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Spectral Tur\'{a}n problem of non-bipartite graphs: Forbidden books

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Pith's one-line read The paper finds the exact maximum spectral radius among non-bipartite graphs with no book of r+1 triangles, for every r and large n.

desk verdict A genuinely new non-bipartite spectral Turán theorem with a repairable but load-bearing gap in one case of the proof. read the letter →

arxiv 2506.04884 v1 pith:R6D3HBCK submitted 2025-06-05 math.CO

Ruifang Liu , Lu Miao This is my paper
classification math.CO MSC 05C5005C35
keywords spectralextremabookgraphnon-bipartitegraphsradiusextremaladjacencymatrixleadingeigenvectormatchingnumber
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

For a fixed integer $r\geq1$ and every order $n\geq 8(r^2+r+4)$, the paper determines the exact maximum spectral radius (the largest eigenvalue of the adjacency matrix) among all non-bipartite graphs on $n$ vertices that contain no book $B_{r+1}$, meaning no set of $r+1$ triangles sharing a common edge. The unique extremal graph is $K^{r,r}_{\lfloor(n-1)/2\rfloor,\lceil(n-1)/2\rceil}$: take the most balanced complete bipartite graph on $n-1$ vertices and add one new vertex that has exactly $r$ neighbours in each part. This answers the non-bipartite version of the extremal spectral problem for books when $r\geq1$, and it exposes a sharp change from the known $r=0$ case, where the extremal graph is a balanced complete bipartite graph with one edge subdivided. A direct corollary is a spectral forcing condition: any non-bipartite graph whose spectral radius exceeds that of the candidate must contain a book $B_{r+1}$.

What carries the argument

The proof is carried out on a partition $V(G^*)=\{u^*\}\cup A\cup B$, where $u^*$ is a vertex of largest leading-eigenvector entry, $A=N(u^*)$ is its neighbourhood, and $B$ is the set of remaining vertices. The load-bearing tool is the residual index $\Gamma_w=d_A(w)(x_{u^*}-x_w)$ for $w\in B$, which measures how much the leading eigenvector drops on vertices of $B$ with many neighbours in $A$; accumulating these drops and comparing with a lower bound on $\rho(G^*)^2$ rules out configurations in which $A$ has no edges. Additional machinery includes a degree-versus-matching-number bound (Lemma 2.1) that controls $e(A)$, a local edge-swap lemma (Lemma 2.2) that shows certain one-edge modifications increase the spectral radius while preserving both non-bipartiteness and $B_{r+1}$-freeness, and an equitable-partition computation that gives the quintic equation for the spectral radius of the candidate graph.

What would settle it

Take $r=1$ and $n=48$, enumerate or heuristically maximize the spectral radius over non-bipartite graphs on 48 vertices with no two triangles sharing an edge, and check whether any graph beats $K^{1,1}_{23,24}$; a single counterexample would refute the theorem. More broadly, for each $r\geq1$ and $n\geq8(r^2+r+4)$, compare the candidate against a balanced complete bipartite graph with one extra edge inside a part, with a pendant triangle, or with a different apex-degree pattern, since any such graph with larger spectral radius would disprove the claimed uniqueness.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: for $r\geq1$ and $n\geq 8(r^2+r+4)$, if $G$ is a non-bipartite $B_{r+1}$-free graph of order $n$, then $\rho(G)\leq\rho(K^{r,r}_{\lfloor(n-1)/2\rfloor,\lceil(n-1)/2\rceil})$, with equality if and only if $G$ is isomorphic to $K^{r,r}_{\lfloor(n-1)/2\rfloor,\lceil(n-1)/2\rceil}$. Here the forbidden book $B_{r+1}$ consists of $r+1$ triangles sharing a common edge, and the extremal graph is obtained from the complete bipartite graph $K_{\lfloor(n-1)/2\rfloor,\lceil(n-1)/2\rceil}$ by adding a new vertex $v_0$ adjacent to exactly $r$ vertices in each part. The paper also computes $\rho(K^{r,r}_{s,t})$ as the largest root of the quintic $x^5-(2r+st)x^3-2r^2x^2+(2str-sr^2-tr^2)x=0$, and proves structural rigidity of the extremal graph: with respect to the vertex of largest leading-eigenvector entry, the neighbourhood set and its complement each have matching number at most one, and the sum of the two matching numbers is exactly one.

Load-bearing premise

The load-bearing premise is that a maximal counterexample can always be locally rearranged — by moving or adding an edge incident to a vertex with smaller leading-eigenvector entry — without accidentally creating the forbidden book or making the graph bipartite; if one of these rearrangements fails in an unexamined subcase, the uniqueness conclusion does not follow.

Editorial extensions

If this is right

  • If the theorem is correct, any non-bipartite $n$-vertex graph with spectral radius greater than $\rho(K^{r,r}_{\lfloor(n-1)/2\rfloor,\lceil(n-1)/2\rceil})$ must contain a book $B_{r+1}$; this is a sharp spectral forcing condition for books.
  • The extremal graph is uniquely characterized for every $r\geq1$, so the non-bipartite spectral extremal problem for books is now completely solved in the regime $n\geq8(r^2+r+4)$.
  • The result completes the comparison across book sizes: for $r=0$ the extremal graph is a balanced complete bipartite graph with one edge subdivided, while for every $r\geq1$ it is the apex graph with $r$ neighbours in each part.
  • The proof's structural lemmas imply that in any extremal graph the two sides of the partition are almost completely joined, with exactly one internal edge inside the neighbourhood set or inside its complement, yielding the rigid shape of the candidate.

Reading between the lines

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

  • Beyond the paper's range, the bound $n\geq8(r^2+r+4)$ is forced by inequalities rather than by an observed phase transition, so the same extremal graph likely persists for smaller $n$; locating the true threshold is a concrete next question.
  • The residual-index technique should transfer to other 3-chromatic color-critical forbidden subgraphs, yielding extremal graphs shaped as balanced complete bipartite graphs with a small non-bipartite cap; testing odd wheels or books with pendant structure would show how general the mechanism is.
  • The paper's closing fixed-size conjecture, if true, implies the extremal non-bipartite book-free graph is a star plus one edge regardless of $r$; a small computational search for fixed edge counts would test whether the order-based and size-based extremal graphs really diverge this sharply.
  • Since the candidate spectral radius is the largest root of an explicit quintic, one gets the asymptotic $\rho\sim(n-1)/2$; a higher-order expansion could sharpen the spectral forcing condition and benchmark near-extremal graphs.
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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 the spectral Turan problem for non-bipartite graphs excluding the book graph B_{r+1}, for r>=1. The main theorem (Theorem 1.3) states that for n>=8(r^2+r+4), every non-bipartite B_{r+1}-free graph G of order n satisfies rho(G) <= rho(K^{r,r}_{floor((n-1)/2),ceil((n-1)/2)}), with equality only for that graph. The proof uses a Perron-vector partition into A=N(u*) and B, the Chvatal-Hanson bound on e(A), the residual index Gamma_w, and a long structural case analysis. The lower bound is derived from an equitable quotient matrix (Lemma 2.4), and the upper bound is obtained by a series of contradiction inequalities that narrow the structure of the extremal graph to the claimed extremal graph.

Significance. If the proof is correct, the result resolves Problem 1.2 and complements the known bipartite extremal result of Zhai and Lin by showing that the non-bipartite extremal graph is a complete bipartite graph K_{floor((n-1)/2),ceil((n-1)/2)} with r extra neighbours of a new vertex in each part. This is a natural and nontrivial extension of the r=0 result of Lin, Ning and Wu. The paper gives explicit parameter-free lower bounds, uses the Chvatal-Hanson theorem in a clean way, and the quotient-matrix computation in Lemma 2.4 is checkable. The main weakness is that the structural proof is long and terse, and one step in Theorem 3.11 is genuinely incomplete. Because that step is load-bearing, the current version cannot be accepted as is, although the gap appears local and likely repairable.

major comments (1)
  1. [Theorem 3.11, Case 1.1, Claim 1 (page 11)] The proof that G' remains non-bipartite is incomplete. The inference 'If G' is bipartite, then ... C must be a 5-cycle' is false under the stated local hypotheses. For r=1, take u*, A={u0,u1,u2}, B={w0,w1,z}, with edges u*ui for each i, w0w1, w0u0, w1u1, zu0, zu2. This graph is non-bipartite (it contains the 7-cycle u0-w0-w1-u1-u*-u2-z-u0), is B_2-free, satisfies e(A)=0, G*[B]=K2 plus an isolated vertex, and NA(w0)={u0} is not a subset of NA(w1)={u1}. However G'=G*-u0w0+u0w1 is bipartite. Thus the assertion that every odd cycle must be a 5-cycle is contradicted by a graph satisfying exactly the hypotheses used at that point. The subsequent restriction to 5-cycles is used to prove the non-bipartiteness of G', and hence Claim 1's conclusion NA(w0) is a subset of NA(w1); this conclusion is needed to establish dA(w0)=r and ultimately to prove e(A)!=0 in Theorem 3.11. The gap is load-bearing. A repair through edge-maximality, for instance adding a missing edge from w0 to an A-neighbour of w1 after the swap, is plausible, but it is not present in the manuscript.
minor comments (4)
  1. [Section 3, notation before Lemma 3.6] The notation dA(u), dB(u) is overloaded: earlier it denotes the number of neighbours, while from Lemma 3.6 onward it sometimes denotes the number of non-neighbours. The overlines are not consistently rendered in the text, which makes parts of Theorem 3.11 very hard to follow. A clean distinction, e.g. d_bar_A(u), would improve readability.
  2. [Theorem 3.11, Case 1.1, after (9)] The sentence 'By (9), it follows that dA(w1)>=1' does not follow from dA(w1)<=|A|-1 alone; in the context it relies on the earlier observation that NA(w1) is nonempty because G* is non-bipartite. That earlier observation should be restated or the implication clarified.
  3. [Throughout] There are several typographical issues: for example, 'G Tn,2' appears for non-isomorphism symbols in the abstract and introduction, and 'outplanar' in the introduction should be 'outerplanar'. These do not affect the mathematics.
  4. [Lemma 2.5, even n case] The line 'f((n-2)/2, n/2, r, (n-1)/2 - 1/(4(n-1))) < 0' is asserted without showing the computation. The inequality is believable and consistent with the subsequent bound, but a short verification would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the theorem is derived from external inequalities and equitable-quotient computations; the flagged 5-cycle inference is a correctness concern, not a circular reduction.

full rationale

The paper's central claim is a spectral Turan-type upper bound with a uniquely characterized extremal graph. The candidate K^{r,r} is constructed directly, its spectral radius is computed from an equitable partition in Lemma 2.4, and the required lower bounds are derived by evaluating the resulting characteristic polynomial in Lemma 2.5. The upper-bound side is driven by the Chvatal-Hanson lemma, the Wu-Xiao-Hong Perron-vector edge-swap lemma, and the quotient-matrix spectral lemma, all of which are standard external results and are not restatements of the target theorem. No parameter is fitted to a subset of the extremal data and then renamed as a prediction: the extremal graph is not inferred from spectral measurements but is the output of structural contradictions. Prior theorems by Zhai-Lin and Lin-Ning-Wu are used only as motivational context and do not carry the proof. The one passage worth flagging is in Theorem 3.11, Case 1.1, where the proof asserts that if G' is bipartite then every odd cycle of G* must be a 5-cycle; a 7-cycle may exist under the stated local hypotheses, so this appears to be a possible correctness gap in the manuscript rather than a circular step. Because a mathematical error of that kind is not an equivalence-by-construction, self-citation, or fitted-input reduction, it does not raise the circularity score. None of the seven enumerated circularity patterns occurs in the derivation.

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

The central claim rests on standard external theorems (Perron-Frobenius, Chvátal-Hanson, equitable quotient matrix, spectral radius edge-swap lemma) and on the finite-existence/edge-maximality setup. There are no free parameters fitted to data and no invented entities; K^{r,r}_{s,t} is the proposed extremal object, not an ad hoc postulate.

assumptions (5)
  • standard math Chvátal-Hanson bound: f(ν,Δ) ≤ ν(Δ+1) for all positive integers ν,Δ
    Invoked in Lemma 3.2 to bound e(G*[A]) by ν(G*[A])(r+1).
  • standard math Perron-Frobenius spectral radius and Perron vector properties
    Used throughout: positive Perron vector, eigen-equations ρx_u=Σx_v, spectral radius of connected graph.
  • standard math Lemma 2.2 (Wu-Xiao-Hong): edge rotation increases spectral radius when Perron entry of target vertex is at least source
    Used repeatedly to replace edges u0w0 with u0w1 and to prove maximality contradictions.
  • standard math Equitable quotient matrix lemma: largest eigenvalue of nonnegative irreducible symmetric matrix equals largest eigenvalue of equitable quotient
    Used in Lemma 2.4 to compute ρ(K^{r,r}_{s,t}) as largest root of the 5x5 quotient characteristic polynomial.
  • domain assumption Existence and edge-maximality of an extremal graph on the finite set of n-vertex graphs
    The proof selects G* maximizing ρ among non-bipartite B_{r+1}-free graphs; finite set guarantees existence, and maximality is used to force neighborhoods in structural claims.

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Pith. "Pith review of Spectral Tur\'{a}n problem of non-bipartite graphs: Forbidden books." pith.science (2026). https://pith.science/paper/R6D3HBCK

@misc{pith2026250604884,
  author       = {Pith},
  title        = {Pith review of: Spectral Tur\'an problem of non-bipartite graphs: Forbidden books},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6D3HBCK}},
  note         = {Machine review of arXiv:2506.04884}
}
abstract

A book graph $B_{r+1}$ is a set of $r+1$ triangles with a common edge, where $r\geq0$ is an integer. Zhai and Lin [J. Graph Theory 102 (2023) 502-520] proved that for $n\geq\frac{13}{2}r$, if $G$ is a $B_{r+1}$-free graph of order $n$, then $\rho(G)\leq\rho(T_{n,2})$, with equality if and only if $G\cong T_{n,2}$. Note that the extremal graph $T_{n,2}$ is bipartite. Motivated by the above elegant result, we investigate the spectral Tur\'{a}n problem of non-bipartite $B_{r+1}$-free graphs of order $n$. For general $r\geq1$, let $K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}$ be the graph obtained from $K_{\lceil\frac{n-1}{2}\rceil,\lfloor\frac{n-1}{2}\rfloor}$ by adding a new vertex $v_{0}$ such that $v_{0}$ has exactly $r$ neighbours in each part of $K_{\lceil\frac{n-1}{2}\rceil,\lfloor\frac{n-1}{2}\rfloor}$. By adopting a different technique named the residual index, Chv\'{a}tal-Hanson theorem and typical spectral extremal methods, we in this paper prove that: If $G$ is a non-bipartite $B_{r+1}$-free graph of order $n$, then $\rho(G)\leq\rho\Big(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}\Big)$ , with equality if and only if $G\cong K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}$. An interesting phenomenon is that the spectral extremal graphs are completely different for $r=0$ and general $r\geq1$.

Figures

Figures reproduced from arXiv: 2506.04884 by the authors.

Figure 1
Figure 1. The graph K r,r ⌊ n−1 2 ⌋,⌈ n−1 2 ⌉ . r ≥ 0. It is easy to see that Br+1 is color-critical with χ(Br+1) = 3. It follows from Nikiforov’s result that Tn,2 attains the maximum spectral radius among all Br+1-free graphs of order n. Note that n0(H) is exponential with |V(H)|. Very recently, Zhai and Lin [48] improved n0(H) to be a linear function on |V(H)| for H being a book graph. Theorem 1.1 ( [48]). Let n and r be in… view at source ↗
Figure 2
Figure 2. The graph K r1,r2 s,t . The quotient matrix of M with respect to Π is defined as the k × k matrix BΠ = (bi, j) k i, j=1 , where bi, j is the average value of all row sums of Mi, j . The partition Π is called equitable if each block Mi, j of M has constant row sum bi, j . Also, we say that the quotient matrix BΠ is equitable if Π is an equitable partition of M. Lemma 2.3 ( [1, 18]). Let M be a real symmetric matrix a… view at source ↗
Figure 3
Figure 3. The graph G ∗ with NA(w0) ⊈ NA(w1). Claim 1. NA(w0) ⊆ NA(w1). Proof. Suppose to the contrary that NA(w0) ⊈ NA(w1). Then there exists a vertex u0 ∈ NA(w0)\NA(w1). Let G ′ = G ∗ − u0w0 + u0w1. Then G ′ is still Br+1-free. Otherwise, G ′ contains a Br+1 and u0w1 ∈ E(Br+1), and hence u0 and w1 have at least one common neighbour in G ′ , which contradicts that NG′(u0) ∩ NG′(w1) = ∅. Next we prove that G ′ is still non-bi… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The graph G ∗ with e(B) = 1. have dA(w0) = r, and hence dA(w0) = |A| − r. By (9), it follows that dA(w1) ≥ 1. Now, let NA(w0) = {u1, . . . , ur}. Since ρxw0 = xw1 + Pr i=1 xui ≤ (r + 1)xu ∗ , xw0 ≤ r+1 ρ xu ∗ . Hence Γw0 = dA(w0)(xu ∗ − xw0 ) = r(xu ∗ − xw0 ) ≥ r [PIT…
Figure 5
Figure 5. Figure 5: The graph G ∗ with e(A) = 1. Proof. If NB(u0) ⊈ NB(u1), then there exists a vertex w ∈ NB(u0)\NB(u1). Let G ′ = G ∗ − wu0 + wu1. Then G ′ is still Br+1-free. If not, G ′ contains a Br+1 and wu1 ∈ E(Br+1), which contradicts that NG′(u1)∩NG′(w) = ∅. Note that G ′ is stil…
Figure 6
Figure 6. Figure 6: The graph G ∗ with e(A) ≥ 2. Moreover, we have ρxu0 = xu ∗ + axu1 + X w∈B0∪B1 xw ≤ [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

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