pith. sign in

arxiv: 2606.04963 · v1 · pith:ROKAY57Pnew · submitted 2026-06-03 · 🧮 math.CO

Localization of spectral Tur\'an theorems for signed graphs

Linfeng Xie , Xiaogang Liu This is my paper

Pith reviewed 2026-06-28 05:31 UTC · model grok-4.3

classification 🧮 math.CO
keywords signed graphsTurán theoremsspectral radiuslocalized inequalitiesextremal graphseigenvalue boundswalk counts
0
0 comments X

The pith

Signed graphs satisfy a vertex-localized Turán inequality that gives a sharp upper bound on the largest eigenvalue, with equality only for identified extremal signed graphs.

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

The paper extends localized Turán theorems from ordinary graphs to signed graphs. It proves a vertex-localized inequality that relates local structure measures directly to the spectral radius of the signed graph. The work characterizes the signed graphs that attain the bound. It also carries over a walk-based local spectral version of the inequality. The results strengthen and generalize prior bounds known for both unsigned and signed graphs.

Core claim

For signed graphs, a vertex-localized Turán-type inequality holds that produces a sharp upper bound on the largest eigenvalue expressed through localized Turán parameters, with equality attained exactly on the extremal signed graphs that are characterized in the paper. The same localization technique also yields a generalized walk-based local spectral Turán inequality that applies to signed graphs.

What carries the argument

Vertex-localized Turán parameters, which act as local structural measures to bound the largest eigenvalue of a signed graph.

If this is right

  • The largest eigenvalue of any signed graph is bounded above by a function of its vertex-localized Turán parameters.
  • Equality holds precisely for the extremal signed graphs identified in the paper.
  • The walk-based local spectral Turán inequality extends directly to the signed setting.
  • Several earlier spectral Turán results for unsigned graphs and signed graphs are recovered or strengthened as special cases.

Where Pith is reading between the lines

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

  • The same localization technique may apply to other spectral invariants of signed graphs beyond the largest eigenvalue.
  • Local computations of Turán parameters could yield practical approximations for spectral radii in large signed networks.
  • The approach suggests that signed-graph spectral problems can be reduced to local checks in a manner parallel to the unsigned case.

Load-bearing premise

The localization parameters can be defined and applied consistently to signed graphs without additional structural restrictions that would invalidate the equality cases or bounds.

What would settle it

A concrete signed graph in which the largest eigenvalue strictly exceeds the value predicted by the localized Turán parameters, or an extremal graph that lies outside the characterized family.

read the original abstract

In this paper, we extend localized Tur\'an theorems to signed graphs and study the corresponding spectral Tur\'an problems. Firstly, we establish a vertex-localized Tur\'an-type inequality for signed graphs and characterize the extremal graphs reaching this bound. Secondly, we derive a sharp upper bound for the largest eigenvalue of a signed graph via localized Tur\'an parameters, and identify the extremal graphs attaining equality. Finally, we generalize a walk-based local spectral Tur\'an inequality to signed graphs. Our results generalize and improve several existing theorems for unsigned and signed 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.

Referee Report

0 major / 1 minor

Summary. The manuscript extends localized Turán theorems to signed graphs. It establishes a vertex-localized Turán-type inequality for signed graphs and characterizes the extremal graphs attaining the bound. It derives a sharp upper bound on the largest eigenvalue of a signed graph in terms of localized Turán parameters and identifies the equality cases. It also generalizes a walk-based local spectral Turán inequality to signed graphs. The results are presented as generalizations and improvements of existing theorems for both unsigned and signed graphs.

Significance. If the derivations hold, the work supplies localized spectral bounds for signed graphs, which refines extremal spectral graph theory in the signed setting by adapting vertex-localized parameters and walk counts to the signed adjacency matrix. The explicit characterization of extremal graphs and the generalization of walk-based inequalities are potentially useful for further developments in signed spectral extremal problems.

minor comments (1)
  1. The abstract states that the results 'generalize and improve several existing theorems,' but the introduction should explicitly list the specific prior results (with citations) that are being improved and in what quantitative sense.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript defines vertex-localized Turán parameters and walk counts directly on the signed adjacency matrix, then adapts standard proof techniques from the unsigned case to obtain the stated inequalities and equality characterizations. No equation reduces a claimed prediction or bound to a fitted input by construction, no uniqueness theorem is imported solely via self-citation, and no ansatz is smuggled through prior work by the same authors. The central claims therefore rest on independent combinatorial arguments rather than tautological re-labeling of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are described or can be extracted.

pith-pipeline@v0.9.1-grok · 5610 in / 1096 out tokens · 26975 ms · 2026-06-28T05:31:09.509850+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 1 linked inside Pith

  1. [1]

    R. Adak, L. S. Chandran, Vertex-Based Localization of Tur´ an’s Theorem, arXiv: 2504.02806 (2025)

    arXiv 2025

  2. [2]

    N. Alon, M. Krivelevich, B. Sudakov, Tur´ an numbers of bipartite graphs and related Ramsey-type questions, Combin. Probab. Comput. 12 (2003) 477–494

    2003

  3. [3]

    M. Beck, T. Zaslavsky, The number of nowhere-zero flows on graphs and signed graphs, J. Combin. Theory Ser. B 96 (2006) 901–918

    2006

  4. [4]

    Bradaˇ c, A generalization of Tur´ an’s theorem, arXiv:2205.08923 (2022)

    D. Bradaˇ c, A generalization of Tur´ an’s theorem, arXiv:2205.08923 (2022)

    arXiv 2022

  5. [5]

    B. Bukh, M. Tait, Tur´ an numbers of theta graphs, Combin. Probab. Comput. 29 (2020) 495–507

    2020

  6. [6]

    J. A. Bondy, M. Simonovits, Cycles of even length in graphs, J. Combin. Theory Ser. B 16 (1974) 97–105. 17

    1974

  7. [7]

    Chaiken, A combinatorial proof of the all minors matrix tree theorem, SIAM J

    S. Chaiken, A combinatorial proof of the all minors matrix tree theorem, SIAM J. Algebraic Discrete Methods, 3 (1982) 319–329

    1982

  8. [8]

    Erd˝ os, T

    P. Erd˝ os, T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar. 10 (1959) 337–356

    1959

  9. [9]

    Harary, On the notion of balance of a signed graph, Michigan Math

    F. Harary, On the notion of balance of a signed graph, Michigan Math. J. 2 (2) (1953) 143–146

    1953

  10. [10]

    Heider, Attitudes and cognitive organization, J

    F. Heider, Attitudes and cognitive organization, J. Psychol. 21 (1) (1946) 107–112

    1946

  11. [11]

    Y. Hou, Z. Tang, D. Wang, On signed graphs with just two distinct Laplacian eigenvalues, Appl. Math. Comput. 351 (2019) 1–7

    2019

  12. [12]

    M. R. Kannan, S. Pragada, Signed spectral Tura´ n type theorems, Linear Algebra Appl. 663 (2023) 62–79

    2023

  13. [13]

    M. R. Kannan, H. Kumar, S. Pragada, Localization of spectral Tur´ an-type theorems, arXiv:2512.01409 (2025)

    arXiv 2025

  14. [14]

    K. Lan, J. Li, F. Liu, Remarks on the Largest Eigenvalue of a Signed Graph, Bull. Malays. Math. Sci. 46 (5) (2023) 157

    2023

  15. [15]

    B. Li, B. Ning, Localized and weighted versions of extremal problems, arXiv:2509.17055 (2025)

    arXiv 2025

  16. [16]

    F. Liu, S. Sun, Y. Wang, Q. Wu, A local Tur´ an inequality for walks and the spectral radius, arXiv:2605.02191 (2026)

    Pith/arXiv arXiv 2026

  17. [17]

    L. Liu, B. Ning, Local properties of the spectral radius and Perron vector in graphs, J. Combin. Theory Ser. B 176 (2026) 241–253

    2026

  18. [18]

    L. Liu, B. Ning, A new spectral Tur´ an theorem for weighted graphs and consequences, arXiv:2510.26410 (2025)

    arXiv 2025

  19. [19]

    Malec, C

    D. Malec, C. Tompkins, Localized versions of extremal problems, European J. Combin. 112 (2023) 103715

    2023

  20. [20]

    J. M. Moon, On independent complete subgraphs in a graph, Canadian J. Math. 20 (1968) 95–102

    1968

  21. [21]

    G. Sun, F. Liu, K. Lan, A note on eigenvalues of signed graphs, Linear Algebra Appl. 652 (2022) 125–131

    2022

  22. [22]

    Tur´ an, Eine Extremalaufgabe aus der Graphentheorie, Mat

    P. Tur´ an, Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941) 436– 452

    1941

  23. [23]

    W. Wang, Z. Yan, J. Qian, Eigenvalues and chromatic number of a signed graph, Linear Algebra Appl. 619 (2021) 137–145

    2021

  24. [24]

    L. Xie, X. Liu, Bounds for the largest eigenvalue and sum of Laplacian eigenvalues of signed graphs, arXiv:2512.01736 (2025)

    arXiv 2025

  25. [25]

    Zaslavsky, Signed graphs, Discrete Appl

    T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1) (1982) 47–74

    1982

  26. [26]

    Zaslavsky, Balanced decompositions of a signed graph, J

    T. Zaslavsky, Balanced decompositions of a signed graph, J. Combin. Theory Ser. B 43 (1987) 1–13. 18

    1987

  27. [27]

    Zaslavsky, Matrices in the theory of signed simple graphs, in Advances in Discrete Mathematics and Applications: Mysore, 2008, 207–229, Ramanujan Math

    T. Zaslavsky, Matrices in the theory of signed simple graphs, in Advances in Discrete Mathematics and Applications: Mysore, 2008, 207–229, Ramanujan Math. Soc. Lect. Notes Ser., 13, Ramanujan Math. Soc., Mysore, 2010

    2008

  28. [28]

    Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas, Electron

    T. Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas, Electron. J. Combin. 5 (2018), Dynamic Surveys 8, 524 pages

    2018

  29. [29]

    Zhao, X.-D

    K. Zhao, X.-D. Zhang, A localized approach for Tur´ an number of long cycles, J. Graph Theory 108 (2025) 582–607. 19

    2025