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arxiv: 2606.21233 · v1 · pith:QCLBFITYnew · submitted 2026-06-19 · 🧮 math.CO

Upper Bounds for the Largest Laplacian Eigenvalue of Simplicial Complexes

Huan-Zhi Zhang , Yi-Zheng Fan This is my paper

Pith reviewed 2026-06-26 14:16 UTC · model grok-4.3

classification 🧮 math.CO
keywords simplicial complexesup-Laplacianlargest eigenvalueupper boundshomologycomplement complexpartite complexescombinatorial topology
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The pith

The largest eigenvalue of the (r-1)-up Laplacian on an r-dimensional simplicial complex is bounded by the maximum number of vertices completing the boundary of any r-face.

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

The paper establishes a sharper upper bound on the largest eigenvalue of the combinatorial (r-1)-up Laplacian of a finite r-dimensional simplicial complex K on n vertices. The new bound replaces the known estimate of n with the maximum, taken over all r-faces F, of the size of the union of the neighbor sets N_K(E) for each (r-1)-face E in the boundary of F. It also gives an exact homological criterion for when the eigenvalue reaches the absolute maximum n, namely when the r-dimensional complement of K has nonzero reduced homology in dimension r-1 over the reals. For graphs this recovers the classical disconnection condition on the complement graph.

Core claim

We prove that λ_max(L^up_{r-1}(K)) ≤ max_{F∈S_r(K)} |∪_{E∈∂F} N_K(E)| ≤ n, where N_K(E) is the set of vertices u outside E such that E ∪ {u} is an r-face. Equality holds in the bound n if and only if the r-dimensional complement K^c has nonzero reduced homology ilde H_{r-1}(K^c, ℝ). We characterize the equality case in the sharper bound and construct a family of partite semiregular complexes with admissible additions that attain it.

What carries the argument

The quantity max_{F∈S_r(K)} |∪_{E∈∂F} N_K(E)|, which counts the largest collection of vertices that complete the boundary faces of an r-face and serves as the high-dimensional analog of the maximum-degree bound for graph Laplacians.

If this is right

  • The bound is attained by partite semiregular complexes with admissible additions.
  • Equality holds in the universal bound n precisely when the complement has nonzero reduced (r-1)-homology over the reals.
  • For r=1 the result reduces to a known bound and equality criterion for the Laplacian of a graph.
  • The paper supplies an explicit characterization of equality in the sharper bound.

Where Pith is reading between the lines

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

  • The local counting quantity could be used to obtain tighter estimates on other spectral invariants of simplicial complexes.
  • The direct link between the eigenvalue and homology of the complement suggests a route to constructing complexes with controlled spectra via topological modifications.
  • Numerical checks on small random or geometric complexes would reveal how often the new bound is strictly tighter than n.

Load-bearing premise

The up-Laplacian is defined via the standard combinatorial coboundary operator on the chain complex of the simplicial complex, and the complement K^c is the standard r-dimensional complement.

What would settle it

A concrete r-dimensional simplicial complex in which the numerically computed largest eigenvalue of L^up_{r-1} exceeds the value of max_{F} |∪_{E∈∂F} N_K(E)| would disprove the claimed bound.

read the original abstract

Let $K$ be a finite $r$-dimensional simplicial complex with vertex set $V$ of size $n$. We study the largest eigenvalue of the combinatorial $(r-1)$-up Laplacian $L^{\operatorname{up}}_{r-1}(K)$. It is known that \[ \lambda_{\max}\bigl(L^{\operatorname{up}}_{r-1}(K)\bigr)\le n. \] We first give a homological equality criterion for this universal bound, namely, the equality holds if and only if the $r$-dimensional complement $K^c$ of $K$ has a nonzero reduced homology $\widetilde H_{r-1}(K^c,\mathbb{R})$. For $r=1$, this is the classical graph condition that the complement graph is disconnected. Secondly, we prove a sharper upper bound for $\lambda_{\max}(L^{\operatorname{up}}_{r-1}(K))$: \[ \lambda_{\max}(L^{\operatorname{up}}_{r-1}(K)) \le \max_{F\in S_r(K)} \bigl|\bigcup_{E \in \partial F} N_K(E) \bigr| \le n,\] where, for an $(r-1)$-face $E$, $N_K(E)$ denotes the set of vertices $u$ outside $E$ such that the union $E \cup \{u\}$ is an $r$-face of $K$. This is the high-dimensional analog of the graph Laplacian bound. We give an explicit characterization of the equality case, and construct a broad family attaining the bound, namely, the partite semiregular complexes with admissible additions.

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 proves that for a finite r-dimensional simplicial complex K on n vertices, the largest eigenvalue of the combinatorial (r-1)-up Laplacian satisfies λ_max(L^up_{r-1}(K)) ≤ max_{F∈S_r(K)} |∪_{E∈∂F} N_K(E)| ≤ n, where N_K(E) is the set of vertices u ∉ E such that E ∪ {u} is an r-face. It also gives a homological equality criterion for the universal bound n: equality holds if and only if the reduced homology ilde H_{r-1}(K^c, ℝ) is nonzero, where K^c is the r-dimensional complement. The r=1 case recovers the classical graph-Laplacian statement, and the paper constructs partite semiregular complexes attaining the sharper bound.

Significance. If the proofs hold, the result supplies a strictly combinatorial sharpening of the known λ_max ≤ n bound together with an explicit homological characterization of equality, directly generalizing the graph case. The construction of a broad family of examples (partite semiregular complexes with admissible additions) that attain the bound is a concrete strength.

minor comments (3)
  1. [§2] §2 (definitions): the notation S_r(K) is used in the statement of the main bound but its precise definition (presumably the set of r-faces) should be recalled explicitly before the theorem to avoid any ambiguity for readers unfamiliar with the authors' prior notation.
  2. [§4] The equality case in the homological criterion is stated cleanly, but the manuscript should include a short verification that the constructed partite semiregular examples indeed satisfy ilde H_{r-1}(K^c, ℝ) ≠ 0 when equality is claimed.
  3. Figure captions and the statement of the main theorem should consistently use the same symbol for the up-Laplacian (L^up_{r-1}(K)) to prevent minor notation drift.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary accurately reflects the main contributions, including the sharpened combinatorial bound and the homological characterization of equality. As no specific major comments are listed in the report, there are no individual points to address.

Circularity Check

0 steps flagged

No circularity; derivation uses standard definitions and known universal bound

full rationale

The paper states the universal bound λ_max(L^up_{r-1}(K)) ≤ n as previously known and derives a strictly sharper combinatorial upper bound max_F |∪_{E∈∂F} N_K(E)| directly from the definition of the up-Laplacian and the neighborhood sets N_K(E). The equality criterion invokes the standard reduced homology of the complement K^c over ℝ, recovering the classical graph case for r=1 exactly. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain; all objects match textbook combinatorial and homological definitions. The construction of partite semiregular examples is explicit and independent of the bound itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

This is a pure-mathematics proof paper that relies on established definitions from algebraic topology and combinatorial spectral theory rather than introducing new fitted quantities or entities.

axioms (2)
  • domain assumption Standard definition and properties of the combinatorial up-Laplacian on simplicial complexes
    Invoked throughout to define L^up_{r-1}(K) and its eigenvalues.
  • standard math Reduced homology functor over the reals on simplicial complexes
    Used in the equality criterion for the bound n.

pith-pipeline@v0.9.1-grok · 5836 in / 1324 out tokens · 20685 ms · 2026-06-26T14:16:02.538808+00:00 · methodology

discussion (0)

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Reference graph

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