$-spectral radius $λ^{(p)}(G)$ is defined as \[ λ^{(p)}(G):=\max_{|x_1|^p+\cdots+|x_n|^p=1} r\sum_{\{i_1,\ldots,i_r\}\in E(G)}x_{i_1}\cdots x_{i_r}. \] The $p$-spectral radius was introduced by Keevash-Lenz-Mubayi, and subsequently studied by Nikiforov in 2014. The most extensively studied case is when $p=r$, and $λ^{(r)}(G)$ is called the spectral radius of $G$. The $α$-normal labeling method, which was introduced by Lu and Man in 2014, is effective method for computing the spectral radii of uniform hypergraphs. It labels each corner of an edge by a positive number so that the sum of the corner labels at any vertex is $1$ while the product of all corner labels at any edge is $α$. Since then, this method has been used by many researchers in studying $λ^{(r)}(G)$. In this paper, we extend Lu and Man's $α$-normal labeling method to the $p$-spectral radii of uniform hypergraphs for $p e r$; and find some applications.

Motivation & Objective

• To generalize the α-normal labeling method from spectral radii (p = r) to p-spectral radii for p ≠ r in r-uniform hypergraphs.
• To establish a theoretical framework linking p-spectral radii to weighted incidence matrices and α-normal labelings.
• To derive new analytical results—such as monotonicity, convexity, and tight upper bounds—on p-spectral radii using the extended method.
• To characterize the p-spectral radius of hypergraph products, extensions, and induced subhypergraphs.
• To provide a theoretical tool for analyzing p-spectral radii analogous to the established method for p = r.

Proposed method

• Introduce a generalized α-normal labeling for p > r, where vertex sums of edge corner labels equal 1 and edge product of labels equals α = λ^(p)(G)^{-p/r} · r^{p-r} / (λ^(p)(G[S]))^{p-r} for induced subhypergraphs.
• Define a consistent labeling system using weighted incidence matrices B(v,e) such that the ratio of labels along any cycle equals 1, ensuring well-defined vertex weights.
• Construct a vector x ∈ ℝⁿ₊ with x_v = (w(e)/(rB(v,e)))^{1/p} for v ∈ e, ensuring ||x||_p = 1 and enabling evaluation of P_G(x) = λ^(p)(G).
• Prove that λ^(p)(G) = r^{1−r/p} · max_i {α_i^{−1/p}} for hypergraphs that are consistently α_i-normal on induced subhypergraphs G[S_i].
• For p < r, define α-subnormal labeling with inequalities: ∑_e∋v B(v,e) ≤ 1 and m^{r−p} ∏_{v∈e} B(v,e) ≥ α.
• Derive upper bounds for λ^(p)(G) using α-subnormal labeling: λ^(p)(G) ≤ (r/m)^{1−r/p} / α^{1/p} for p < r.

Experimental results

Research questions

• RQ1 How can the α-normal labeling method be generalized to compute p-spectral radii for p ≠ r in r-uniform hypergraphs?
• RQ2 What are the structural and analytical properties of p-spectral radii that can be derived using the extended α-normal labeling framework?
• RQ3 How does the p-spectral radius behave under hypergraph operations such as products, extensions, and induced subhypergraphs?
• RQ4 Can monotonicity and convexity properties of λ^(p)(G) be established for p ≠ r using the new method?
• RQ5 What are the tightest possible upper bounds on λ^(p)(G) in terms of degrees and hypergraph structure?

Key findings

• The p-spectral radius λ^(p)(G) is exactly characterized as λ^(p)(G) = r^{1−r/p} · max_i {α_i^{−1/p}} for hypergraphs that are consistently α_i-normal on induced subhypergraphs G[S_i].
• A new monotonicity result is established: λ^(p)(G) is monotonic in the maximum and minimum degrees of G, with explicit growth rates derived for p > r.
• For p > r, the p-spectral radius is shown to be convex in certain hypergraph parameters, with two convexity theorems (Theorem 3.4 and 3.5) proven using the α-normal labeling framework.
• A tight upper bound on λ^(p)(G) is derived using vertex degrees, valid for all r-uniform hypergraphs and p ≥ 1.
• The p-spectral radius of the hypergraph join G₁ ∗ G₂ and the Cartesian product G₁ × G₂ are computed exactly using the extended method (Theorems 3.6 and 3.7).
• The p-spectral radius of a hypergraph extension is characterized via the α-normal labeling method, with a formula provided in Theorem 3.8.