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Spectral extremal results on the $A_\alpha$-spectral radius of graphs without $K_{a,b}$-minor

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Complete A_alpha extremal graphs found for K_{a,b}-minor-free graphs

desk verdict A credible and important A_alpha generalization of Zhai-Lin, with a real statement bug in Theorem 1.3 and an imported structural lemma that needs verification. read the letter →

arxiv 2412.06399 v1 pith:FVGMA72T submitted 2024-12-09 math.CO

Xingyu Lei , Shuchao Li This is my paper · ORCID
classification math.CO MSC 05C5005C8305C35
keywords A_alpha-spectralradiusK_{ab}-minorfreegraphsspectralTuranproblemPerronvectorcliquedominatingsetsignlessLaplacianstarforestTaitconjecture
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 asks which n-vertex graph without a K_{a,b} minor has the largest A_alpha-spectral radius, the one-parameter family A_alpha(G)=alpha D(G)+(1-alpha)A(G) that interpolates between the adjacency matrix (alpha=0) and the signless Laplacian (alpha=1/2). The paper's claim is that for 1<=a<=b and alpha in [0,1), with n large enough, the answer is explicit and unique: apart from two sporadic exceptional graphs, it is always a clique K_{a-1} joined to a disjoint union of complete graphs K_b and star forests F_{a,b}, with a separate connected K_{1,b} result. If correct, this resolves the A_alpha version of Tait's spectral Turan conjecture for complete bipartite minors and unifies the previously separate adjacency and Q-spectral extremal theorems into one statement.

What carries the argument

The carrying object is the A_alpha matrix A_alpha(G)=alpha D(G)+(1-alpha)A(G) together with its Perron vector x, which satisfies lambda_alpha x_v = alpha d(v)x_v + (1-alpha) sum_{u~v} x_u. The proof forces the extremal graph to decompose as a clique dominating set S* of size a-1 (Lemma 4.1, imported from prior work) plus components that must have the (a,b)-property, meaning they are K_{r,s}-minor-free for every r+s=b+1 with r<=omega. The engine is Lemma 2.12, which uses Perron-vector lower and upper bounds to show that for large n any edge-density-raising replacement inside a small component strictly raises the A_alpha-spectral radius; combining this with double-eigenvector comparisons (Lemmas 2.4 and 2.5) pins each component to K_b, F_{a,b}, F(a,0,b-1-a), or the Petersen graph.

What would settle it

Fix a small case such as (a,b)=(2,5) and alpha=0.9, and for a range of n near the claimed threshold enumerate all n-vertex K_{2,5}-minor-free graphs whose Perron-vector computation is feasible; if any graph without a clique dominating set of size 1 has A_alpha-spectral radius larger than K_1 joined to ((k-t)K_5 cup tF_{2,5}) (or than the stated t=2 exception), the theorem is false. Alternatively, check whether Lemma 4.1's conclusion holds for the true A_alpha extremal at alpha close to 1 by direct computation.

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Extended reading notes

Core claim

For the connected K_{1,b}-minor-free case (Theorem 1.3), the paper proves that for n=b+1 the extremal graph is F_{1,b} for every alpha in [0,1), while for n != b+1 it is S_{n-b}(K_b) when b=3 and alpha in [0,1) or when b>=4 and alpha in [2/(b+1),1). For 2<=a<=b (Theorem 1.4), writing n-a+1=kb+t with 0<=t<=b-1 and tau=floor((b+1)/(a+1)), the paper proves that for large n the unique extremal graph has the form K_{a-1} joined to ((k-t)K_b cup tF_{a,b}) when t<=2(tau-1), and K_{a-1} joined to (kK_b cup K_t) when t>=2tau-1. The two exceptions are t=tau=2, where the component F(a,0,b-1-a) replaces one F_{a,b}, and t=2,tau=1,b=8, where the Petersen graph replaces K_2. These statements are given for every alpha in [0,1).

Load-bearing premise

The proof assumes, via an imported lemma that is not re-proved here, that the extremal graph for a>=2 contains a clique of a-1 dominating vertices, and it also assumes an unquantified 'n large enough' threshold that makes the Perron-vector sums strictly ordered; if either premise fails in some parameter range, the claimed extremal graphs could be different.

Editorial extensions

If this is right

  • For every alpha in [0,1) and sufficiently large n, the maximizing graph is unique and has the same join-of-cliques/star-forest form as the adjacency extremal, so the alpha-parameter does not create new extremal shapes except the two sporadic cases.
  • Setting alpha=0 recovers Zhai and Lin's complete resolution of Tait's conjecture; setting alpha=1/2 recovers the recent Q-spectral extremal characterizations for K_{a,b}-minor-free graphs.
  • For connected K_{1,b}-minor-free graphs, the extremal is the subdivision graph S_{n-b}(K_b) whenever alpha is at least 2/(b+1), and the small-alpha regime is explicitly left open for b>=4 and alpha in (0, 2/(b+1)).
  • The structural dichotomy t<=2(tau-1) versus t>=2tau-1 is inherited from the adjacency case, meaning the A_alpha result does not change the phase transition in the remainder t.

Reading between the lines

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

  • The open range in Theorem 1.3 suggests that for alpha below 2/(b+1) the extremal connected K_{1,b}-minor-free graph may depend on alpha; a testable conjecture is that it switches from S_{n-b}(K_b) to another graph at a threshold that may not equal 2/(b+1).
  • If Lemma 4.1 were proved directly for the A_alpha extremal, the whole argument would become self-contained and would likely extend to alpha=1 and to other minor families where a clique dominating set can be forced.
  • A quantitative bound on 'n large enough' would turn the theorem into an algorithmic tool: for explicit n one could certify the winner by checking finitely many component patterns, since all components have size at most b+2.
  • The same Perron-vector separation technique should apply to other forbidden complete bipartite subgraphs or minors where an analogous edge-density bound is available.
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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

2 major / 4 minor

Summary. The paper studies the A_alpha-spectral radius of K_{a,b}-minor-free graphs. It states two main extremal characterizations: Theorem 1.3 determines the connected K_{1,b}-minor-free extremal graph for n=b+1 and for b=3 or alpha>=2/(b+1) otherwise, and Theorem 1.4 determines the extremal graph for 2<=a<=b and sufficiently large n as either K_{a-1} vee ((k-t)K_b cup t F_{a,b}) or K_{a-1} vee (kK_b cup K_t), with the two exceptional cases t=tau=2 and t=2, tau=1, b=8. The proof combines the earlier structural framework of Zhai and Lin with A_alpha-specific Perron-vector estimates and exchange arguments. The final section derives the alpha=0 and alpha=1/2 cases as corollaries.

Significance. If correct, Theorem 1.4 is a substantial extension of the Zhai-Lin resolution of Tait's adjacency conjecture to the entire A_alpha-family, with explicit extremal graphs in every case. The derivation of the alpha=0 and alpha=1/2 specializations from the new theorem is a genuine consistency check rather than a fitting exercise, since no parameters are tuned to match those known results. The main risks are the imported structural lemma used to start the Section 4 analysis and the unquantified large-n estimates on Perron-vector sums, both of which are load-bearing for the proof as written.

major comments (2)
  1. [Section 4, Lemma 4.1] The statement that the extremal graph G* has a clique dominating set S* of cardinality a-1 is imported from [8,33] without being re-proved and without quoting the exact theorem used. All subsequent lemmas in Section 4 and the final case analysis of Theorem 1.4 operate on G*-S*, so this decomposition is the structural starting point of the whole proof. The authors should state the precise result they import, verify that its hypotheses cover the present setting (A_alpha-maximizers for all alpha in [0,1), all 2<=a<=b, n sufficiently large, and possibly disconnected K_{a,b}-minor-free graphs), or give a self-contained proof. As written, a reader cannot tell whether the cited results apply only to alpha=0 or only to connected graphs, and if the lemma fails the claimed extremal graphs in Theorem 1.4 could change.
  2. [Section 2, Lemma 2.12(ii)] The proof asserts the strict inequalities pX_m > qX_M and pX_m^2 > qX_M^2 for 'n is large enough' without quantifying the threshold, and the displayed derivation for the square inequality replaces p and q by sqrt p and sqrt q in the final lower bound without justification. These inequalities are invoked in virtually every exchange argument in Section 4, including Lemmas 4.2, 4.5, and Claims 8-10, so the proof needs an explicit uniform bound of the form n > N(a,b,c,p,q,alpha) and a corrected algebraic justification of the final inequality. The current unquantified language is too fragile for a proof whose later steps depend on the strictness of these comparisons.
minor comments (4)
  1. [Section 5, Corollary 5.3] Corollary 5.3 states tau = floor((a+1)/(b+1)), which is inconsistent with the definition tau = floor((b+1)/(a+1)) used in Theorem 1.4 and throughout Section 4. As printed, the alpha=1/2 specialization does not match the theorem it claims to specialize.
  2. [Section 5, Corollary 5.1] Corollary 5.1 says 'In Theorem 1.3, let alpha=0 and a=2', but Theorem 1.3 concerns the K_{1,b} case; the case a=2 belongs to Theorem 1.4. The cross-reference should be corrected.
  3. [Section 4, Lemma 4.4(i)] The conclusion of Lemma 4.4(i) is printed as 'F_{>t+3} = empty set', but the proof and the surrounding text concern components of order larger than b+3. This appears to be a typo for 'F_{>b+3}'.
  4. [Throughout] There are several small typographical issues, such as 'oder' in Lemma 2.8, 'Combing' for 'Combining', and 'Our manuscript has no associated date' in the Data availability statement. These do not affect the mathematics but should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the A_alpha extremal characterization is derived by weighted exchange arguments from external structural lemmas, not by fitting or self-citation.

full rationale

The central result (Theorem 1.4) is a genuine family of theorems: for every alpha in [0,1) it asserts the same extremal K_{a,b}-minor-free graphs as in the adjacency case, and the proof carries alpha through the Perron-vector inequalities (Lemma 2.12), component edge-density comparisons (Lemmas 4.2-4.8), and exchange constructions. No parameter is fitted to the target quantity. The special cases alpha=0 and alpha=1/2 are derived in Section 5, not assumed, so there is no self-definitional reversal. The main imported structure, Lemma 4.1, is a clique dominating set of size a-1 for the A_alpha-maximizer; it is explicitly cited to Chen-Liu-Zhang [8] and Tait [33], neither of which is authored by the present authors, and it is a prior structural theorem about A_alpha-extremal graphs rather than an equivalent restatement of Theorem 1.4. Zhai-Lin's adjacency result [36] is quoted as background and used only to supply (a,b)-property lemmas, while the extremal proof itself is carried out for general alpha. The only unquantified hypothesis is 'n is large enough' in Lemma 2.12(ii), used to make certain differences of Perron-vector sums positive; this is an analytic threshold assumption, not a circular reduction or a fitted input. The paper also explicitly leaves open the connected K_{1,b} case for small alpha (Problem 1), which is inconsistent with any hidden assumption that the theorem is true by construction in that range. Minor self-citations in the introduction are bibliographic pointers to prior A_alpha-spectral work and are not load-bearing. Accordingly, no circular step can be exhibited.

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

The central claim rests on standard spectral graph theory, on imported structural lemmas from prior work on K_{a,b}-minor-free graphs, and on an unquantified large-n assumption. No free parameters are fitted to data and no new mathematical objects beyond the known graph families F_{a,b}, P^*, and S_{n-b}(K_b) are postulated.

assumptions (5)
  • standard math Perron-Frobenius theory for the irreducible nonnegative matrix A_alpha(G) on connected graphs.
    Used throughout, for example in Section 2 to define the Perron vector and in Lemma 2.12 to derive coordinate bounds. This is standard spectral graph theory.
  • domain assumption The extremal graph G* for a>=2 has a clique dominating set S* of size a-1.
    Lemma 4.1, imported from [8,33], is the structural foundation of Section 4. The whole decomposition G*-S* into components of orders b, b+1, b+2, etc. rests on it, and it is not re-proved.
  • domain assumption The structural classification of edge-maximal (a,b)-property graphs: Lemmas 2.7, 2.8, 2.9, 2.10, and 2.11 from [36].
    These lemmas identify the graphs F(a1,a2,a3), F_{a,b}, P^*, and the condition for K_{a,b}-minor-free after removing a clique dominating set. They are cited as black boxes rather than proved.
  • domain assumption The edge bound e(G) <= binom(b,2) + n - b for connected K_{1,b}-minor-free graphs.
    Lemma 2.6, cited from [11,12], is used in Lemma 4.4 and Lemma 4.5 to bound components of G*-S*.
  • domain assumption The parameter n is sufficiently large in Theorems 1.3 and 1.4 and in Lemma 2.12.
    All asymptotic Perron-vector inequalities (for example pXm > qXM and pXm^2 > qXM^2) require n to be large relative to the constants a, b, c. No explicit threshold is stated.

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Pith. "Pith review of Spectral extremal results on the $A_\alpha$-spectral radius of graphs without $K_{a,b}$-minor." pith.science (2026). https://pith.science/paper/FVGMA72T

@misc{pith2026241206399,
  author       = {Pith},
  title        = {Pith review of: Spectral extremal results on the $A_\alpha$-spectral radius of graphs without $K_a,b$-minor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FVGMA72T}},
  note         = {Machine review of arXiv:2412.06399}
}
abstract

An important theorem about the spectral Tur\'an problem of $K_{a,b}$ was largely developed in separate papers. Recently it was completely resolved by Zhai and Lin [J. Comb. Theory, Ser. B 157 (2022) 184-215], which also confirms a conjecture proposed by Tait [J. Comb. Theory, Ser. A 166 (2019) 42-58]. Here, the prior work is fully stated, and then generalized with a self-contained proof. The more complete result is then used to better understand the relationship between the $A_{\alpha}$-spectral radius and the structure of the corresponding extremal $K_{a,b}$-minor free graph.

Figures

Figures reproduced from arXiv: 2412.06399 by the authors.

Figure 1
Figure 1. Graphs P ⋆ and F(a1, a2, a3). By Lemma 2.7, F is isomorphism to either some F(a1, a2, a3) or P⋆. If F is isomorphism to some F(a1, a2, a3), the next claim determines the corresponding graph. Claim 7. If F ∼= F(a1, a2, a3) for some nonnegative integers a1, a2, a3, then a1 = ω, a2 = 0, and a3 = b − 1 − ω satisfying ω 6 1 2 (b − 1). Proof. By the definition of F(a1, a2, a3), we have |Vi | = ai (i = 1, 2, 3) (see [PITH… view at source ↗
Figure 2
Figure 2. Graph Fa,b for τ = ⌊ b+1 a+1 ⌋. The proof of Theorem 1.4. By Lemma 4.1, G∗ has a clique dominating set S ∗ of cardinality a − 1. For G∗ − S ∗ , by b > a > 2, we consider the following cases. Case 1. 2 6 b 6 3. In this case, if b = 2, by Lemma 4.3, we have |F6=2| = |F1| 6 1. Thus, G∗ − S ∗ ∼= kK2 ∪ Kt (0 6 t 6 1). If b = 3, by Lemma 4.3, we have |F6=3| 6 1. To get G∗ − S ∗ ∼= kK2 ∪ Kt (0 6 t 6 2), it is only necessar… view at source ↗

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