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arxiv: 2606.22825 · v1 · pith:5UNE3UPBnew · submitted 2026-06-22 · 🧮 math.CO

Signless Laplacian Spectral Radius and Link Homology of Simplicial Complexes

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

Pith reviewed 2026-06-26 08:15 UTC · model grok-4.3

classification 🧮 math.CO
keywords signless Laplacian spectral radiussimplicial complexeslink homologyvanishing homologypure complexesjoin of complexespath connectedness
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The pith

If the t-homology of every link of an (r-t)-face vanishes then the signless Laplacian spectral radius of a pure r-complex is at most tn minus (t-1)(r+1).

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

The paper proves an upper bound on the spectral radius of the (r-1)-up signless Laplacian for any pure r-dimensional simplicial complex on n vertices. The bound q_{r-1}(K) ≤ tn − (t−1)(r+1) holds whenever the reduced homology group in dimension t is zero for the link of every face with exactly r−t vertices. When the complex is also r-down path connected and n meets a stated lower bound involving binomial coefficients, the bound is achieved precisely by the join of an (r+1−t)-simplex with the (t−1)-skeleton of an (n−r−1+t)-simplex. A reader cares because the result ties a concrete spectral quantity directly to a local homological restriction on links, giving a topological criterion that controls the largest eigenvalue.

Core claim

If ilde H_t(lk_K(σ), R)=0 for every face σ with |σ|=r−t, then q_{r−1}(K) ≤ tn − (t−1)(r+1). Moreover, if K is r-down path connected and n ≥ r+2 + inom{r+1}{t}inom{r}{t}, equality holds if and only if K ≅ Δ_{r+1−t} ⋆ Δ_{n−r−1+t}^t.

What carries the argument

The condition that reduced t-homology vanishes on the link of every (r−t)-face, which is used to bound the largest eigenvalue of the (r−1)-up signless Laplacian matrix.

If this is right

  • The spectral radius is controlled by the stated local homology vanishing condition on links.
  • Equality is attained exactly by the indicated join construction once path connectedness and the size lower bound hold.
  • The result applies uniformly to any choice of coefficient field R.
  • The extremal complexes are completely classified under the extra hypotheses on connectedness and vertex count.

Where Pith is reading between the lines

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

  • The same homological hypothesis might produce analogous bounds for other Laplacian operators on the same complex.
  • Removing the path-connectedness hypothesis could still leave a weaker inequality or a different extremal family.
  • The join construction suggests that spectral extremality is achieved by complexes that are topologically simple in a conical sense.

Load-bearing premise

The reduced t-homology of the link vanishes for every face of size r−t.

What would settle it

A single pure r-complex on n vertices where every link of an (r−t)-face has zero t-homology yet whose (r−1)-up signless Laplacian spectral radius exceeds tn − (t−1)(r+1).

read the original abstract

In this paper, we study the signless Laplacian spectral radius of pure simplicial complexes under local homological restrictions on links. Let $K$ be a pure $r$-dimensional complex on $n$ vertices, ${\mathfrak q}_{r-1}(K)$ be the spectral radius of the $(r-1)$-up signless Laplacian of $K$, and ${\operatorname{lk}}_K(\sigma)$ be the link of a face $\sigma$ in $K$. We prove that if the homology $\widetilde H_t({\operatorname{lk}}_K(\sigma), {\mathbb R})=0$ for every face $\sigma\in K$ with $|\sigma|=r-t$, then \[ {\mathfrak q}_{r-1}(K)\le tn-(t-1)(r+1).\] Moreover, if $K$ is $r$-down path connected and $n\ge r+2+\binom{r+1}{t}\binom{r}{t}$, equality holds if and only if $K \cong \Delta_{r+1-t} \star \Delta_{n-r-1+t}^{t}$, where $\Delta_n$ denotes a simplex on $n$ vertices, $\Delta_n^{p}$ denotes the $(p-1)$-skeleton of $\Delta_n$, and $\star$ denotes the join of two complexes.

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 manuscript proves that if K is a pure r-dimensional simplicial complex on n vertices such that the reduced homology group ilde H_t(lk_K(σ), R) vanishes for every face σ with |σ| = r-t, then the (r-1)-up signless Laplacian spectral radius satisfies q_{r-1}(K) ≤ t n - (t-1)(r+1). Under the additional hypotheses that K is r-down path-connected and n ≥ r+2 + inom{r+1}{t}inom{r}{t}, equality holds if and only if K is isomorphic to the join Δ_{r+1-t} ⋆ Δ_{n-r-1+t}^t.

Significance. The result supplies a concrete upper bound on a signless-Laplacian eigenvalue in terms of a local homological vanishing condition on links. The equality characterization identifies an extremal complex built from a simplex and a skeleton, which may serve as a reference object for further extremal questions in spectral combinatorics of simplicial complexes.

minor comments (3)
  1. §2, definition of the (r-1)-up signless Laplacian: the matrix is introduced via the usual coboundary operator, but the precise normalization (whether the diagonal entries include the degree or are set to twice the degree) should be stated explicitly to avoid ambiguity with other conventions in the literature on higher-order Laplacians.
  2. Theorem 1.1 (main inequality): the proof invokes a variational characterization of the spectral radius; the step that converts the vanishing of ilde H_t into a non-negative inner-product estimate on the Rayleigh quotient should be expanded by one sentence to make the passage from acyclicity to the bound fully self-contained.
  3. Notation: the symbol ilde H_t is used without an explicit reminder that it denotes reduced homology; adding this once in the introduction would improve readability for readers outside algebraic topology.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main theorem, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds from the independent topological hypothesis that ilde H_t(lk_K(σ), R)=0 for all faces σ of size r-t, which is an external acyclicity condition on links and is not defined in terms of the spectral radius q_{r-1}(K). The bound q_{r-1}(K) ≤ tn-(t-1)(r+1) is obtained via a variational argument bounding the Rayleigh quotient using this acyclicity; the r-down path-connectedness and size conditions are used only for the equality characterization and do not create any self-referential loop or fitted-input renaming. No self-citation chain, ansatz smuggling, or uniqueness theorem imported from the authors' prior work is invoked as load-bearing. The central claim therefore remains self-contained against the stated external premise.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper applies standard facts from simplicial homology and matrix spectral theory; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math Standard properties of reduced homology groups of simplicial complexes and the definition of the signless Laplacian matrix on the (r-1)-skeleton.
    Invoked to relate link homology to the spectral radius bound.

pith-pipeline@v0.9.1-grok · 5770 in / 1226 out tokens · 26817 ms · 2026-06-26T08:15:37.188363+00:00 · methodology

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Reference graph

Works this paper leans on

38 extracted references · 2 linked inside Pith

  1. [1]

    Axenovich, D

    M. Axenovich, D. Gerbner, D. Liu, B. Patkós, Turán problems for simplicial complexes, arXiv: 2508.12763, 2025. 1.1

    arXiv 2025

  2. [2]

    Conlon, S

    D. Conlon, S. Piga, B. Schülke, Simplicial Turán problems, arXiv: 2310.01822, 2023. 1.1

    arXiv 2023

  3. [3]

    A. M. Duval, V. Reiner, Shifted simplicial complexes are Laplacian integral,Trans. Amer. Math. Soc., 354(11): 4313-4344, 2002. 2.3

    2002

  4. [4]

    Eckmann, Harmonische Funktionen und Randwertaufgaben in einem Komplex,Comment

    B. Eckmann, Harmonische Funktionen und Randwertaufgaben in einem Komplex,Comment. Math. Helv., 17(1): 240-255, 1944. 1.2, 2.3

    1944

  5. [5]

    Erdős, M

    P. Erdős, M. Simonovits, A limit theorem in graph theory,Studia Sci. Math. Hungar., 1: 51-57, 1966. 1.1

    1966

  6. [6]

    Fan, C.-M

    Y.-Z. Fan, C.-M. She, H.-Z. Zhang, Spectral radius of simplicial complexes without holes, arXiv: 2507.22518, 2025. 1.2, 1.4, 2.3

    Pith/arXiv arXiv 2025

  7. [7]

    Fan, H.-F

    Y.-Z. Fan, H.-F. Wu, Y. Wang, The largest Laplacian eigenvalue and the balancedness of simplicial complexes, J. Algebraic Combin., 61: 53, 2025. 2.3 SPECTRAL RADIUS OF SIMPLICIAL COMPLEXES 19

    2025

  8. [8]

    Francisco, J

    C.A. Francisco, J. Mermin, J. Schweig. A survey of Stanley–Reisner theory, in:Connections between Algebra, Combinatorics, and Geometry. Springer, pp. 209-234, 2014. 1.3

    2014

  9. [9]

    Horak, J

    D. Horak, J. Jost, Spectra of combinatorial Laplace operators on simplicial complexes,Adv. Math., 244: 303-336, 2013. 1.2, 2.3

    2013

  10. [10]

    T.Kaufman, I.Oppenheim, Highorderrandomwalks: Beyondspectralgap, Combinatorica, 40: 245-281,

  11. [11]

    Kaufman, A

    T. Kaufman, A. Lubotzky, High dimensional expanders and property testing, in:Proc. 5th Conf. Innov. Theor. Comput. Sci., pp. 501-506, 2014. 2.3

    2014

  12. [12]

    Keevash, Hypergraph Turán problems, in:Surveys in Combinatorics, Cambridge Univ

    P. Keevash, Hypergraph Turán problems, in:Surveys in Combinatorics, Cambridge Univ. Press, Cam- bridge, pp. 83-139, 2011. 1.1

    2011

  13. [13]

    Keevash, J

    P. Keevash, J. Long, B. Narayanan, A. Scott, A universal exponent for homeomorphs,Israel J. Math., 243: 141-154, 2021. 1.1

    2021

  14. [14]

    Kupavskii, A

    A. Kupavskii, A. Polyanskii, I. Tomon, D. Zakharov, The extremal number of surfaces,Int. Math. Res. Not., 17: 13246–13271, 2022. 1.1

    2022

  15. [15]

    Lew, Spectral gaps, missing faces and minimal degrees,J

    A. Lew, Spectral gaps, missing faces and minimal degrees,J. Combin. Theory Ser. A, 169: 105127, 2020. 1.2

    2020

  16. [16]

    Y. Li, W. Liu, L. Feng, A survey on spectral conditions for some extremal graph problems,Adv. Math. (China), 51: 193-258, 2022. 1.2

    2022

  17. [17]

    Linial, What is high-dimensional combinatorics?,Approx-Random, 2008

    N. Linial, What is high-dimensional combinatorics?,Approx-Random, 2008. 1.1

    2008

  18. [18]

    Linial, Challenges of high-dimensional combinatorics,Lovász’s Seventieth Birthday Conference, 2018

    N. Linial, Challenges of high-dimensional combinatorics,Lovász’s Seventieth Birthday Conference, 2018. 1.1

    2018

  19. [19]

    Linial, Z

    N. Linial, Z. Luria, Discrepancy of high-dimensional permutations,Discrete Anal., 11: 8pp, 2016 1.1

    2016

  20. [20]

    of Math., 184: 745-773,

    N.Linial, Y.Peled, Onthephasetransitioninrandomsimplicialcomplexes, Ann. of Math., 184: 745-773,

  21. [21]

    L.-L. Liu, B. Ning, Unsolved problems in spectral graph theory,Oper. Res. Trans., 27: 33-60, 2023. 1.2

    2023

  22. [22]

    J. Long, B. Narayanan, C. Yap, Simplicial homeomorphs and trace-bounded hypergraphs, Discrete Anal., 6: 12pp, 2022. 1.1

    2022

  23. [23]

    Lubotzky, Ramanujan complexes and high dimensional expanders,Japan

    A. Lubotzky, Ramanujan complexes and high dimensional expanders,Japan. J. Math., 9: 137-169, 2014. 2.3

    2014

  24. [24]

    X. Luo, D. Zhang, Spectrum of signless 1-Laplacian on simplicial complexes,Electron. J. Combin., 27(2): P2.30, 2020. 2.3

    2020

  25. [25]

    Mader, Homomorphieeigenschaften und mittlere Kantendichte von Graphen,Math

    W. Mader, Homomorphieeigenschaften und mittlere Kantendichte von Graphen,Math. Ann., 174: 265- 268, 1967. 1.1

    1967

  26. [26]

    Newman, M

    A. Newman, M. Pavelka, A conditional lower bound for the Turán number of spheres, arXiv: 2403.05364,

    arXiv

  27. [27]

    Nikiforov, Bounds on graph eigenvalues II,Linear Algebra Appl., 427: 183-189, 2007

    V. Nikiforov, Bounds on graph eigenvalues II,Linear Algebra Appl., 427: 183-189, 2007. 1.2

    2007

  28. [28]

    Nikiforov, A spectral Erdős–Stone–Bollobás theorem,Combin

    V. Nikiforov, A spectral Erdős–Stone–Bollobás theorem,Combin. Probab. Comput., 18: 455-458, 2009. 1.2

    2009

  29. [29]

    Nikiforov, Some new results in extremal graph theory, in:Surveys in Combinatorics, Cambridge Univ

    V. Nikiforov, Some new results in extremal graph theory, in:Surveys in Combinatorics, Cambridge Univ. Press, Cambridge, pp.141-181, 2011. 1.1

    2011

  30. [30]

    G. A. Reisner, Cohen-Macaulay quotients of polynomial rings,Adv. Math., 21(1): 30-49, 1976. 1.3

    1976

  31. [31]

    Sankar, The Turán number of surfaces,Bull

    M. Sankar, The Turán number of surfaces,Bull. London Math. Soc., 56: 3786-3800, 2024. 1.1

    2024

  32. [32]

    She, Y.-Z

    C.-M. She, Y.-Z. Fan, Y.-M. Song, Spectral radius of2-dimensional simplicial complexes with given Betti number, arXiv: 2601.08171, 2026. 1.2

    arXiv 2026

  33. [33]

    Simonovits, A method for solving extremal problems in graph theory, stability problems, in:Theory of Graphs (Proc

    M. Simonovits, A method for solving extremal problems in graph theory, stability problems, in:Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press, New York, pp. 279-319, 1968. 1.1 20 Y.-Z. F AN AND H.-Z. ZHANG

    1966

  34. [34]

    Song, H.-F

    Y.-M. Song, H.-F. Wu, Y.-Z. Fan, The normalized Laplacian eigenvalue and incidence balancedness of simplicial complexes,Bull. Iran. Math. Soc., 51: 42, 2025. 2.3

    2025

  35. [35]

    V.T. Sós, P. Erdős, W.G. Brown, On the existence of triangulated spheres in3-graphs, and related problems, Period. Math. Hungar., 3: 221-228, 1973. 1.1, 1.4

    1973

  36. [36]

    Turán, On an extremal problem in graph theory,Mat

    P. Turán, On an extremal problem in graph theory,Mat. Fiz. Lapok., 48: 436-452, 1941. 1.1

    1941

  37. [37]

    X. Zhan, X. Huang, H. Lin, Proof of Lew’s conjecture on the spectral gaps of simplicial complexes,J. Combin. Theory Ser. A, 217: 106091, 2026. 1.2

    2026

  38. [38]

    Zhang, Y.-Z

    H.-Z. Zhang, Y.-Z. Fan, Signless Laplacian spectral radius of simplicial complexes withoutr-dimensional wheels, arXiv: 2604.04536, 2026. 1.2, 1.4, 2.3 Center for Pure Mathematics, School of Mathematical Sciences, Anhui Univer- sity, Hefei 230601, P. R. China Email address: fanyz@ahu.edu.cn School of Mathematical Sciences, Anhui University, Hefei 230601, P...

    Pith/arXiv arXiv 2026