A note on the spectral radius and $[a,b]$-factor of graphs

Dandan Fan; Huiqiu Lin; Yinfen Zhu

A note on the spectral radius and $[a,b]$-factor of graphs

T0 review · 0 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A sharp spectral radius threshold guarantees [a,b]-factors in large connected graphs with minimum degree at least a.

desk verdict Solves the b>a spectral radius case of Hao-Li's factor problem with a sharp extremal graph; the proof is standard but dense, with minor typos and one unproved side theorem. read the letter →

arxiv 2509.00769 v2 pith:RINAX5SJ submitted 2025-08-31 math.SP

Dandan Fan , Huiqiu Lin , Yinfen Zhu This is my paper · ORCID

classification math.SP MSC 05C50

keywords spectralradius[ab]-factorextremalgraphminimumdegreefractionaleigenvaluessharpbound

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 establishes a sharp spectral-radius condition for the existence of an [a,b]-factor: for b>a≥1, every connected n-vertex graph with minimum degree at least a and n at least 2(a+b+2)(b+2) has an [a,b]-factor as soon as its spectral radius reaches that of a specific extremal graph H^{a,b}_n. The only exception is H^{a,b}_n itself, which is shown to contain no [a,b]-factor. The result answers an open problem about sharp bounds for [a,b]-factors under a minimum-degree assumption. Because a graph with minimum degree below a cannot contain such a factor, the degree hypothesis is natural, and the extremal graph makes the threshold tight.

What carries the argument

The argument runs on three tools. First, a single-set [a,b]-factor criterion (Lemma 2.4) that characterizes factor existence through an inequality involving a set S, the vertices of low degree in G−S, and a,b. Second, a spectral perturbation rule (Lemma 2.1) that compares spectral radii after moving edges from a lower Perron-coordinate vertex to a higher one; this lets the proof deform any maximal factor-free graph into a canonical shape. Third, Lemmas 2.6 and 2.7 identify H^{a,b}_n as the unique spectral-radius maximizer within the relevant family of factor-free candidates. The order lower bound n≥2(a+b+2)(b+2) is used throughout to control the upper-bound estimates.

What would settle it

Constructing a connected graph G with n≥2(a+b+2)(b+2), δ(G)≥a, ρ(G)≥ρ(H^{a,b}_n), and no [a,b]-factor, with G not isomorphic to H^{a,b}_n, would disprove Theorem 1.1. A more targeted check is to verify Lemma 2.4 against a graph that satisfies the displayed inequality but is known to lack an [a,b]-factor for b>a; if such a graph exists, the proof's central criterion fails.

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Extended reading notes

Core claim

The central claim is Theorem 1.1. Let a and b be positive integers with b>a, and let G be a connected graph of order n≥2(a+b+2)(b+2) with minimum degree δ(G)≥a. If ρ(G)≥ρ(H^{a,b}_n), then G contains an [a,b]-factor unless G is isomorphic to H^{a,b}_n. The graph H^{a,b}_n is built by taking a join K_a ∨ (K_{n-a-b-1} ∪ (b+1)K_1) and adding a−1 edges from one isolated vertex to a−1 vertices in the large clique. H^{a,b}_n has no [a,b]-factor, so the spectral threshold cannot be lowered. The proof maximizes spectral radius among factor-free graphs, uses a single-subset version of the [a,b]-factor existence criterion, and shows that any graph beating the threshold must satisfy that criterion.

Load-bearing premise

The proof relies on the single-set [a,b]-factor criterion being exactly right for all b>a; if that criterion admits exceptions, the step where a factor-free graph must violate the inequality falls apart.

Editorial extensions

If this is right

Every connected graph with minimum degree at least a and order above the stated threshold either contains an [a,b]-factor or is exactly the extremal graph H^{a,b}_n.
The threshold ρ(H^{a,b}_n) is sharp: H^{a,b}_n attains it but has no [a,b]-factor.
The same spectral condition also forces a fractional [a,b]-factor, since every integral [a,b]-factor is fractional.
A companion size condition in the paper gives a sharp edge-count threshold for the same factor problem.
This resolves the spectral-radius side of the open problem by supplying the sharp lower bound.

Reading between the lines

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

A natural next step is to test whether the order threshold 2(a+b+2)(b+2) can be substantially lowered; the proof's estimates likely leave room for a smaller true extremal order.
The same extremal construction may guide sharp spectral conditions for related factor types, such as [a,b]-parity factors, where the factor criterion changes.
One could probe whether replacing the minimum-degree condition δ(G)≥a with a weaker average-degree hypothesis changes the extremal graph or the threshold.

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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

Dear colleague,

The headline is that this is a solid, incremental resolution of a named open problem. Fan, Lin and Zhu prove a sharp threshold: for connected n-vertex graphs with δ≥a and n large, ρ(G)≥ρ(H^{a,b}_n) forces an [a,b]-factor unless G is H^{a,b}_n. That graph, built from K_a joined to a large clique plus b+1 isolated vertices, is genuinely new and the bound is tight. This covers exactly the b>a case that Tang-Zhang's a=b result leaves open, so the problem from Hao-Li is answered.

The paper does what it claims. It uses the standard single-set Heinrich et al. factor criterion, Perron-vector switching, and the Hong-Shu-Fang/Nikiforov bound. The structure is clear, and the citations to Hao-Li and Tang-Zhang are appropriate. I did not verify every inequality, but I checked the logic at the joints and found no load-bearing flaw.

The soft spots are minor and fixable. Lemma 2.7 has a typo: 'p(p1+a+1)' should be 'q(p1+a+1)' in the displayed definition of f(ρ,ρ'). In Case 2, the write-up says ρ(G)>n−b−1 where n−b−2 would be the correct strict lower bound; this is harmless because n−b−2>a−1. The reader's worry about the inequality direction in Subcase 2.2 is, on reading, not real: 0=∑_{v∈W2} d_{G*−S2}(v) ≤ a|W2|−b|S2|−1 = a−1 is exactly the violation of Lemma 2.4 for a≥1, so it correctly shows G* has no [a,b]-factor.

The bigger soft spot is Theorem 3.1 in the concluding remarks. It states a size threshold without proof, citing only 'a technique analogous to [17]'. That is not enough. Either give the proof or downgrade it to a remark/conjecture.

Who is this for? Anyone working on spectral conditions for factors. It is not a conceptual breakthrough, but it is a correct-looking resolution of an open problem with the right extremal construction. A serious referee should go through the algebra in Lemma 2.7 carefully; the paper deserves that effort. I would send it out, and require the typo fix and the Theorem 3.1 issue addressed before publication.

Referee Report

0 major / 5 minor

Summary. The paper gives a sharp spectral-radius condition for the existence of an [a,b]-factor in connected graphs with minimum degree at least a, for b > a >= 1. The main result, Theorem 1.1, states that for n >= 2(a+b+2)(b+2), if rho(G) >= rho(H^{a,b}_n), then G contains an [a,b]-factor unless G is isomorphic to H^{a,b}_n, a specific graph shown to contain no such factor. The proof combines Perron-vector edge-switching arguments, the Hong-Shu-Fang/Nikiforov spectral-radius bound, the Heinrich et al. single-set version of Tutte's factor theorem, and a detailed case analysis on the vertex set W appearing in the factor-criterion violation. The paper also states a size version and a fractional-factor corollary.

Significance. If the proof is correct, the paper resolves the spectral-radius part of Problem 1 posed by Hao and Li, complementing the known a = b case. The extremal graph is natural and the condition is shown to be sharp. A particular strength is that the extremal graph is not assumed but constructed from the factor-criterion violation, and the proof relies on established external lemmas rather than circular reasoning. The main argument is technically demanding; I could verify the overall structure and the key reduction steps, and the potential issues flagged in the reading report (Lemma 2.4 applicability and the final Subcase 2.2 inequality) are not actual mathematical flaws. However, several typos and terse inequality checks make the paper harder to read and should be corrected.

minor comments (5)

[Lemma 2.7 (p. 5)] In the displayed definition of f(rho,rho'), the term '(rho^2 + rho + p(p1 + a + 1))' should read 'q(p1 + a + 1)'. The preceding denominator and inequality (5) use q, and the later bounds rely on 2 <= q <= a-1. As printed, the expression is inconsistent and cannot be verified. Please correct this typo throughout the displayed formula.
[Theorem 1.1, Case 2 (p. 7)] The sentence 'we have rho(G) > n - b - 1 > a - 1 >= q' overstates what is known: (13) gives only rho(G) > n - b - 2. The needed conclusion is rho(G) > n - b - 2 > a - 1 >= q, which is sufficient. Please correct the displayed inequality.
[Theorem 1.1, Subcase 2.2 (p. 8-9)] The final check that G* has no [a,b]-factor is written as '0 = sum_{v in W2} d_{G*-S2}(v) <= a|W2| - b|S2| - 1 = a - 1'. This is confusing because Lemma 2.4 requires the opposite inequality, a|W2| - sum d_{G*-S2}(v) >= b|S2| + 1, to certify non-existence of a factor. Since d_{G*-S2}(v)=0 for every v in W2, the violation condition is a(b+1) >= ab+1, i.e. a >= 1; the displayed line is an equivalent rearrangement but the implication should be stated explicitly. Please rewrite this step.
[Theorem 1.1, after (13)] The deduction 'by the maximality of rho(G), G[V(G)\ W] is K_{n-t} and e(S,W)=st' is used later in the proof but is not justified. Please add a sentence explaining that adding any missing edge with at least one endpoint outside W (or between S and W) does not alter the certificate from Lemma 2.4 for S and W, so the no-factor property and the minimum-degree condition are preserved, contradicting maximality. This is a small but important justification.
[Theorem 1.1, after (13)] The statement 'H^{a,b}_n contains no [a,b]-factor' is used to justify rho(G) >= rho(H^{a,b}_n) and is plausible, but no proof is given in the text. A one-line counting argument (the b+1 vertices in W have total degree demand a(b+1), while the capacity from the a vertices of K_a plus the a-1 extra edges is at most ab+a-1) would make the sharpness claim self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is proved from external spectral and factor criteria; the extremal graph is constructed and analyzed, not assumed.

full rationale

The paper's central result, Theorem 1.1, is derived by contradiction using established external lemmas: Lemma 2.1 and Lemma 2.5 (Perron-vector comparison), Lemma 2.2 and Lemma 2.3 (Hong–Shu–Fang/Nikiforov spectral radius bound), and Lemma 2.4 (Heinrich et al. [a,b]-factor criterion). The extremal graph H^{a,b}_n is explicitly constructed, and its spectral radius is bounded in Lemma 2.6 and Lemma 2.7 by direct computation and comparison within the class G^{a,b}_n. The proof does not fit any parameter to the target conclusion, does not define the spectral condition in terms of factor existence, and does not rely on the authors' own prior work as the load-bearing ingredient. The cited papers [9], [12], and [17] are contextual or related results, not substitutes for the proof. The apparent typo in Lemma 2.7 (p(p1+a+1) versus q(p1+a+1)) and the minor overstatement in Case 2 do not amount to circularity; they are presentation errors in an otherwise self-contained argument. The final check in Subcase 2.2 is a direct application of Lemma 2.4's violation form, not a circular invocation of the theorem being proved. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The proof rests on standard spectral graph theory results (Perron-Frobenius, eigenvalue comparisons) and a known factor characterization. No free parameters or new postulates are introduced; the graph H^{a,b}_n is a concrete construction, not an invented entity.

assumptions (3)

standard math Perron-Frobenius theorem for the adjacency matrix of a connected graph
Used throughout Section 2 to choose Perron vectors and apply eigenvalue comparison lemmas (Lemma 2.1, Lemma 2.5).
domain assumption Lemma 2.4 (Heinrich et al.) characterization of [a,b]-factors via a single set S
This is the main existence criterion used to prove non-existence of [a,b]-factors in the extremal graph and in the contradiction argument.
domain assumption Lemma 2.2 (Hong-Shu-Fang, Nikiforov) sharp spectral radius upper bound
Used to bound the spectral radius of extremal graphs in Lemma 2.6 and Theorem 1.1, Case 1.

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Pith. "Pith review of A note on the spectral radius and $[a,b]$-factor of graphs." pith.science (2026). https://pith.science/paper/RINAX5SJ

@misc{pith2026250900769,
  author       = {Pith},
  title        = {Pith review of: A note on the spectral radius and $[a,b]$-factor of graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RINAX5SJ}},
  note         = {Machine review of arXiv:2509.00769}
}

abstract

The investigation of eigenvalue conditions for the existence of an $[a,b]$-factor originates in the work of Brouwer and Haemers (2005) on perfect matchings. In the decades since, spectral extremal problems related to $[a,b]$-factors have attracted considerable attention. In this paper, we establish a spectral radius condition that ensures the existence of an $[a,b]$-factor in a graph $G$ with minimum degree $\delta(G) \geq a$, where $b > a \geq 1$. This result resolves a problem posed by Hao and Li [Electron. J. Combin. (2024)].

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [17]

Y. Li, D. Fan, Y. Zhu, Spectral radius and fractional [ a, b]-factor of graphs, Linear Algebra Appl. 715 (2025) 32–45

work page 2025

[2] [1]

Brouwer, W

A. Brouwer, W. Haemers, Eigenvalues and perfect matchings, Linear Algebra Appl.395 (2005) 155–162

work page 2005

[3] [2]

Cioab˘ a, D

S. Cioab˘ a, D. Gregory, W. Haemers, Matchings in regular graphs from eigenvalues,J. Combin. Theory Ser. B99 (2) (2009) 287–297

work page 2009

[4] [3]

Cioab˘ a, Perfect matchings, eigenvalues and expansion,C

S. Cioab˘ a, Perfect matchings, eigenvalues and expansion,C. R. Math. Acad. Sci. Soc. R. Can. 27 (4) (2005) 101–104. 10

work page 2005

[5] [4]

Cioab˘ a, D

S. Cioab˘ a, D. Gregory, Large matchings from eigenvalues,Linear Algebra Appl.422 (1) (2007) 308–317

work page 2007

[6] [5]

E. Cho, J. Hyun, S. O, J. Park, Sharp conditions for the existence of an even [ a, b]-factor in a graph, Bull. Korean Math. Soc.58 (1) (2021) 31–46

work page 2021

[7] [6]

A. Fan, R. Liu, G. Ao, Spectral radius, odd [1 , b]-factor and spanning k-tree of 1-binding graphs, Linear Algebra Appl.705 (2025) 1–16

work page 2025

[8] [7]

A. Fan, R. Liu, G. Ao, Spectral radius, fractional [ a, b]-factor and ID-factor-critical graphs, Discrete Math. 347 (2024), no. 7, Paper No. 113976, 11 pp

work page 2024