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arxiv: 1906.10614 · v1 · pith:BNT6YOHEnew · submitted 2019-06-25 · 🧮 math.CO

On the spectral radii of the unicyclic hypergraphs with fixed matching number

Guanglong Yu , Chao Yan , Yarong Wu , Hailiang Zhang This is my paper

Pith reviewed 2026-05-25 16:26 UTC · model grok-4.3

classification 🧮 math.CO
keywords spectral radiusunicyclic hypergraphsmatching numberk-uniform hypergraphsextremal hypergraphsadjacency tensor
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The pith

Connected k-uniform unicyclic hypergraphs with maximum spectral radius are uniquely determined for matching number at least z or exactly fixed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the unique hypergraphs that attain the maximum spectral radius among all connected k-uniform unicyclic hypergraphs with matching number at least z. It also identifies the unique maximizer when the matching number takes any prescribed value. These statements apply for every uniformity k at least 3. A sympathetic reader cares because the spectral radius of the adjacency tensor controls expansion and other global properties, so the extremal examples make the effect of the matching number constraint explicit.

Core claim

We determine the unique hypergraphs with maximum spectral radius among all connected k-uniform (k≥3) unicyclic hypergraphs with matching number at least z, and among all connected k-uniform (k≥3) unicyclic hypergraphs with a given matching number, respectively.

What carries the argument

The spectral radius of the adjacency tensor, maximized by isolating a unique structure inside the family of connected k-uniform unicyclic hypergraphs under a matching-number constraint.

If this is right

  • A unique maximizer exists in the class with matching number at least z.
  • A unique maximizer exists in the class with any fixed matching number.
  • The two classes share the same extremal structure when the lower bound is active.
  • The maximum spectral radius is therefore realized by an explicitly describable hypergraph in each case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same uniqueness statements would remain valid if the matching-number lower bound were replaced by an upper bound.
  • The identified maximizers supply concrete test objects for any algorithm that computes the spectral radius of a hypergraph.
  • Parallel results could be sought when the unicyclic condition is relaxed to a fixed number of cycles.

Load-bearing premise

The family of connected k-uniform unicyclic hypergraphs remains closed under the structural changes used to compare spectral radii, and a maximum exists inside each matching-number slice.

What would settle it

A connected k-uniform unicyclic hypergraph whose matching number is at least z yet whose spectral radius exceeds that of the claimed unique maximizer would falsify the result.

Figures

Figures reproduced from arXiv: 1906.10614 by Chao Yan, Guanglong Yu, Hailiang Zhang, Yarong Wu.

Figure 1.1
Figure 1.1. Figure 1.1: G1 − G6 q q q Theorem 1.3 Let G ∈ H and ρ(G) = ρmax. Then m ≥ z + 1, and (1) if m = z + 1, then G ∼= U (n, k; 0; 0, z − 1; 0, 0) (see G1 in [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: K and K ′ u u v1,1 v1,2 v1,4 v1,5 v1,1 v1,2 v2,1 v2,3 v2,1 v2,2 v2,2 v1,4 v1,5 q v2,3 Let K be a connected k-uniform hypergraph (k ≥ 3) with at least 2 edges and u ∈ V (K), e1 = {v1,1, v1,2, . . ., v1,k−1, u} be a nonpendant edge incident with u, and e1,1, e1,2, . . ., e1,s (1 ≤ s ≤ k − 2) be the pendant edges incident with vertices v1,1, v1,2, . . ., v1,s of e1 respectively, where e1,i ∩ e1 = {v1,i}, de… view at source ↗
read the original abstract

We determine the unique hypergraphs with maximum spectral radius among all connected $k$-uniform ($k\geq 3$) unicyclic hypergraphs with matching number at least $z$, and among all connected $k$-uniform ($k\geq 3$) unicyclic hypergraphs with a given matching number, respectively.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper determines the unique connected k-uniform (k≥3) unicyclic hypergraphs maximizing the spectral radius among all such hypergraphs with matching number at least z, and separately among those with a prescribed matching number.

Significance. If the characterizations hold, the results extend classical extremal spectral results from graphs to uniform hypergraphs, providing explicit maximizers under matching-number constraints. The finiteness of the classes ensures existence, and the use of standard edge-moving or grafting operations (preserving uniformity, connectedness, and the matching bound) would constitute a standard but useful contribution to spectral hypergraph theory.

minor comments (3)
  1. The abstract asserts uniqueness without indicating the proof strategy or the explicit form of the extremal hypergraphs; a brief description of the candidate hypergraphs (e.g., a specific unicyclic structure with pendant edges) should be added to the abstract or introduction.
  2. Notation for the adjacency tensor and the spectral radius (e.g., definition of the Rayleigh quotient or the eigenvalue equation) should be recalled or referenced in §2 to make the comparison arguments self-contained.
  3. The paper should explicitly verify that the proposed operations (edge relocation or grafting) preserve both the unicyclic property and the matching-number bound; a short lemma or remark would suffice.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation and recommendation of minor revision. The report provides no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claim is the determination of unique maximizers for spectral radius within finite classes of connected k-uniform unicyclic hypergraphs under matching-number constraints. No equations, derivations, or self-citations appear in the provided abstract or description that reduce any prediction or uniqueness result to a fitted input, self-definition, or load-bearing prior work by the same authors. Standard edge-moving or grafting arguments are expected to remain within the class and are independent of the target result. The derivation is therefore self-contained against external benchmarks in extremal hypergraph theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or new entities introduced in any proof.

pith-pipeline@v0.9.0 · 5568 in / 1033 out tokens · 32308 ms · 2026-05-25T16:26:43.835038+00:00 · methodology

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Reference graph

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