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arxiv: 2604.11428 · v1 · submitted 2026-04-13 · 🧮 math.CO

Spectral Tur\'an problem for tmathcal{K}₄⁻-free unbalanced signed graphs

Linfeng Xie , Xiaogang Liu This is my paper

Pith reviewed 2026-05-10 16:09 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C5005C35
keywords spectral Turán problemunbalanced signed graphstK4-extremal graphsspectral radiusindexsigned graphs
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The pith

The extremal unbalanced signed graphs maximizing the index and spectral radius among tK4--free ones with fixed order are characterized.

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

This paper establishes a characterization of the graphs that attain the maximum possible index and spectral radius in the class of unbalanced signed graphs on a given number of vertices that avoid the forbidden family tK4-. The family tK4- consists of unbalanced signed graphs whose underlying graphs contain t copies of K4, possibly sharing vertices. Knowing these extremal examples matters because it solves the spectral Turán problem in this signed setting, giving the precise upper bound on the spectral radius under the avoidance condition. A reader interested in extremal graph theory would see this as a natural extension of classical results to signed graphs where the spectrum depends on the signing.

Core claim

The authors characterize the extremal graphs that achieve the maximum index and spectral radius among all tK4--free unbalanced signed graphs with given order. Here tK4 denotes the family of all graphs consisting of t copies of K4 that are allowed to share vertices, and tK4- is the set of all unbalanced signed graphs whose underlying graphs are in tK4.

What carries the argument

The tK4- family of unbalanced signed graphs, which acts as the forbidden substructure whose avoidance constrains the graphs whose spectral radius is to be maximized.

If this is right

  • The maximum value of the spectral radius is attained precisely by the characterized graphs for each n.
  • These results apply uniformly to all orders n.
  • The characterization identifies the structures that are both free of tK4- and optimal in terms of spectral properties.

Where Pith is reading between the lines

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

  • If the characterization is complete, it could be used to derive asymptotic growth rates for the maximum spectral radius as n tends to infinity.
  • Similar methods might extend to other forbidden families in signed graphs or to other spectral parameters.
  • Checking the result computationally for small values of t and n would provide verification.

Load-bearing premise

The extremal characterization is assumed to hold for every order n independently of particular sign patterns or graph connectivity.

What would settle it

Finding a tK4--free unbalanced signed graph on n vertices with a spectral radius strictly larger than that of the identified extremal graphs would disprove the characterization.

Figures

Figures reproduced from arXiv: 2604.11428 by Linfeng Xie, Xiaogang Liu.

Figure 1
Figure 1. Figure 1: The signed graphs Γs,n and Σk,n. As shown in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

Let $tK_4$ denote the family of all graphs consisting of $t$ copies of $K_4$ that are allowed to share vertices and $t\mathcal{K}_{4}^{-}$ be the set of all unbalanced signed graphs whose underlying graphs are elements of $tK_4$. In this paper, we characterize the extremal graphs that achieve the maximum index and spectral radius among all $t\mathcal{K}_{4}^{-}$-free unbalanced signed graphs with given order.

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 characterizes the extremal unbalanced signed graphs on n vertices that are free of tK4- (the family of unbalanced signed graphs whose underlying graphs consist of t, possibly vertex-overlapping, copies of K4) and that maximize the spectral radius of the signed adjacency matrix (also called the index). The characterization identifies specific signed graphs, obtained from signed Turán constructions, as the unique maximizers for each n.

Significance. If the claimed characterization holds, the result provides a precise spectral extremal function and graph description for this forbidden family in the signed-graph setting. This extends classical and spectral Turán theory to unbalanced signed graphs and supplies a concrete, falsifiable prediction for the maximum index. The manuscript's use of standard signed-graph tools (eigenvalue interlacing, Perron-Frobenius analysis on the signed adjacency matrix, and domination arguments for disconnected graphs) is a methodological strength that makes the central claim verifiable in principle.

minor comments (3)
  1. §2, Definition 2.3: the precise notion of 'unbalanced' for a signed graph containing multiple K4 copies is stated only informally; an explicit sign-pattern condition or reference to the balance equation would remove ambiguity for readers.
  2. Theorem 1.1 and the statement of the extremal graphs: the uniqueness claim is asserted for all n ≥ some n0, but the small-n cases are not separately verified or tabulated; adding a short table or remark for n < 10 would strengthen the presentation.
  3. Figure 1 (or the corresponding construction diagram): the edge signs are not labeled on the drawing; explicit +/− annotations or a caption clarifying the signing would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on the spectral Turán problem for tK4--free unbalanced signed graphs and for recommending minor revision. No specific major comments were provided in the report, so we have no point-by-point responses to major issues. We are pleased that the referee recognized the methodological strengths of the paper, including the use of eigenvalue interlacing, Perron-Frobenius analysis, and domination arguments.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper characterizes extremal tK4--free unbalanced signed graphs via standard signed-graph spectral methods (eigenvalue bounds, Perron-Frobenius on the signed adjacency matrix, and constructions from signed Turán graphs). Definitions of tK4 (overlapping copies) and tK4- (unbalanced signed realizations) are taken as given without internal derivation. No equations or steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the central claim extends prior independent results uniformly to arbitrary n, with disconnected cases dominated by largest components. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all such elements are implicit in the definitions of signed graphs and the forbidden family.

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Reference graph

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