The multiplicity of the laplacian eigenvalue 1 of a tree

Dein Wong; Songnian Xu; Wenhao Zhen

arxiv: 2605.30982 · v1 · pith:NBXDLMXMnew · submitted 2026-05-29 · 🧮 math.SP

The multiplicity of the laplacian eigenvalue 1 of a tree

Songnian Xu , Dein Wong , Wenhao Zhen This is my paper

Pith reviewed 2026-06-28 20:19 UTC · model grok-4.3

classification 🧮 math.SP MSC 05C50

keywords Laplacian matrixeigenvalue multiplicitytreesreduced treespendant P3spectral graph theory

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The pith

Reduced trees without pendant P3 have m_T(1) at most (n-6)/4, or at most (n-7)/4 if that maximum is missed, or at most (n-8)/4 if both are missed.

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

The paper refines the upper bound on the multiplicity of the Laplacian eigenvalue 1 for reduced trees on n vertices that contain no pendant P3. Starting from the known bound of (n-6)/4, it proves that if this value is not attained then the multiplicity is at most (n-7)/4, with all such trees characterized. If neither value is attained then the multiplicity drops further to at most (n-8)/4, again with full characterization of the achieving trees. A reader would care because these results give the next possible maximum values and identify exactly which trees reach them.

Core claim

For a reduced tree T with n ≥ 7 vertices and no pendant P3, if m_T(1) ≠ (n-6)/4 then m_T(1) ≤ (n-7)/4 and the equality cases are completely characterized; moreover if m_T(1) ≠ (n-6)/4 and m_T(1) ≠ (n-7)/4 then m_T(1) ≤ (n-8)/4 with the equality cases also completely characterized.

What carries the argument

The assumption that T is reduced with no pendant P3, used to apply structural restrictions and case distinctions on how the multiplicity can fall short of the primary maximum.

If this is right

The multiplicity m_T(1) cannot lie strictly between (n-6)/4 and (n-7)/4.
The multiplicity m_T(1) cannot lie strictly between (n-7)/4 and (n-8)/4 when it also misses the first maximum.
Complete structural descriptions exist for all trees attaining each of these three successive bounds.

Where Pith is reading between the lines

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

The stepwise nature of the bounds indicates that m_T(1) takes values in a discrete set determined by n mod 4.
Similar successive bounds could be derived for other forbidden substructures or for different eigenvalues.
These characterizations allow enumeration of all possible multiplicities for trees satisfying the conditions.

Load-bearing premise

The tree T must be reduced and contain no pendant P3.

What would settle it

A single reduced tree with no pendant P3 on n≥7 vertices where the multiplicity m_T(1) exceeds (n-7)/4 but is less than (n-6)/4.

read the original abstract

Let $G$ be a connected, undirected simple graph. Denote by $L(G)$ the Laplacian matrix of $G$, and let $m_{G}(\lambda)$ be the multiplicity of an eigenvalue $\lambda$ of $L(G)$. When $G$ is a tree $T$ with $n \ge 6$ vertices, Tian et al. [Discrete Mathematics, 2026] proved that if $T$ is reduced and contains no pendant $P_3$, then \[ m_{T}(1) \le \frac{n-6}{4}, \] and they gave a complete characterization of the graphs for which equality holds. In this paper, we further investigate the above problem. Still assuming that $T$ is a tree with $n \ge 7$ vertices which is reduced and has no pendant $P_3$, we prove the following results. If $m_T(1) \neq \frac{n-6}{4}$, then \[ m_{T}(1) \le \frac{n-7}{4}, \] and we give a complete characterization of the graphs for which equality holds. If, moreover, $m_T(1) \neq \frac{n-6}{4}, \frac{n-7}{4}$, then \[ m_{T}(1) \le \frac{n-8}{4}, \] and we also give a complete characterization of the extremal graphs.

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

This paper refines the Tian et al. bound on m_T(1) by two further steps with equality cases for the same narrow class of reduced trees.

read the letter

The main point is that the authors extend the existing upper bound on the multiplicity of Laplacian eigenvalue 1 for reduced trees with no pendant P3. Starting from the Tian et al. result that m_T(1) is at most (n-6)/4, they show that if equality is missed then the multiplicity drops to at most (n-7)/4, and if that is also missed then it is at most (n-8)/4, all for n at least 7. They supply complete lists of the trees that achieve equality at each stage.

The characterizations of the extremal graphs are the part that adds concrete value. Once you have the exact trees that hit each bound, it becomes easier to see the pattern and perhaps use the result in later work on the same family.

The limitations are straightforward. The setting is inherited directly from the prior paper and remains restrictive, so the result does not broaden the scope. The numerical improvement is small, and the paper introduces no new techniques beyond case analysis on the existing structural assumptions. Everything rests on the correctness of the proofs, which follow the standard direct approach in this subfield.

This is a paper for specialists already working on Laplacian multiplicities for trees under these conditions. A reader outside that narrow track will not gain much. The claims are specific and the extension is honest, so the manuscript deserves to go through peer review rather than a desk rejection.

Referee Report

1 major / 0 minor

Summary. The paper refines the bound on the multiplicity m_T(1) of the Laplacian eigenvalue 1 for reduced trees T with n≥7 vertices containing no pendant P3. Building on Tian et al., it proves that if m_T(1) ≠ (n-6)/4 then m_T(1) ≤ (n-7)/4 with a complete characterization of equality cases, and further that if m_T(1) ≠ (n-6)/4 and ≠ (n-7)/4 then m_T(1) ≤ (n-8)/4, again with complete characterization of the extremal graphs.

Significance. If correct, the result sharpens the possible values of m_T(1) under the standing assumptions and supplies explicit extremal families, which strengthens the structural understanding of Laplacian spectra on trees. The case-distinction approach and characterizations are potentially useful for further work in spectral graph theory.

major comments (1)

The manuscript consists only of the abstract stating the claims; no proofs, derivations, lemmas, or supporting arguments appear in any section. Without these, the central inequalities and characterizations cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. The sole major comment is addressed below; we agree it requires a revision to the manuscript.

read point-by-point responses

Referee: The manuscript consists only of the abstract stating the claims; no proofs, derivations, lemmas, or supporting arguments appear in any section. Without these, the central inequalities and characterizations cannot be verified.

Authors: We acknowledge that the version under review contained only the abstract. The full proofs, including the case analysis for the successive bounds m_T(1) ≤ (n-7)/4 and m_T(1) ≤ (n-8)/4 together with the equality characterizations, were omitted from the submitted file. The revised manuscript will contain the complete arguments, lemmas, and derivations needed to verify the claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states and proves refined multiplicity bounds for the Laplacian eigenvalue 1 on reduced trees without pendant P3, building directly on the base inequality from the cited Tian et al. result via explicit case distinctions and graph characterizations. All load-bearing steps are standard mathematical derivations from the Laplacian matrix definition and tree structure properties; no parameters are fitted, no quantities are defined in terms of themselves, and the single external citation supplies only the starting bound rather than the new refinements. The derivation chain is therefore self-contained and independent of any self-referential reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard definitions of the Laplacian matrix, trees, reduced graphs, and pendant P3 from prior literature, plus the domain assumption that the trees satisfy those restrictions.

axioms (2)

standard math Laplacian matrix L(G) = D(G) - A(G) where D is the degree matrix and A the adjacency matrix
Invoked as the definition underlying all eigenvalue statements.
domain assumption T is reduced and contains no pendant P3
Explicit standing assumption required for the multiplicity bounds to apply.

pith-pipeline@v0.9.1-grok · 5794 in / 1292 out tokens · 32499 ms · 2026-06-28T20:19:11.759469+00:00 · methodology

The multiplicity of the laplacian eigenvalue 1 of a tree

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)