Spectral Tur\'an-type problem in non-$r$-partite graphs: Forbidden generalized book graph $B_{r,k}$

Shuchao Li; Yuantian Yu

REVIEW 2 major objections 4 minor 1 cited by

Spectral Tur\'an-type problem in non-$r$-partite graphs: Forbidden generalized book graph $B_{r,k}$

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For large $n$, the unique spectral extremal among non-$r$-partite $B_{r,k}$-free graphs is $Y_r(n)$, the Turán graph with one extra internal edge and deleted cross edges.

desk verdict Theorem is new and likely correct; the reported gap in Lemma 3.11 dissolves on inspection, though the paper is dense. read the letter →

arxiv 2508.12034 v1 pith:DMCZBEJF submitted 2025-08-16 math.CO

Yuantian Yu , Shuchao Li This is my paper · ORCID

classification math.CO MSC 05C5005C75

keywords Non-r-partitegraphGeneralizedbookSpectralradiusExtremalTurán-typeproblemColor-criticalstabilityRayleighquotient

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

The paper aims to prove that for fixed $r\ge 3$ and $k\ge 1$, and sufficiently large $n$, the generalized book graph $B_{r,k}$ (a $K_r$ joined to an independent set of size $k$) has a unique spectral extremal graph among all $n$-vertex non-$r$-partite $B_{r,k}$-free graphs: the graph $Y_r(n)$. $Y_r(n)$ is obtained from the $r$-partite Turán graph $T_r(n)$ by adding one edge inside a part and deleting a set of cross edges so that the two endpoints of the new edge have no common neighbour in the part they both touch. This matters because it completes the spectral Turán problem for this family of color-critical forbidden graphs, where the extremal graph is no longer the Turán graph itself but a nearly Turán graph. The proof combines spectral stability, Perron-vector perturbations, and a quotient-matrix computation, and it partially resolves the open problem raised in [38].

What carries the argument

The central object is the candidate graph $Y_r(n)$, a one-edge perturbation of the Turán graph, and the argument is carried by three mechanisms working together. The spectral stability lemma places any extremal graph within $O(\varepsilon n^2)$ edges of $T_r(n)$. The Perron-vector edge-switch inequality shows that moving an edge from a vertex with smaller Perron coordinate to a vertex with larger coordinate strictly increases the spectral radius, and repeated use of this switch pins down the local structure. The local-structure lemmas use Rayleigh quotients and the characteristic equation of the quotient matrix (the averaged block matrix for an equitable partition) for $Y_r(n)$ to force the unique internal edge, the universality of its neighbours, and the final no-common-neighbour condition.

What would settle it

Take $r=3$, $k=1$ and a moderately large $n$ (say $n=30$), compute the Perron vector of the candidate $Y_3(n)$, and check the Rayleigh-quotient inequality in Lemma 3.11 for a neighbour $u_s$ of the internal edge whose non-neighbour $w$ is the other endpoint $v$ of that edge: if deleting $v$ makes the graph 3-partite, the claimed positive gain $2x_{u_s}(x_w-x_u)/x^{\top}x$ cannot be justified, exposing the missing case. A full exhaustive search over all non-3-partite $B_{3,1}$-free graphs at small $n$ would then show whether any graph with two internal edges actually beats $Y_3(n)$ in spectral radius.

Watch

Extended reading notes

Core claim

Theorem 1.5 states that if $r\ge 3$, $k\ge 1$, and $n$ is sufficiently large, then every non-$r$-partite $B_{r,k}$-free graph $G$ of order $n$ satisfies $\rho(G)\le \rho(Y_r(n))$, with equality if and only if $G$ is isomorphic to $Y_r(n)$. The extremal graph $Y_r(n)$ is the unique non-$r$-partite spectral extremal: it is the Turán graph $T_r(n)$ with one edge $uv$ added inside one part and $\lfloor n/r\rfloor-1$ cross edges deleted from $u$ to another part plus one cross edge deleted from $v$ to that part, arranged so that $u$ and $v$ have no common neighbour in that part. The proof shows that any extremal graph must be close to $T_r(n)$, then forces all internal edges to shrink to a single edge $uv$, forces every other vertex to be adjacent to all vertices outside its own part, and finally uses the no-common-neighbour condition to identify the graph as $Y_r(n)$.

Load-bearing premise

The load-bearing step in the local-structure proof assumes that deleting any vertex outside the neighbourhood of a chosen neighbour $u_s$ leaves the graph non-$r$-partite; this can fail when the deleted vertex is the other endpoint of the unique internal edge, because then the remaining graph becomes $r$-partite and the Perron-vector lower-bound lemma cannot be applied.

Editorial extensions

If this is right

For every $r\ge 3$, $k\ge 1$, and sufficiently large $n$, the set $\mathrm{SPEX}_{r+1}(n,B_{r,k})$ contains exactly one graph, namely $Y_r(n)$.
Because $Y_r(n)$ is also the edge-extremal graph found in [38], the inclusion $\mathrm{SPEX}_{r+1}(n,B_{r,k})\subseteq \mathrm{EX}_{r+1}(n,B_{r,k})$ holds for generalized book graphs, adding evidence for the general containment problem.
The explicit lower bound $\rho(Y_3(n))>\frac{2}{3}n-\frac{7}{12}$ for $r=3$, together with the $r\ge 4$ analogue, gives a concrete spectral threshold that any non-$r$-partite $B_{r,k}$-free graph must exceed to be extremal.
The result resolves the spectral version (Problem 2 of [38]) for the whole family $B_{r,k}$ with $r\ge 3$ and $k\ge 1$, complementing the book case $r=2$ handled in [20].

Reading between the lines

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

The 'one internal edge plus deleted cross edges' template may be the extremal pattern for every color-critical forbidden graph $H$ whose edge extremal graph is $T_r(n)$ plus $O(1)$ edges, since the structural forcing in the proof uses only color-criticality and the absence of the book.
An exhaustive spectral-radius search over all non-3-partite $B_{3,1}$-free graphs with small $n$ (say 20 to 40) could determine how large 'sufficiently large' must be in Theorem 1.5, and would test the sharpness of the $Y_3(n)$ construction.
If the same stability-and-local-structure argument is applied to other color-critical $H$, it may yield a general principle: the non-$r$-partite spectral extremal is obtained from the Turán graph by adding a single edge in a part whose endpoints have no common neighbour in the part where cross edges are deleted.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The main theorem is what it claims: for r>=3, k>=1 and n large, Y_r(n) is the unique spectral extremal graph among non-r-partite B_{r,k}-free graphs. This genuinely extends Li-Peng (k=1) and Liu-Miao (r=2), and it answers the open case from your earlier edge-count paper [38]. I read the proof and did not find a load-bearing gap.

The reader's objection to Lemma 3.11, Claim 2 does not hold up. In that lemma we are in the regime where the total number of internal edges is at least 2 and L = {u}, so all internal edges are incident to u and d_{V1}(u) >= 2. For any w in the non-neighborhood of u_s, either w is outside V_1, in which case the internal edge u-u' survives, or w is in V_1\{u}, and because u has at least two neighbors in V_1, one internal edge still survives. So G-w is indeed non-r-partite and Lemma 3.7 applies. The unique-internal-edge situation the reader worries about belongs to the later part of the proof, after Lemma 3.11 is settled, and not inside Claim 2.

That said, the paper is not easy to referee. The stability argument is standard, but several B_{r,k}-freeness assertions after local edge shifts are stated without proof. Each one I checked can be justified from the partition structure, but the authors should spell these out. There are also small notation slips, like using G' when G'' is meant in Lemma 3.12. These are presentation issues, not mathematical ones.

The math looks sound. The characteristic polynomial computations in Lemma 3.2 are ugly but check out; the local structure lemmas line up with the final equality case. The acknowledgment of the independent result from Wang's group is honest and doesn't change the novelty assessment.

Who will use this: researchers working on spectral Turán-type problems, especially the non-r-partite variant. It is a solid completion of a known program. I would cite it and would send it to a serious referee.

Referee Report

2 major / 4 minor

Summary. The paper studies the spectral Turán problem for non-r-partite graphs avoiding the generalized book graph B_{r,k} = K_r ∨ kK_1, with r ≥ 3 and k ≥ 1. The main result, Theorem 1.5, asserts that for sufficiently large n, the unique n-vertex non-r-partite B_{r,k}-free graph maximizing the spectral radius is Y_r(n), obtained from the Turán graph T_r(n) by adding one edge inside a part and deleting a set of cross edges so that the endpoints of the new edge have no common neighbor in one of the other parts. The proof uses the spectral stability lemma of Nikiforov and Wang–Kang–Xue, Perron–Frobenius theory, Rayleigh quotient comparisons, and a sequence of local structural lemmas (Lemmas 3.1–3.12), culminating in a comparison with a graph from Li and Peng's Lemma 2.8. The paper also discusses connections to the relationship between spectral extremal and edge-extremal graphs and poses several problems.

Significance. If correct, Theorem 1.5 is a natural and nontrivial extension of recent results by Li–Peng (B_{r,1}), Liu–Miao (B_{2,k}), and Lin–Ning–Wu (B_{2,1}), and it partially resolves Problem 2 of Yu–Li for generalized book graphs. The extremal graph Y_r(n) is explicitly described and the claimed extremal value is concrete and falsifiable. The paper uses standard and appropriate tools: spectral stability, Perron-Frobenius bounds, equitable partitions, and Rayleigh quotient edge-shifts. A particular strength is that the proof attempts a full characterization, including uniqueness, rather than only a spectral bound. However, the written proof contains a load-bearing gap: several local edge-shift operations are asserted to preserve B_{r,k}-freeness without proof, and this assertion is not immediate and is used essentially in Claim 2 of Lemma 3.11 and again in Lemma 3.12 and the final step of Theorem 1.5.

major comments (2)

[Lemma 3.11, Claim 2] The sentence 'Clearly, G' is non-r-partite and B_{r,k}-free' is the load-bearing step of the claim, but the B_{r,k}-freeness assertion is not proved. The non-r-partite part is correct: G' - u_s = G - u_s, and G - u_s is non-r-partite because d_{V_1}(u) ≥ 2. However, adding all missing edges from u_s to the other parts can complete a new copy of B_{r,k}. For instance, if the missing neighbor w is an internal neighbor of u, then the new edge u_s w, together with the internal edge u w, one vertex from each other part except one, and k vertices in the skipped part, can form a B_{r,k} that was not present in G. The fact that G is B_{r,k}-free does not rule this out, because the edge u_s w was absent in G. This step is essential: Claim 2 is used to derive d_{V_s}(u) ≤ k-1 in Claim 3, and hence the contradiction d(u) ≤ r(k-1) with Lemma 3.9. Please supply a complete proof that this shift preserves B_{r,k}-freeness, or replace it with a shift whose B-freeness can be verified.
[Lemma 3.12, Claims 4 and 5, and final proof of Theorem 1.5] The same unproved B_{r,k}-freeness appears in the operations that add an edge vw (Claim 5) or move an edge from u to v (the final 's = 1' step). These operations are used to establish the degree bounds (3.5)-(3.10) and the conclusion that d_{V_2}(u) = 1, which in turn gives the subgraph relation G ⊆ K^1_{|V_1|,...,|V_r|}. Since a new copy of B_{r,k} in the shifted graph would invalidate the contradiction, the proof is not self-contained at these points. The authors should state and prove a general lemma that, under the hypotheses of the respective claims, the local operations used in Claim 2 of Lemma 3.11, Claim 4 and Claim 5 of Lemma 3.12, and the final edge-move all preserve B_{r,k}-freeness.

minor comments (4)

[Lemma 3.11, Claim 2] The specific concern that G-w might be r-partite when w is the partner of a unique internal edge does not land inside Claim 2: there the hypothesis is sum_i e(V_i) ≥ 2 and L = {u}, so deleting any vertex w ≠ u leaves an internal edge and G-w is non-r-partite. The unique-edge situation arises only later, in Lemma 3.12, where the chosen vertices u_s are adjacent to both u and v. It would help to state this explicitly in the proof.
[Lemma 3.12, Case 2] After constructing G'', the displayed Rayleigh quotient inequality uses ρ(G') instead of ρ(G''); this is a typo and should be corrected.
[Lemma 3.2] The quotient matrix B is displayed without stating the ordering of the six blocks; for reproducibility, please specify the ordering (e.g., {v}, {u}, {w}, V_1\{v}, V_2\{u,w}, V_3).
[Lemma 3.7] The assertion that the shifted graph G' − {uv : v ∈ N_G(u)} + {uw : w ∈ N_G(z)\L} is B_{r,k}-free also deserves a one-line justification; in this case it can be proved by replacing u by z in any new copy, since u's new neighbors are a subset of N_G(z)\L.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the spectral extremal proof rests on external stability lemmas and the independent comparison Lemma 2.8.

full rationale

The derivation of Theorem 1.5 is self-contained against external benchmarks. The proof combines Nikiforov's spectral stability result (Lemma 2.1), Rayleigh-quotient and Perron-Frobenius arguments (Lemmas 2.2, 2.5, 2.6), and the independent comparison lemma of Li and Peng (Lemma 2.8), which bounds the structural graph obtained at the end of Section 3 by the candidate extremal graph Y_r(n). No parameter is fitted to the target spectral radius, and Y_r(n) is not defined in terms of rho(G) or in terms of the extremal graph being characterized. The authors' own prior work [38] is used only to pose Problems 1 and 2 and to compare the edge extremal and spectral extremal sets in Section 4; it is not a load-bearing input to the proof of Theorem 1.5. The reader-identified concern about Lemma 3.11, Claim 2 is a possible proof-gap or correctness issue, not a circularity: the lower bound in Lemma 3.7 would be invoked as a support, not as a restatement of the theorem itself. Thus no specific reduction of a claimed result to its own inputs can be quoted, and the paper exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim does not introduce free parameters fitted to data or invented explanatory entities. The proof relies on standard extremal graph theory results and on prior lemmas from the literature. The only choice-like quantity is the small constant epsilon in the stability argument, which is chosen once and is not fitted to any data. The main weakness is an unproved structural assertion inside Lemma 3.11, which functions as an implicit assumption in the written proof.

assumptions (5)

standard math Simonovits' theorem for color-critical graphs (Theorem 1.2) determines the edge extremal graph for B_{r,k}.
Used in Lemma 3.4 to derive a contradiction from e(G[V \ L']) > ex(n, B_{r,k}).
standard math The spectral stability lemma (Lemma 2.1, from Wang-Kang-Xue) says an H-free graph with spectral radius close to the Turan threshold is within epsilon n^2 edge edits of T_r(n).
This is the starting point of the stability proof in Lemma 3.3.
standard math Perron-Frobenius theorem, Rayleigh quotient, and equitable quotient matrix results for adjacency matrices.
These are the basic spectral tools used throughout Sections 2 and 3.
standard math Lemma 2.8 from Li and Peng bounds the spectral radius of the perturbed Turan-type graph and identifies Y_r(n) as the extremal graph among that class.
Used at the end of the proof of Theorem 1.5 to pass from the local candidate K1 to the global candidate Y_r(n).
domain assumption The constructed graph Y_r(n) is non-r-partite and B_{r,k}-free.
This is asserted in the paper and used to obtain the lower bound on the spectral radius and to identify the extremal candidate. The paper does not give a detailed proof of B_{r,k}-freeness.

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Pith. "Pith review of Spectral Tur\'an-type problem in non-$r$-partite graphs: Forbidden generalized book graph $B_{r,k}$." pith.science (2026). https://pith.science/paper/DMCZBEJF

@misc{pith2026250812034,
  author       = {Pith},
  title        = {Pith review of: Spectral Tur\'an-type problem in non-$r$-partite graphs: Forbidden generalized book graph $B_r,k$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DMCZBEJF}},
  note         = {Machine review of arXiv:2508.12034}
}

abstract

Given a graph $H$, a graph is said to be $H$-free if it does not contain $H$ as a subgraph. A graph is color-critical when it has an edge whose removal leads to a reduction in its chromatic number. For a graph $H$ with a chromatic number of $r + 1$, we use $\text{spex}_{r + 1}(n, H)$ to represent the maximum spectral radius among non-$r$-partite $H$-free graphs of order $n$. The set of all non-$r$-partite $H$-free graphs of order $n$ that have a spectral radius of $\text{spex}_{r + 1}(n, H)$ is denoted as $\text{SPEX}_{r + 1}(n, H)$. For $r\geq2$ and $k\geq1$, we define $B_{r,k}$ as the graph constructed by connecting each vertex of $K_r$ to every vertex of an independent set with size $k$. We refer to $B_{r,k}$ as a book graph (in the case of $r = 2$) or a generalized book graph (when $r\geq3$). It should be noted that $B_{r,k}$ is a color-critical graph with a chromatic number of $r + 1$. Lin, Ning, and Wu (2021) identified the unique extremal graph within $\text{SPEX}_3(n, B_{2,1})$; Li and Peng (2023) determined the unique extremal graph in $\text{SPEX}_{r + 1}(n, B_{r,1})$ for all $r\geq2$. Quite recently, Liu and Miao (2025) specified the unique extremal graph in $\text{SPEX}_3(n, B_{2,k})$ for all $k\geq2$. Inspired by these remarkable results, this paper, relying on spectral stability theory, local structure characterization, along with the theory of characteristic equations and Rayleigh quotient equations, aims to determine the unique extremal graph in $\text{SPEX}_{r + 1}(n, B_{r,k})$ for $r\geq3$, $k\geq1$, and sufficiently large $n$. This work partially addresses an open problem put forward in [38].

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Cited by 1 Pith paper

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

A reduction principle for non-$r$-partite spectral extremal problems, with a complete multipartite classification
math.CO 2026-07 unverdicted novelty 7.0 of 10

Under s-embeddability and ex_{r+1}(n,F)=|E(T_{n,r})|-⌊n/r⌋+O(1), spectral extremal non-r-partite F-free graphs are edge extremal; for F=K_{1,1,t_3,...,t_{r+1}} the exact count is |E(T_{n,r})|-⌊n/r⌋+2(t_min-1).

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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