[Paper Review] Connected Hypergraphs with Small Spectral Radius

Linyuan Lü, Shoudong Man|arXiv (Cornell University)|Feb 21, 2014

Tensor decomposition and applications8 references7 citations

TL;DR

This paper generalizes Smith's classical classification of connected graphs with spectral radius at most 2 to r-uniform hypergraphs. By introducing a novel weighted incidence matrix method to compute spectral radii, the authors identify the smallest limit point of spectral radii as ρ_r = (r−1)!⁠√[r]{4}, and fully classify all connected r-uniform hypergraphs with spectral radius ≤ ρ_r, extending Dynkin and extended Dynkin diagrams to hypergraphs for r ≥ 3.

ABSTRACT

In 1970 Smith classified all connected graphs with the spectral radius at most $2$. Here the spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Recently, the definition of spectral radius has been extended to $r$-uniform hypergraphs. In this paper, we generalize the Smith's theorem to $r$-uniform hypergraphs. We show that the smallest limit point of the spectral radii of connected $r$-uniform hypergraphs is $ρ_r=(r-1)!\sqrt[r]{4}$. We discovered a novel method for computing the spectral radius of hypergraphs, and classified all connected $r$-uniform hypergraphs with spectral radius at most $ρ_r$.

Motivation & Objective

• To extend Smith’s 1970 classification of connected graphs with spectral radius ≤ 2 to r-uniform hypergraphs.
• To identify the smallest limit point of spectral radii for connected r-uniform hypergraphs.
• To classify all connected r-uniform hypergraphs with spectral radius at most ρ_r = (r−1)!⁠√[r]{4}.
• To develop a new computational method for spectral radii in hypergraphs based on weighted incidence matrices.
• To establish hypergraph analogues of Dynkin and extended Dynkin diagrams for r ≥ 3.

Proposed method

• Introduce a spectral radius definition for r-uniform hypergraphs using the maximum of a homogeneous polynomial over the r-norm unit sphere.
• Define α-normal and α-supernormal labelings on hypergraphs to bound spectral radii via product conditions on edge labels.
• Use reduction and extension techniques to classify hypergraphs by recursively simplifying or building from base structures.
• Apply a novel weighted incidence matrix method to compute spectral radii, which generalizes naturally to r = 2 and provides an alternative proof of Smith’s theorem.
• Construct explicit α-normal and strictly α-supernormal labelings to prove spectral radius bounds for candidate hypergraphs.
• Leverage the limit behavior of r-uniform paths: lim_{n→∞} ρ(A_n^{(r)}) = ρ_r = (r−1)!⁠√[r]{4}.

Experimental results

Research questions

• RQ1 What is the smallest limit point of the spectral radii of connected r-uniform hypergraphs?
• RQ2 Which connected r-uniform hypergraphs have spectral radius at most ρ_r = (r−1)!⁠√[r]{4}?
• RQ3 Can Smith’s classification of graphs with spectral radius ≤ 2 be generalized to r-uniform hypergraphs?
• RQ4 What are the structural analogues of Dynkin and extended Dynkin diagrams in the context of r-uniform hypergraphs?
• RQ5 How can spectral radii of hypergraphs be effectively computed and bounded using labeling techniques?

Key findings

• The smallest limit point of spectral radii for connected r-uniform hypergraphs is ρ_r = (r−1)!⁠√[r]{4}, with lim_{n→∞} ρ(A_n^{(r)}) = ρ_r.
• All connected 3-uniform hypergraphs with spectral radius ≤ ρ_3 = 2⁠√[3]{4} are completely classified, including those with spectral radius exactly ρ_3.
• For r ≥ 4, all connected r-uniform hypergraphs with spectral radius < ρ_r are classified into two families: extensions of 3-uniform hypergraphs from Theorem 2 and specific 4-branch hypergraphs H^{(r)}_{1,1,1,l} for l ≤ 4.
• For r ≥ 4, all connected r-uniform hypergraphs with spectral radius exactly ρ_r are classified into two families: extensions of 3-uniform hypergraphs from Theorem 1 and the hypergraph H^{(r)}_{1,1,2,2}.
• The hypergraph H^{(4)}_{1,1,2,2} has spectral radius exactly ρ_4, and its r-uniform extension H^{(r)}_{1,1,2,2} also has spectral radius ρ_r.
• The spectral radius of H^{(4)}_{1,1,1,5} exceeds ρ_4, so all extensions H^{(r)}_{1,1,1,l} with l ≥ 5 have spectral radius > ρ_r, confirming the classification is complete.