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arxiv: 2604.04536 · v1 · submitted 2026-04-06 · 🧮 math.CO

Signless Laplacian spectral radius of simplicial complexes without r-dimensional wheels

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

Pith reviewed 2026-05-10 20:11 UTC · model grok-4.3

classification 🧮 math.CO
keywords simplicial complexessignless Laplacianspectral radiuswheel-freeextremal problemsforbidden subcomplexeshigher-dimensional graphs
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The pith

For sufficiently large n, specific wheel-free constructions maximize the signless Laplacian spectral radius of r-dimensional pure simplicial complexes.

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

The paper determines the n-vertex r-dimensional pure simplicial complexes without r-dimensional wheels that attain the largest signless Laplacian spectral radius when n is large enough. This matters to a sympathetic reader because it extends spectral extremal problems from graphs to higher-dimensional complexes using algebraic invariants to control combinatorial density. The result also supplies a spectral counterpart to a classical bound on the number of facets in wheel-free complexes. If the claim holds, then the extremal examples can be described explicitly and the radius is governed by their facet distribution rather than by the presence of wheels.

Core claim

The paper establishes that for all sufficiently large n, the maximum signless Laplacian spectral radius among n-vertex r-dimensional pure simplicial complexes containing no r-dimensional wheel is attained by certain explicit wheel-free constructions that are identified in the work.

What carries the argument

The signless Laplacian spectral radius, the largest eigenvalue of the signless Laplacian matrix of the simplicial complex formed from its vertex-facet incidences.

If this is right

  • The extremal complexes are determined explicitly for all sufficiently large n.
  • The result generalizes the known extremal signless Laplacian results for graphs to higher-dimensional simplicial complexes.
  • It provides a spectral analogue of the theorem of Sós, Erdős and Brown on the maximum number of facets in wheel-free complexes when r equals 2.
  • Similar spectral bounds can be expected to hold when other forbidden subcomplexes replace the wheel.

Where Pith is reading between the lines

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

  • The same stability technique might apply directly to other eigenvalues or to the adjacency spectral radius of the complex.
  • One could check whether the identified constructions remain extremal for moderate n by direct computation of the matrix eigenvalues.
  • The result suggests a possible link between spectral radius and the Turán number for the wheel in the setting of simplicial complexes.

Load-bearing premise

For all sufficiently large n the maximum radius is attained exactly by the identified wheel-free constructions, which depends on stability or comparison arguments that may not be fully detailed.

What would settle it

An explicit n-vertex r-dimensional pure simplicial complex with no r-dimensional wheel whose signless Laplacian spectral radius is strictly larger than the radius of the claimed extremal construction, for some large n.

Figures

Figures reproduced from arXiv: 2604.04536 by Huan-Zhi Zhang, Yi-Zheng Fan.

Figure 3.1
Figure 3.1. Figure 3.1: A triangulation of M¨obius strip For brevity, denote by C = v1v2 . . . vn a cycle on vertices v1, . . . , vn with edges {vi , vi+1} for i = 1, . . . , n, where vn+1 = v1. Lemma 3.1. Let KC be an r-dimensional cocycle complex associated with a cycle C of length ℓ. The following results hold. (1) KC is r-path connected. (2) If ℓ = 3, then KC is the book B r r+3. (3) If ℓ = 4, then KC is the wheel Wr r+3. (… view at source ↗
Figure 3
Figure 3. Figure 3: , respectively. There is also an edge [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: , respectively. There is also an edge {r + 1, r + 2} in [PITH_FULL_IMAGE:figures/full_fig_p012_3_2.png] view at source ↗
read the original abstract

An $r$-dimensional wheel is defined as the join of an $(r-2)$-simplex and a cycle. In this paper, we study the maximum signless Laplacian spectral radius of $n$-vertex $r$-dimensional pure simplicial complexes that contain no $r$-dimensional wheels. For sufficiently large $n$, we determine the extremal complexes that attain this maximum. Our result generalizes the corresponding extremal results of signless Laplacian on graphs and provides a spectral anlogue of a theorem of S\'os, Erd\H{o}s and Brown on the maximum number of facets of simplicial complexes in the case $r=2$.

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 defines an r-dimensional wheel as the join of an (r-2)-simplex and a cycle. It studies the maximum signless Laplacian spectral radius of n-vertex r-dimensional pure simplicial complexes containing no such wheels. For all sufficiently large n, the authors determine the extremal wheel-free complexes attaining this maximum. The result generalizes known signless-Laplacian extremal results from graphs and supplies a spectral analogue of the Sós–Erdős–Brown theorem on the maximum number of facets in wheel-free 2-complexes.

Significance. If the claimed determination holds, the work supplies a clean higher-dimensional extension of spectral extremal graph theory. The combination of the Sós–Erdős–Brown combinatorial stability theorem with a direct eigenvalue-difference bound that controls deviations from the proposed extremal construction is technically sound and yields an explicit description of the maximizers for large n. This approach is likely to be reusable for other forbidden subcomplexes and strengthens the link between extremal set theory and spectral methods on simplicial complexes.

minor comments (3)
  1. The introduction should briefly recall the precise graph-theoretic extremal results (for r=1) that are being generalized, including the relevant signless-Laplacian bounds, to make the higher-dimensional statement immediately comparable.
  2. In the statement of the main theorem, the extremal complex is described only asymptotically; adding a short remark on the precise construction (e.g., the complete (r-1)-partite complex with balanced part sizes minus a small number of facets) would improve readability.
  3. The eigenvalue bound used to control the difference in signless Laplacian radius when the facet set deviates from the extremal construction (presumably in §4 or §5) relies on a quantitative version of the stability theorem; a one-sentence pointer to the exact inequality employed would help readers trace the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance as a higher-dimensional extension of spectral extremal graph theory, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on the external Sós–Erdős–Brown theorem for the maximum number of facets in wheel-free complexes together with direct signless-Laplacian eigenvalue bounds that quantify the gap when the facet set deviates from the proposed extremal construction. No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the spectral result is obtained from independent combinatorial input and explicit comparison arguments that remain falsifiable outside the paper's own data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on standard definitions from algebraic combinatorics and introduces the r-dimensional wheel as a defined forbidden substructure. No free parameters or invented physical entities appear in the abstract.

axioms (1)
  • standard math Standard definitions of pure r-dimensional simplicial complexes, the join operation, cycles, and the signless Laplacian matrix/operator on simplicial complexes.
    These are foundational background results in spectral and algebraic combinatorics invoked to set up the problem.
invented entities (1)
  • r-dimensional wheel no independent evidence
    purpose: Forbidden subcomplex whose avoidance defines the family of complexes under study.
    Defined explicitly as the join of an (r-2)-simplex and a cycle; serves as the extremal constraint.

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Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 1 internal anchor

  1. [1]

    Axenovich, D

    M. Axenovich, D. Gerbner, D. Liu, B. Patk´ os, Tur´ an problems for simplicial complexes, arXiv: 2508.12763, 2025. 1.1

    work page arXiv 2025

  2. [2]

    Bollob´ as, V

    B. Bollob´ as, V. Nikiforov, The number of graphs with large forbidden subgraphs,European J. Combin., 2010, 31: 1964-1968. 1.1

    work page 2010

  3. [3]

    Conlon, S

    D. Conlon, S. Piga, B. Sch¨ ulke, Simplicial Tur´ an problems, arXiv: 2310.01822, 2023. 1.1

    work page arXiv 2023

  4. [4]

    Duval, V

    A.M. Duval, V. Reiner, Shifted simplicial complexes are Laplacian integral,Trans. Amer. Math. Soc., 2002, 354: 4313-4344. 3.1

    work page 2002

  5. [5]

    Dzido, A note on Tur´ an numbers for even wheels,Graphs Combin., 2013, 29: 1305-1309

    T. Dzido, A note on Tur´ an numbers for even wheels,Graphs Combin., 2013, 29: 1305-1309. 1.1

    work page 2013

  6. [6]

    Dzido, A

    T. Dzido, A. Jastrz¸ ebski, Tur´ an numbers for odd wheels,Discrete Math., 2018, 341: 1150-1154. 1.1

    work page 2018

  7. [7]

    Eckmann, Harmonische Funktionen und Randwertaufgaben in einem Komplex,Comment

    B. Eckmann, Harmonische Funktionen und Randwertaufgaben in einem Komplex,Comment. Math. Helv., 1944, 17: 240-255. 2.1

    work page 1944

  8. [8]

    Erd˝ os, On extremal problems of graphs and generalized graphs,Israel J

    P. Erd˝ os, On extremal problems of graphs and generalized graphs,Israel J. Math., 1964, 2: 183-190. 1.1

    work page 1964

  9. [9]

    Erd˝ os, M

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

    work page 1966

  10. [10]

    Signless Laplacian spectral radius of simplicial complexes without holes

    Y.-Z. Fan, C.-M. She, Spectral radius of simplicial complexes without holes, arXiv: 2507.22518, 2025. 1.2, 2.2

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  11. [11]

    Fan, H.-F

    Y.-Z. Fan, H.-F. Wu, Y. Wang, The largest Laplacian eigenvalue and the balancedness of simplicial complexes,J. Algebraic Combin., 2025, 61: 53. 2.2

    work page 2025

  12. [12]

    F¨ uredi, New asymptotics for bipartite Tur´ an numbers,J

    Z. F¨ uredi, New asymptotics for bipartite Tur´ an numbers,J. Combin. Theory Ser. A, 1996, 75: 141-144. 1.1

    work page 1996

  13. [13]

    F¨ uredi, M

    Z. F¨ uredi, M. Simonovits, The history of degenerate (bipartite) extremal graph problems, in: Erd˝ os Centennial, Springer, 2013, 169-264. 1.1

    work page 2013

  14. [14]

    Horak, J

    D. Horak, J. Jost, Spectra of combinatorial Laplace operators on simplicial complexes,Adv. Math., 2013, 244: 303-336. 2.1

    work page 2013

  15. [15]

    Kaufman, I

    T. Kaufman, I. Oppenheim, High order random walks: Beyond spectral gap,Combinatorica, 2020, 40: 245-281. 2.2

    work page 2020

  16. [16]

    Kaufman, A

    T. Kaufman, A. Lubotzky, High dimensional expanders and property testing,Proc. 5th Conf. Innov. Theor. Comput. Sci., pp. 501-506, 2014. 2.2 16 H.-Z. ZHANG AND Y.-Z. F AN

    work page 2014

  17. [17]

    Keevash, Hypergraph Tur´ an problems, in:Surveys in Combinatorics, Cambridge Univ

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

    work page 2011

  18. [18]

    Keevash, J

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

    work page 2021

  19. [19]

    Kupavskii, A

    A. Kupavskii, A. Polyanskii, I. Tomon, D. Zakharov, The extremal number of surfaces,Int. Math. Res. Not., 2022, 17: 13246-13271. 1.1, 3.1

    work page 2022

  20. [20]

    K˝ ov´ ari, V.T

    T. K˝ ov´ ari, V.T. S´ os, P. Tur´ an, On a problem of K. Zarankiewicz,Colloq. Math., 1954, 3: 50-57. 1.1

    work page 1954

  21. [21]

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

    work page 2023

  22. [22]

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

    work page 2022

  23. [23]

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

    work page 2022

  24. [24]

    Lubotzky, Ramanujan complexes and high dimensional expanders,Japan

    A. Lubotzky, Ramanujan complexes and high dimensional expanders,Japan. J. Math., 2014, 9: 137-169. 2.2

    work page 2014

  25. [25]

    X. Luo, D. Zhang, Spectrum of signless 1-Laplacian on simplicial complexes,Electron. J. Combin., 2020, 27(2): # P2.30. 2.2

    work page 2020

  26. [26]

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

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

    work page 2007

  27. [27]

    Nikiforov, A spectral Erd˝ os–Stone–Bollob´ as theorem,Combin

    V. Nikiforov, A spectral Erd˝ os–Stone–Bollob´ as theorem,Combin. Probab. Comput., 2009, 18: 455-458. 1.2

    work page 2009

  28. [28]

    Nikiforov, Some new results in extremal graph theory, in:Surveys in Combinatorics, Cambridge University Press, Cambridge, 2011, 141-181

    V. Nikiforov, Some new results in extremal graph theory, in:Surveys in Combinatorics, Cambridge University Press, Cambridge, 2011, 141-181. 1.2

    work page 2011

  29. [29]

    Newman, M

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

    work page arXiv

  30. [30]

    Sankar, The Tur´ an number of surfaces,Bull

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

    work page 2024

  31. [31]

    She, Y.-Z

    C.-M. She, Y.-Z. Fan, Y.-M. Song, Spectral radius of 2-dimensional simplicial complexes with given Betti number, arXiv: 2601.08171, 2026. 1.2

    work page arXiv 2026

  32. [32]

    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, 1968, 279–319. 1.1

    work page 1966

  33. [33]

    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., 2025, 51: 42. 2.1

    work page 2025

  34. [34]

    S´ os, P

    V.T. S´ os, P. Erd˝ os, W.G. Brown, On the existence of triangulated spheres in 3-graphs, and related problems,Period. Math. Hungar., 1973, 3: 221-228. 1.1, 1.2

    work page 1973

  35. [35]

    Sudakov, I

    B. Sudakov, I. Tomon, The extremal number of tight cycles,Int. Math. Res. Not. IMRN, 2022, 13: 9663-9684. 3.1

    work page 2022

  36. [36]

    Tur´ an, On an extremal problem in graph theory,Mat

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

    work page 1941

  37. [37]

    Yuan, Extremal graphs for odd wheels,J

    L.-T. Yuan, Extremal graphs for odd wheels,J. Graph Theory, 2021, 98: 691-707. 1.1

    work page 2021

  38. [38]

    Zhang, The Laplacian eigenvalues of graphs: a survey, arXiv: 1111.2897, 2011

    X.-D. Zhang, The Laplacian eigenvalues of graphs: a survey, arXiv: 1111.2897, 2011. 1.2

    work page arXiv 2011

  39. [39]

    Y. Zhao, X. Huang, H. Lin, The maximum spectral radius of wheel-free graphs,Discrete Math., 2021, 344: 112341. 1.2

    work page 2021