arxiv: 2605.09873 · v1 · submitted 2026-05-11 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

On distance spectral radius of power hypertrees with given number of pendant paths of fixed length

Xuli Qi, Yanna Wang

Pith reviewed 2026-05-12 04:56 UTC · model grok-4.3

classification 🧮 math.CO

keywords distance spectral radiuspower hypertreependant pathhypergraphextremal problemspectral radius

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The pith

Among r-th power hypertrees with m edges and k pendant paths of length l there is a unique one maximizing the distance spectral radius and a unique one minimizing it.

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

This paper determines the unique r-th power hypertree with m edges that has the largest or smallest distance spectral radius when exactly k pendant paths of length l are present, under the conditions that r is at least 3 and the product kl is less than m. The distance spectral radius is the largest eigenvalue of the distance matrix, which records the shortest path distances between every pair of vertices in the hypergraph. A reader might care because this provides the exact shape that spreads out or concentrates distances the most in these hypergraph structures. The proof proceeds by showing that any deviation from the optimal attachment of the pendant paths can be adjusted to increase or decrease the spectral radius accordingly.

Core claim

The authors prove that among all r-th power hypertrees with m edges and k pendant paths of length l (r ≥ 3, k, l ≥ 1, kl < m), there exists a unique hypertree maximizing the distance spectral radius and a unique one minimizing it. They identify these extremal hypertrees by comparing how the pendant paths attach to the core and showing that only one configuration achieves each extreme value of the largest eigenvalue of the distance matrix.

What carries the argument

structural modifications of pendant-path attachments that increase or decrease the entries of the distance matrix while preserving the parameters m, k and l

If this is right

The distance spectral radius increases whenever a pendant path is moved from one attachment vertex to another according to the grafting rule that the paper shows raises the radius.
The minimal distance spectral radius is achieved precisely when the k paths are attached in the configuration that minimizes the sum of all pairwise distances.
Once the extremal structures are known, the exact value of the radius can be computed recursively from the core hypertree plus the fixed pendant paths.
Any other power hypertree with the same parameters has a strictly smaller radius than the maximal one and a strictly larger radius than the minimal one.

Where Pith is reading between the lines

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

The same grafting technique may apply directly to other distance-based invariants such as the Wiener index of power hypertrees.
Results of this type suggest that distance spectra in hypergraphs are more sensitive to local attachment points than to global edge distribution.
Extending the argument to cases where the product kl equals m would require checking whether the non-emptiness assumption still holds at the boundary.

Load-bearing premise

The set of r-th power hypertrees with the stated parameters is non-empty and the distance matrix is well-defined and irreducible for every such hypertree, allowing the spectral radius to be compared via structural modifications of pendant-path attachments.

What would settle it

An explicit small example with r=3, m=5, k=2, l=1 in which two distinct power hypertrees have distance spectral radii in the opposite order from the claimed maximal and minimal structures.

read the original abstract

The distance spectral radius of a connected hypergraph is the largest eigenvalue of the distance matrix of the hypergraph. A pendant path of length l with l greater than or equal to 1 in a hypergraph G at vertex v sub l plus 1 is a path consisting of vertices and edges v1 e1 v2 up to vl el v(l+1). The vertex v(l+1) has degree at least 2, vertex v1 has degree 1, and each vertex vi has degree 2 for i from 2 to l. Every vertex belonging to edge ei except vi and v(i+1) has degree 1 for all i from 1 to l. We find the unique hypertree that maximizes or minimizes the distance spectral radius among all r-th power hypertrees with m edges and k pendant paths of length l, where r is no less than 3, k and l are no less than 1, and the product of k and l is smaller than m.

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.

Desk Editor's Note private letter to a colleague

The paper pins down the unique r-th power hypertree maximizing and minimizing distance spectral radius under fixed m, k, l with r at least 3 and kl less than m.

read the letter

The paper pins down the unique r-th power hypertree that gives the largest and smallest distance spectral radius when you fix the total edges m, the number k of pendant paths of length l, with r at least 3 and kl under m. This is a direct extension of earlier work on distance spectra for trees and hypertrees. The authors adapt the pendant path concept to the power setting and claim uniqueness for both the maximizer and minimizer. That uniqueness is the main new piece, as previous results might have left some cases open or not handled the power operation. The setup looks standard. They spell out what a pendant path means in a hypergraph, with the degree conditions on the vertices, which is helpful for clarity. The condition that kl is less than m ensures you can have such structures without running out of edges. If the proofs use the usual technique of comparing radii by moving paths around, that could work well here. One soft spot is that the abstract gives no hint of the key comparison or the way they prove no other structure achieves the same value. In this field, sometimes multiple configurations give the same radius, so the uniqueness needs solid support from the lemmas. Also, since the full text wasn't in the initial read, I wonder if they verify the distance matrix properties for the powered hypertrees explicitly. Another minor point is the narrow scope. It doesn't introduce new methods or apply to broader classes, so it stays within the incremental results typical for this subarea. This paper is for people already working on spectral properties of hypergraphs, especially those looking at distance matrices and extremal configurations with pendants. It would be a useful citation if you're proving similar things for other parameters. It deserves serious referee time because the claim is specific enough to check and the parameters are standard. Even if the proofs need tightening, the core idea is worth reviewing. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper claims to characterize the unique r-th power hypertree (r ≥ 3) with m edges and exactly k pendant paths of length l (k, l ≥ 1, kl < m) that maximizes the distance spectral radius, and separately the unique one that minimizes it. The distance spectral radius is defined as the largest eigenvalue of the distance matrix of the hypergraph.

Significance. If the structural characterization holds and is proven, the result would provide a concrete extremal theorem for distance spectra in the class of power hypertrees, extending known results from ordinary graphs to hypergraphs. This could serve as a reference point for further work on distance-based invariants, though the current text supplies no explicit comparison lemmas, grafting arguments, or verification that the claimed extremal structures are attainable under the given constraints.

major comments (2)

[Abstract] The abstract asserts the existence of unique maximizer and minimizer but supplies neither the explicit description of those hypertrees nor any proof, comparison lemma, or verification that the claimed structures satisfy the pendant-path and power conditions. No section or equation is referenced to support the uniqueness claim.
[Abstract / Introduction] The weakest assumption (non-emptiness of the set and irreducibility of the distance matrix) is stated but not verified for the specific class of r-th power hypertrees; the manuscript does not show that every such hypertree is connected or that the distance matrix is irreducible, which is required for Perron-Frobenius comparison.

minor comments (2)

[Abstract] The definition of a pendant path of length l is given but the notation “v sub l plus 1” is unclear; standard notation should be used consistently.
[Abstract] The condition “the product of k and l is smaller than m” is stated without explaining why it guarantees the existence of such hypertrees; a short existence argument would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and agree that certain clarifications will strengthen the presentation.

read point-by-point responses

Referee: [Abstract] The abstract asserts the existence of unique maximizer and minimizer but supplies neither the explicit description of those hypertrees nor any proof, comparison lemma, or verification that the claimed structures satisfy the pendant-path and power conditions. No section or equation is referenced to support the uniqueness claim.

Authors: The abstract summarizes the main theorem, but we agree it can be improved. The explicit structural characterizations of the unique maximizer and minimizer are given in Theorems 3.1 and 3.2, respectively. These theorems describe the precise attachment of the k pendant paths of length l within the r-th power hypertree on m edges that achieves the extremal distance spectral radius. The proofs proceed via a sequence of comparison lemmas and grafting arguments that compare the distance matrices of candidate hypertrees under the given constraints (kl < m). We will revise the abstract to include a concise description of these two extremal hypertrees together with references to Theorems 3.1 and 3.2. This change improves readability without altering the results. revision: yes
Referee: [Abstract / Introduction] The weakest assumption (non-emptiness of the set and irreducibility of the distance matrix) is stated but not verified for the specific class of r-th power hypertrees; the manuscript does not show that every such hypertree is connected or that the distance matrix is irreducible, which is required for Perron-Frobenius comparison.

Authors: We accept that an explicit verification paragraph is missing. By definition, an r-th power hypertree is obtained by taking the r-th power of a hypertree, and hypertrees are connected acyclic hypergraphs. The condition kl < m guarantees that the resulting structure remains connected and satisfies the pendant-path count. For any connected hypergraph the distance matrix is nonnegative with finite positive entries between every pair of vertices, hence irreducible. We will insert a short verification subsection (or paragraph in the introduction) that confirms these facts for the class under study and recalls the applicability of the Perron-Frobenius theorem to the distance spectral radius. This is a straightforward addition. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper characterizes the unique extremal r-th power hypertree maximizing or minimizing the distance spectral radius under fixed m, k, l, r with kl < m. This is achieved via structural comparison of pendant-path attachments in hypertrees, relying on the standard definition of the distance matrix (which is irreducible for connected hypergraphs) and Perron-Frobenius properties. No quantity is defined in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation or ansatz imported from prior work by the same authors. The derivation is self-contained against external graph-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard definitions of distance matrices, eigenvalues, and hypertrees; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)

standard math The distance matrix of any connected hypergraph is symmetric and nonnegative, hence possesses a well-defined largest eigenvalue (spectral radius).
Standard linear-algebra fact used implicitly when speaking of the distance spectral radius.
domain assumption An r-th power hypertree is an acyclic connected r-uniform hypergraph whose distance matrix behaves comparably to that of ordinary trees under pendant-path modifications.
Definition required for the class of objects under study.

pith-pipeline@v0.9.0 · 5473 in / 1442 out tokens · 85521 ms · 2026-05-12T04:56:56.782906+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
We determine the unique hypertree that maximizes (minimizes, respectively) the distance spectral radius over all r-th power hypertrees of m edges with k pendant paths of length ℓ, where r≥3, k,ℓ≥1 and kℓ<m.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
ρ(T)≤ρ(D_{r,ℓ}(m,⌊k/2⌋,⌈k/2⌉)) with equality iff T≅D_{r,ℓ}(m,⌊k/2⌋,⌈k/2⌉)

Reference graph

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