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Supersaturation in Nosal graphs: Triangles and books

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Pith's one-line read Every m-edge graph contains at least m(λ−√m) triangles, with equality only for complete bipartite graphs, and the same spectral surplus forces large books and sharp kite counts.

desk verdict Core spectral supersaturation results are new and sound; the general book-count theorem needs a completed proof of Claim 4.12, but this paper deserves a serious referee. read the letter →

arxiv 2607.16746 v1 pith:RGWDAK5R submitted 2026-07-18 math.CO

Hongzhang Chen , Yongtao Li , Quanyu Tang This is my paper · ORCID
classification math.CO MSC 05C5005C3505C30
keywords NosalgraphsspectralradiussupersaturationtrianglecountingbooksinkitesPerronvectoredge-spectralextremaltheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies what happens when a graph's spectral radius λ rises above the Nosal threshold √m. Its central result is a linear supersaturation law: every m-edge graph has at least m(λ−√m) triangles, with equality exactly for complete bipartite graphs. From this single inequality, the paper derives a spectral analogue of the Lovász–Simonovits theorem, a third discrete jump in the triangle count, and sharp bounds for the size of books and the number of kites in Nosal graphs. The paper also proves that every Nosal graph contains a book of size greater than ¼√m and at least (1/8−o(1))m copies of the kite C₄⁺, with the constant 1/8 best possible. If correct, these results establish the correct order and constants for edge-spectral supersaturation of triangles and books across all edge densities.

What carries the argument

The load-bearing object is the Perron-vector surplus identity: after normalizing the Perron eigenvector so the maximum coordinate is 1 at a vertex u*, with U = N(u*) and W the remaining vertices, λ²−m equals Σ_{uv∈E(U)}(x_u+x_v−1) − Φ, where Φ is a nonnegative 'deficit' supported on W. The half-missing-weights estimate Φ ≥ ½ Σ_{u∈U} x_u Y(u), where Y(u) is the total Perron weight of W-vertices not adjacent to u, lets the authors convert local bounds on bad edges (those with x_u+x_v>1) into an absorption argument: the positive contribution of E(U) is dominated by Φ, forcing λ ≤ √m in B_{r+1}-free graphs. For triangles, the machinery is a third-moment count combined with Cauchy interlacing and

What would settle it

Run an exhaustive search over all graphs with up to, say, 30 edges and check whether any non-bipartite graph violates t(G) ≥ m(λ−√m); a single counterexample would refute Theorem 1.3. More narrowly, for t = 3, directly verify the coefficient bound asserted in Claim 4.12 on the local configurations described in the proof—if it fails, the general book count and the kite constant lose their support.

Watch

Extended reading notes

Core claim

The paper's central claim is an exact linear supersaturation law: for every m-edge graph G with spectral radius λ, the triangle count t(G) is at least m(λ−√m), with equality if and only if G is complete bipartite. From this inequality, the paper derives a spectral Lovász–Simonovits analogue (λ ≥ √m + q forces t > qm), a third-layer triangle jump (λ ≥ 1+√(m−2) forces t ≥ m−2), a book of size greater than ¼√m in every graph above the Nosal threshold, and the sharp asymptotic constant 1/8 for counting kites C₄⁺. The proofs rest on a Perron-eigenvector decomposition that localizes the spectral surplus λ²−m around a maximum-weight vertex and shows that the surplus is absorbed by half the 'missing

Load-bearing premise

The general book-counting theorem for books of size t ≥ 3 relies on a local coefficient estimate (Claim 4.12) whose proof is only sketched as 'similar to that of Claim 4.8 with two substitutions'; if that estimate fails, the sharp constant 1/(t! 2^t) collapses.

Editorial extensions

If this is right

  • If Theorem 1.3 holds, then every graph with λ ≥ √m + q has more than qm triangles, a spectral Lovász–Simonovits analogue that works for sparse graphs where the classical vertex-density version gives nothing.
  • Theorem 1.1 gives a third discrete layer in the triangle jump phenomenon: exceeding 1+√(m−2) forces m−2 triangles, with extremal split graphs K₃∨((m−3)/3)K₁; a natural k-th layer pattern for larger cliques follows.
  • The book theorem (Theorems 1.5 and 1.8) yields a weak form of the triangular-edge conjecture, giving more than ½√m triangular edges in every Nosal graph.
  • The kite result (Theorem 1.9) determines the sharp asymptotic minimum (1/8)m copies of C₄⁺, and Theorem 4.11 extends this to all books B_t with the sharp constant 1/(t! 2^t).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The linear law t ≥ m(λ−√m) suggests a stability version: graphs with λ slightly above √m and near-minimal triangles should be close to complete bipartite; the exact equality case points to a natural spectral-stability question, possibly approachable by the same Perron-surplus framework.
  • The Perron-surplus and missing-weights machinery is likely exportable to other color-critical subgraphs of chromatic number 3, such as odd cycles, and to further edge-spectral Sidorenko-type counts where sharp constants remain open.
  • The paper's observation that the kite count does not jump at the second layer, with an explicit K_{n,n}-with-a-cycle construction attaining Θ(m^{3/2}) kites, suggests that higher-layer jump behavior is not universal; identifying exactly which subgraphs jump layer-by-layer is a testable programme emerging from this work.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper studies edge-spectral supersaturation in graphs whose spectral radius lies above the Nosal threshold sqrt(m). It proves four main results: (1) a sharp third-layer triangle jump: if lambda(G) >= 1 + sqrt(m-2), then t(G) >= m-2, with equality only for K_3 joined to an independent set; (2) a linear spectral Lovász--Simonovits-type bound t(G) >= m(lambda - sqrt(m)), with equality only for complete bipartite graphs, yielding t(G) > q m whenever lambda >= sqrt(m)+q; (3) every Nosal graph contains a book of size greater than (1/4)sqrt(m); and (4) every Nosal graph contains at least (1/8-o(1))m copies of the kite C_4^+, with 1/8 best possible. A general theorem, Theorem 4.11, claims the sharp asymptotic count (1/(t! 2^t)) m^{t/2} for books B_t for every fixed t. The triangle proofs use interlacing, power-mean inequalities, and Nikiforov's inequality; the book proofs are built on a Perron-vector surplus identity, a half-missing-weights estimate, and local structural decompositions.

Significance. If correct, the results are significant and timely. Theorem 1.3 is an elegant sharp linear bound that improves the Bollobás--Nikiforov inequality near the Nosal threshold and has the right split-graph behavior; Theorem 1.1 gives a clean third-layer jump in the spectral supersaturation hierarchy; Theorem 1.5 advances the book-size problem from 1/9 to 1/4; and Theorem 1.9 determines the sharp asymptotic kite constant. The main proofs are transparent, use standard tools, and contain no fitted parameters or circular reasoning. The general book-counting theorem 4.11 is an ambitious unification, but as written it rests on an only-sketched claim, and this is the main weakness of the manuscript.

major comments (1)
  1. [Section 4.3, Claim 4.12] Claim 4.12 is load-bearing for Theorem 4.11: inequality (28) and the rest of the proof collapse if the claimed local coefficient estimate fails. The proof of the claim is compressed to 'The proof is similar to that of Claim 4.8 with two substitutions'. For t >= 3 the substitutions are not immediate: one must verify that C(d,t)+C(q+1,t) is at least c_xi lambda^t when d+q >= xi lambda, and that the power-mean replacement gives a coefficient kappa < 1 in the bound L_v <= kappa lambda. These are plausible and my own check indicates they work, but the manuscript does not demonstrate them. As submitted, Theorem 4.11 is conditional on an unfinished proof. The claim should be stated and proved as a separate lemma with full details.
minor comments (4)
  1. [Section 4.1, equality case of Theorem 1.8] After Phi = 0, the text says 'Since G has no isolated vertices, every vertex w in W has Perron coordinate 1 and is adjacent to every vertex of U.' The adjacency to every vertex of U does not follow only from lack of isolated vertices; it follows by combining x_w = 1 with the Perron equation at w and lambda = sum_{u in U} x_u. This should be spelled out.
  2. [Section 3, proof of Theorem 1.3] The assertion 'it is well-known that lambda < sqrt(2m)' is used to guarantee r < sqrt(2) and is essential for Lemma 3.2. A one-sentence justification (e.g., from lambda^2 <= 2m with strictness for all graphs) would make the proof self-contained.
  3. [Section 4.2, Example 4.5] The phrase 'the only edge of G_t with at least two common neighbors' is correct for G_t = K_{s,t} plus one edge inside S, but it is easy to misread as applying to all edges of the complete bipartite graph; a brief clarification that 'at least two common neighbors' means two common neighbors that are both adjacent to the edge endpoints inside the same triangle, i.e., codegree at least 2, would help.
  4. [Throughout Section 4] Several claims in the book proofs are described as 'we obtain', 'it follows', or 'this is guaranteed by using (27) again' in places where the intermediate inequality is not displayed. For example, in Claim 4.13 the bound d_U(y) <= lambda should be justified explicitly from (27) and the binomial count. Most of these are fillable, but the presentation would be more rigorous if the displayed inequalities were carried through.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: main theorems are derived from stated spectral hypotheses with standard tools; the only flagged issue is a sketched proof of Claim 4.12, which is a proof gap, not a circular reduction.

full rationale

The paper's central claims are not circular. Theorem 1.3 follows from power-mean monotonicity (Lemma 3.1), the standard eigenvalue identity 6t(G)=Σλ_i^3, and the elementary inequality of Lemma 3.2: after setting λ=r√m, the bound 6t(G)≥λ^3−(2m−λ^2)^{3/2} is reduced to h(r)=r^3−(2−r^2)^{3/2}≥6(r−1), yielding t(G)≥m(λ−√m). This is a genuine analytic derivation, not a renamed input. Theorem 1.1 similarly uses Nikiforov's inequality, Cauchy interlacing from an embedded K4, and the elementary third-moment Lemma 2.2; no fitted parameter or conclusion-equivalent ansatz appears. The book and kite results are built on exact Perron-vector identities (Theorems 4.1 and 4.2), with B_{r+1}-freeness entering only through local degree bounds. The proof of Theorem 1.8 is a self-contained inequality chain, and Theorem 1.9 is a contradiction proof with a fully written local coefficient estimate (Claim 4.8) and a sharpness construction. The general book-count Theorem 4.11 does rely on Claim 4.12, whose proof is only sketched: the manuscript states 'The proof is similar to that of Claim 4.8 with two substitutions.' This is an omitted or compressed proof, and it is load-bearing for Theorem 4.11; however, it is not circular. The indicated substitutions (binomial counts in place of binomial(d,2)+binomial(q+1,2), and power-mean in place of Cauchy-Schwarz) are attempts to replicate an already proved estimate, not a covert use of the conclusion. A failure there would be a correctness gap, not a reduction of the theorem to its assumptions. The paper's self-citations are used for context, benchmarks, and prior bounds (e.g., [26], [49], [58], [28]); none of the central derivations imports a uniqueness theorem or a fitted parameter from those works. I find no step in which a prediction is equivalent by construction to its input, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

Central claims rest on standard spectral graph tools; no free parameters or invented entities. The main risk is not circularity but the compressed proof of Claim 4.12.

assumptions (6)
  • standard math Cauchy interlacing theorem for principal submatrices
    Used in proof of Theorem 1.1 to show three eigenvalues ≤ −1 from an embedded K_4 (Section 2).
  • standard math Perron–Frobenius theorem and existence of a Perron vector with max coordinate 1
    Central to Sections 4.1-4.3 and Theorems 4.1/4.2.
  • standard math Power-mean (Jensen) monotonicity of ℓ_p norms
    Lemma 3.1, used to bound the third-moment of non-principal eigenvalues in Theorem 1.3.
  • standard math Nikiforov's inequality λ^3 ≤ mλ + 2t for K_4-free graphs
    Lemma 2.1 (Nikiforov [45]), used in the K_4-free case of Theorem 1.1.
  • standard math Spectral bound λ < √(2m) for any graph
    Invoked in proof of Theorem 1.3 to set r<√2; standard bound.
  • domain assumption Finite simple graphs; isolated vertices ignored
    Standard convention stated in Section 1; isolated vertices do not affect λ or triangle count.

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Pith. "Pith review of Supersaturation in Nosal graphs: Triangles and books." pith.science (2026). https://pith.science/paper/RGWDAK5R

@misc{pith2026260716746,
  author       = {Pith},
  title        = {Pith review of: Supersaturation in Nosal graphs: Triangles and books},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RGWDAK5R}},
  note         = {Machine review of arXiv:2607.16746}
}
abstract

In this paper, we use the spectral surplus $\lambda(G) - \sqrt{m}$ to measure how far $G$ lies above the Nosal threshold, and prove the following edge-spectral supersaturation results for triangles and books. (a) Every graph $G$ with $m\ge 3$ edges and $\lambda(G) \ge 1 + \sqrt{m-2}$ contains at least $m-2$ triangles, with equality if and only if $G = K_3 \vee \tfrac{m-3}{3} K_1$. This can be viewed as the third-layer supersaturation in the jump phenomenon, after the first layer $t(G) \ge \lfloor \tfrac{1}{2}(\sqrt{m}-1) \rfloor$ proved by Ning and Zhai, and the second layer $t(G) \ge \tfrac{m-1}{2}$ by Zhang and Zhai. (b) Every $m$-edge graph $G$ satisfies $t(G) \ge m\bigl(\lambda - \sqrt{m}\,\bigr)$, with equality if and only if $G$ is complete bipartite. Consequently, $\lambda(G) \ge \sqrt{m} + q$ forces $t(G) > q m$ for every real $q > 0$. This is an edge-spectral counterpart of the Lov\'asz--Simonovits theorem, and it improves the Bollob\'as--Nikiforov bound $t(G) \ge \tfrac13 \lambda(\lambda^2 - m)$ in the range $\sqrt m \le \lambda(G) \le 1.3\sqrt m $. (c) Every $m$-edge Nosal graph $G$ contains a book of size greater than $\tfrac14 \sqrt{m}$. This improves two recent results on the booksize constant: $\tfrac{1}{24}$ proved by Li, Liu and Zhang, and $\tfrac19$ by Zhai, Li and Lou. This narrows the gap toward the conjectured optimal constant $\tfrac13$. (d) Every $m$-edge Nosal graph $G$ contains at least $\bigl(\tfrac{1}{8} - o(1)\bigr) m$ copies of the kite $C_4^+=B_2$, and the constant $\tfrac18$ is best possible. This determines the sharp asymptotic constant for counting $C_4^+$ and strengthens the $\Omega(m)$ bound of Li, Liu and Zhang.

Figures

Figures reproduced from arXiv: 2607.16746 by the authors.

Figure 1
Figure 1. The layer-by-layer jump phenomenon with extremal split graphs. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The flowchart of the proof of Theorem 1.8. Proof of Theorem 1.8. Assume that m ⩾ (4r) 2 and G is a Br+1-free graph with m edges. Throughout the proof, we may assume that G has no isolated vertices. We may assume that λ(G) ⩾ 4r, since otherwise λ(G) < 4r ⩽ √ m, we are done. Our goal is to prove that λ(G) ⩽ √ m, with equality if and only if G is a complete bipartite graph. We denote λ := λ(G). If r = 0, then G is tria… view at source ↗
Figure 3
Figure 3. The flowchart of the lower bound of Theorem [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Two problems on booksize and triangular edges in Nosal graphs

    math.CO 2026-07 accept novelty 8.0 of 10

    Any non-complete-bipartite graph with m edges and spectral radius at least √m has a book of size at least ρ(G)/3 and at least ρ(G) triangular edges.

  2. On a spectral booksize problem fo non bipartite graphs

    math.CO 2026-08 accept novelty 7.0 of 10

    Every sufficiently large m-edge non-bipartite graph without isolated vertices satisfying rho(G)^2 >= m-1+2/(rho(G)-1) is either an exceptional graph S+_{m,s} or contains a book of size at least (1/4-o(1)) sqrt(m), and...

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Reviewed August 1, 2026 · model on record in the stance chip above.