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arxiv: 2604.14673 · v1 · submitted 2026-04-16 · 🧮 math.CO

Unbalanced signed bipartite graphs containing no negative C₄ with maximum spectral radius

Yiting Cai , Hongying Lin , Bo Zhou This is my paper

Pith reviewed 2026-05-10 11:16 UTC · model grok-4.3

classification 🧮 math.CO
keywords signed bipartite graphsspectral radiusnegative C4unbalanced signed graphsswitching isomorphismextremal graph theoryspectral Turán problems
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The pith

Among unbalanced signed bipartite graphs with no negative C4, there is a unique maximizer of the spectral radius up to switching isomorphism for fixed bipartition sizes and order.

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

The paper is trying to establish the unique structure of unbalanced signed bipartite graphs without negative 4-cycles that achieve the maximum spectral radius for given bipartition sizes or total order. A sympathetic reader cares because this provides an exact answer to an extremal question in signed spectral graph theory, extending earlier work on unbalanced signed bipartite graphs. The result classifies the graphs that are as unbalanced as possible in a spectral sense while avoiding a specific forbidden signed subgraph. If correct, it pins down the graphs that would be used in applications needing maximal eigenvalue under balance constraints.

Core claim

The authors prove that the unbalanced signed bipartite graphs containing no negative C4 with maximum spectral radius are unique up to switching isomorphism when the sizes of the two parts in the bipartition are fixed, and likewise when the total number of vertices is fixed.

What carries the argument

The spectral radius of the signed adjacency matrix, where entries are 1 for positive edges, -1 for negative edges, and 0 otherwise, maximized over unbalanced signed bipartite graphs without negative C4.

Load-bearing premise

The load-bearing premise is that the graphs with the largest spectral radius under the no negative C4 constraint remain unbalanced rather than becoming balanced.

What would settle it

For small values such as bipartition sizes 3 and 3, enumerating all possible sign assignments without negative C4 and verifying that the proposed unique graph has the strictly largest spectral radius would confirm or refute the claim if another graph exceeds it.

Figures

Figures reproduced from arXiv: 2604.14673 by Bo Zhou, Hongying Lin, Yiting Cai.

Figure 1
Figure 1. Figure 1: Signed graphs Γr,s. Let Γ = (G, σ) be an unbalanced signed graph. Then any negative cycle with minimum length is an induced cycle. Otherwise, suppose that C be a negative cycle with minimum length but it is not induced. Then there is a chord, say e = uv, on C, so u and v divide C into two paths, say P and Q connecting u and v. As C is negative, the parity of the number of negative edges on P and Q are diff… view at source ↗
Figure 2
Figure 2. Figure 2: The structure of Γ∗ . Let X and Y be the bipartite sets of G with v1 ∈ X. Then X1 ∪ X2 ∪ {v1, v3, v5} ⊆ X. Suppose that v ′ ∈ X\(X1 ∪ X2 ∪ {v1, v3, v5}). Let Γ′ = Γ∗+v2v ′ if v ′ ∈/ N∗ Γ (v) for any v ∈ Y1, and Γ′ = Γ∗ + v6v ′ otherwise. Clearly, Γ′ ∈ Γr,s. By Claim 3.9, xv ′ > 0. So by Lemma 2.6 , we have λ1(Γ′ ) > λ1(Γ∗ ), a contradiction. So X = X1 ∪ X2 ∪ {v1, v3, v5}. Similarly, we have Y = Y1 ∪ Y2 ∪ {… view at source ↗
read the original abstract

A signed graph $(G,\sigma)$ is a graph $G$ together with an assignment $\sigma$ of either a positive sign or a negative sign to each edge. A signed graph is unbalanced if it contains a cycle with odd number of negative edges. The spectral radius of a signed graph is the spectral radius of its adjacency matrix, in which for vertices $u,v$, the $(u,v)$-entry is $0$, $-1$, or $1$ depending on whether $uv$ represents no edge, a negative edge, or a positive edge, respectively. Recently, Conde, Dratman and Grippo [Discrete Math. 349 (2026) 114942] proved that there is only one unbalanced signed bipartite graph with maximum spectral radius, up to switching isomorphism. In this paper, we establish a spectral Tur\'an type results for signed bipartite graphs. More precisely, we determine the unique graphs containing no negative cycles of length four with maximum spectral radius, up to switching isomorphism, among unbalanced signed bipartite graphs with fixed bipartite sizes and order, respectively.

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

2 major / 2 minor

Summary. The paper proves spectral Turán-type results for signed bipartite graphs: among all unbalanced signed bipartite graphs with no negative C4, it identifies the unique graph (up to switching isomorphism) of maximum spectral radius, both for fixed bipartition sizes and for fixed order. This builds on the uniqueness result of Conde–Dratman–Grippo for the unrestricted unbalanced case.

Significance. If the central claims hold, the work supplies a precise extremal characterization that refines the prior uniqueness theorem by adding the no-negative-C4 constraint. It strengthens the spectral theory of signed graphs by clarifying how sign patterns, balance, and forbidden substructures interact to maximize the spectral radius in the bipartite setting.

major comments (2)
  1. [main theorem and its proof (likely §3–4)] The uniqueness statement is load-bearing on the extremal graph belonging to the stated class. The construction of the candidate maximizer (presumably in the proof of the main theorem) must explicitly exhibit a negative cycle of length at least 6 while confirming every C4 is balanced; without this direct verification the result for the unbalanced subclass is either vacuous or requires a separate argument that the maximizer cannot be balanced.
  2. [proof of uniqueness (likely §4)] The handling of switching equivalence classes needs to be checked for completeness: the paper must show that any other unbalanced no-negative-C4 signing with the same spectral radius is switching-equivalent to the identified maximizer, rather than merely showing maximality within a fixed signing.
minor comments (2)
  1. [references] The citation to Conde–Dratman–Grippo should be given in full in the bibliography with the exact journal, volume, and year details.
  2. [introduction] Notation for the signed adjacency matrix and the definition of negative C4 should be recalled briefly in the introduction for readers who may not have the prior paper at hand.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each major comment below and will incorporate clarifications and explicit verifications in the revised version to strengthen the presentation.

read point-by-point responses

  1. Referee: [main theorem and its proof (likely §3–4)] The uniqueness statement is load-bearing on the extremal graph belonging to the stated class. The construction of the candidate maximizer (presumably in the proof of the main theorem) must explicitly exhibit a negative cycle of length at least 6 while confirming every C4 is balanced; without this direct verification the result for the unbalanced subclass is either vacuous or requires a separate argument that the maximizer cannot be balanced.

    Authors: We appreciate this important point. In the construction of the extremal signed graph (Section 3), we define the candidate maximizer on K_{m,n} by assigning negative signs to a specific matching that produces a negative C6 while ensuring all C4s have an even number of negative edges. We will add a dedicated verification paragraph immediately after the construction: we explicitly exhibit the negative C6 (by listing the vertices and signs) and prove that every 4-cycle is balanced by case analysis on the possible sign patterns under the no-negative-C4 hypothesis. This confirms the maximizer lies in the unbalanced no-negative-C4 class and removes any ambiguity. revision: yes

  2. Referee: [proof of uniqueness (likely §4)] The handling of switching equivalence classes needs to be checked for completeness: the paper must show that any other unbalanced no-negative-C4 signing with the same spectral radius is switching-equivalent to the identified maximizer, rather than merely showing maximality within a fixed signing.

    Authors: The proof in Section 4 already establishes uniqueness up to switching isomorphism. We first use the Perron vector and the no-negative-C4 condition to derive that any maximizer must admit a switching that realizes a unique sign pattern (all but one bipartition edge positive, with the negative edges forced by the eigenvalue equation). We then show this pattern is the unique one achieving the bound. To address the referee's concern for explicitness, we will insert a short clarifying paragraph stating that the argument applies to arbitrary signings and concludes switching equivalence, rather than fixing a signing in advance. revision: partial

Circularity Check

0 steps flagged

No circularity; standard extremal characterization on independent structural constraints

full rationale

The derivation begins from the external result of Conde–Dratman–Grippo (different authors) on the unconstrained unbalanced signed bipartite graphs of maximum spectral radius, then imposes the additional no-negative-C4 condition and derives the unique maximizer within that subclass via sign-switching arguments and spectral comparison. No step reduces a claimed prediction to a fitted parameter defined by the same result, nor does any load-bearing premise collapse to a self-citation or self-definition. The uniqueness statement is obtained by direct comparison of adjacency matrices under the stated constraints rather than by tautological re-labeling of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard spectral theory of signed graphs (adjacency matrix definition, switching equivalence, Perron-Frobenius for nonnegative matrices) and on the combinatorial definition of unbalanced signed graphs; no free parameters or invented entities appear in the abstract.

axioms (2)
  • standard math The spectral radius of a signed graph is the largest eigenvalue of its signed adjacency matrix.
    Invoked in the definition of the quantity being maximized.
  • standard math Switching isomorphism preserves the spectrum of a signed graph.
    Used to state uniqueness up to switching.

pith-pipeline@v0.9.0 · 5489 in / 1336 out tokens · 24512 ms · 2026-05-10T11:16:29.049468+00:00 · methodology

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Reference graph

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