An A{\alpha}-spectral radius for the existence of {P3, P4, P5}-factors in graphs

Sizhong Zhou; Yuli Zhang

REVIEW 1 major objections 4 minor 1 cited by

An A{\alpha}-spectral radius for the existence of {P3, P4, P5}-factors in graphs

T0 review · 1 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For connected graphs on $n\ge25$ vertices, passing the $A_\alpha$-spectral radius of $K_1\vee(K_{n-2}\cup K_1)$ forces a $\{P_3,P_4,P_5\}$-factor, except for that graph itself.

desk verdict Solid narrow extension of A_alpha spectral conditions to {P3,P4,P5}-factors, but the s=1 branch of the proof omits the exception graph and the 'unless' clause is actually vacuous. read the letter →

arxiv 2501.05029 v1 pith:23R5PPPC submitted 2025-01-09 math.CO

Yuli Zhang , Sizhong Zhou This is my paper · ORCID

classification math.CO MSC 05C5005C7005C38

keywords A_alpha-matrixA_alpha-spectralradius{P3P4P5}-factorpathfactorisolatedverticesequitablepartitionquotientmatrixspectralextremalgraph

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 establishes a spectral threshold for a spanning path-factor in connected graphs. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph whose components are paths on three, four, or five vertices, so it is a way of tiling the vertex set by short paths. The main theorem says that for $0\le\alpha<2/3$ and $n\ge25$, every connected graph $G$ whose $A_\alpha$-spectral radius $\lambda_\alpha(G)$ is at least $\lambda_\alpha(K_1\vee(K_{n-2}\cup K_1))$ contains such a factor, with one exception: $G=K_1\vee(K_{n-2}\cup K_1)$. The result matters because it converts a nontrivial spanning-substructure question into a single eigenvalue comparison, and the $A_\alpha$-matrix is a one-parameter family interpolating between the adjacency matrix and the signless Laplacian, so the same argument covers both classical spectral settings.

What carries the argument

The central objects are the $A_\alpha$-matrix $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$ and its largest eigenvalue $\lambda_\alpha(G)$, called the $A_\alpha$-spectral radius; here $D(G)$ is the diagonal degree matrix and $A(G)$ the adjacency matrix, so $\alpha=0$ gives the adjacency spectral radius and $\alpha=1/2$ gives half the signless Laplacian spectral radius. The load-bearing mechanism is the isolated-vertex criterion of Lemma 2.1: if every subset $S\subseteq V(G)$ satisfies $i(G-S)\le\frac23|S|$, where $i$ counts isolated vertices, then $G$ has a $\{P_3,P_4,P_5\}$-factor. The proof invokes the contrapositive, converting the absence of a factor into a forbidden subset $S$; with $s=|S|$, the graph is a spanning subgraph of the join $G_1=K_s\vee(K_{n-\lfloor5s/3\rfloor-1}\cup(\lfloor2s/3\rfloor+1)K_1)$. The spectral comparison is carried out by writing the quotient matrix of $G_1$ under its natural equitable partition, whose largest eigenvalue equals $\lambda_\alpha(G_1)$, by bounding the second eigenvalue through interlacing, and by using the fact that $\lambda_\alpha$ strictly increases when edges are added. The proof splits according to $s\bmod3$ and to whether $n$ equals $\lfloor5s/3\rfloor+3$, $+2$, or $+1$, and in every branch shows $\lambda_\alpha(G_1)<\lambda_\alpha(K_1\vee(K_{n-2}\cup K_1))$ unless $s=1$.

What would settle it

A concrete falsifier is a connected graph $G\not\cong K_1\vee(K_{n-2}\cup K_1)$ of order $n\ge25$ that has no $\{P_3,P_4,P_5\}$-factor and satisfies $\lambda_\alpha(G)\ge\lambda_\alpha(K_1\vee(K_{n-2}\cup K_1))$ for some $\alpha\in[0,2/3)$. The proof predicts that every factorless graph lies strictly below the threshold except possibly the exceptional graph, so a search near the exceptional graph (for instance, graphs obtained by rewiring a few edges incident to its isolated vertex) that finds any such $G$ would falsify the theorem.

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Extended reading notes

Core claim

On its own terms, the paper proves Theorem 1.1: for real $\alpha$ with $0\le\alpha<2/3$ and a connected graph $G$ of order $n\ge25$, the inequality $\lambda_\alpha(G)\ge\lambda_\alpha(K_1\vee(K_{n-2}\cup K_1))$ guarantees that $G$ has a $\{P_3,P_4,P_5\}$-factor, unless $G$ is exactly $K_1\vee(K_{n-2}\cup K_1)$. The proof argues by contradiction. Assuming no such factor exists, the isolated-vertex criterion (Lemma 2.1) produces a vertex set $S$ whose deletion leaves more than $\frac23|S|$ isolated vertices; then $G$ is a spanning subgraph of $G_1=K_s\vee(K_{n-\lfloor5s/3\rfloor-1}\cup(\lfloor2s/3\rfloor+1)K_1)$, where $s=|S|$. The $A_\alpha$-spectral radius of $G_1$ is computed exactly through the quotient matrix of an equitable partition, and by interlacing and monotonicity it is shown to be strictly below the threshold value $\lambda_\alpha(K_1\vee(K_{n-2}\cup K_1))$ for every $s\ge2$; the only remaining case $s=1$ forces $G$ to be the exceptional graph itself. This contradiction establishes the theorem.

Load-bearing premise

The load-bearing premise is the external criterion that a graph whose every vertex subset $S$ leaves at most $\frac23|S|$ isolated vertices must contain a $\{P_3,P_4,P_5\}$-factor, together with the implicit exclusion of the exceptional graph from the contrary assumption in the proof's $s=1$ case; if the criterion is not valid or the exclusion is not granted, the spectral comparison does not by itself force the factor.

Editorial extensions

If this is right

Corollary 1.2: the same spectral condition guarantees a $P_{\ge3}$-factor, because every $\{P_3,P_4,P_5\}$-factor is in particular a path factor with components of length at least two.
Setting $\alpha=0$ yields an adjacency-spectral-radius version and setting $\alpha=1/2$ yields the corresponding signless Laplacian version (up to the factor 2), so Theorem 1.1 unifies the two classical spectral theories in one threshold.
The bound applies uniformly for the whole interval $0\le\alpha<2/3$, so the factor guarantee does not depend on choosing a particular matrix parameter.
The theorem isolates $K_1\vee(K_{n-2}\cup K_1)$ as the unique equality case in the spectral comparison, a standard feature of extremal spectral theorems.

Reading between the lines

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

Editorial inference: the constant $\frac23$ in the isolated-vertex criterion is exactly the ratio that makes the threshold graph extremal; the same proof pattern suggests testable analogues for $\{P_3,\dots,P_k\}$-factors, where the corresponding ratio and threshold graph would take the form $K_s$ joined to a clique and roughly $s/(k-1)$ isolated vertices.
Editorial inference: the restriction $\alpha<2/3$ appears to come from polynomial inequalities in the proof rather than from the combinatorial problem itself; for $\alpha\ge2/3$ a different extremal graph may take over, and computing the maximum $A_\alpha$-spectral radius over factorless graphs numerically for a few $n$ would settle whether the threshold is truly linear in $\alpha$.
Editorial inference: because $\lambda_\alpha(G)$ is computable in polynomial time for fixed $\alpha$, the theorem provides a sufficient spectral certificate for the existence of a $\{P_3,P_4,P_5\}$-factor; it does not construct the factor, so a natural next question is whether the certificate can be combined with an efficient extraction algorithm.

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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

Read it. The paper gives a new sufficient spectral condition for {P3,P4,P5}-factors: if λ_α(G) ≥ λ_α(K1∨(K_{n-2}∪K1)) and n≥25, then G has such a factor, with a stated exception. That extends the same group's earlier A_alpha work on P_{\\ge2}-factors. The proof is standard for this subfield: use the Kano-Lu-Yu isolated-vertex criterion, show a hypothetical counterexample is a spanning subgraph of G1=K_s∨(K_{n-⌊5s/3⌋-1} ∪ (⌊2s/3⌋+1)K1), then compare λ_α(G1) against the threshold graph using quotient matrices and interlacing. The case analysis is thorough, and the polynomial inequalities in Cases 1–3, while tedious and not machine-checked, look correct. The central reduction is sound.

There is one real gap, and it is exactly where the reader flags it. The theorem is an implication with an exception: G has a factor unless G=G*. The proof by contradiction must therefore assume both (a) G has no factor and (b) G≠G*. In the s=1 subcase, they correctly get G=G* from equality of spectral radii and then call it a contradiction. But with only (a), G=G* is not a contradiction—it is the allowed exception. The fix is trivial: either add G≠G* to the contrary assumption, or observe that G* itself admits a {P3,P4,P5}-factor for n≥25 (it does—take the universal vertex, the pendant vertex, and one clique vertex as a P3, and partition the remaining clique into paths of orders 3,4,5). The second observation also shows the stated exception is vacuous, so the theorem is weaker than it needs to be and the prose is misleading. But the mathematics is repairable without changing the main argument.

The other soft spot is that the polynomial manipulations are not machine-checked, so there is some residual risk of an algebraic slip. I did not find one, and the style is routine for this area.

Who is this for? People working on A_alpha spectral conditions for path factors. It is honest progress in a narrow area, no new technique. I would send it to review; the s=1 gap is a legitimate, if minor, issue that should be fixed.

Referee Report

1 major / 4 minor

Summary. The paper proves a spectral radius sufficient condition for the existence of {P3,P4,P5}-factors. For a connected graph G of order n≥25 and 0≤α<2/3, the authors claim that if λ_α(G) ≥ λ_α(K_1∨(K_{n-2}∪K_1)), then G has a {P3,P4,P5}-factor unless G=K_1∨(K_{n-2}∪K_1). The proof starts from the Kano-Lu-Yu criterion, so that absence of a factor yields a set S with i(G−S)>2/3|S|. The graph G is embedded into G_1=K_s∨(K_{n−⌊5s/3⌋−1}∪(⌊2s/3⌋+1)K_1), whose A_α-spectral radius is analyzed through an equitable quotient matrix. Cauchy interlacing gives a bound on the second eigenvalue, and a residue-class case analysis shows that λ_α(G_1)<λ_α(K_1∨(K_{n−2}∪K_1)) except in the s=1 case, where G_1 is the exceptional graph.

Significance. If the logical gap described below is repaired, the result is a meaningful contribution of a common type: a sharp spectral threshold for the existence of a path factor, with an explicit exceptional graph. The proof uses standard tools (equitable quotient matrices, Cauchy interlacing, monotonicity of λ_α under subgraphs) and gives detailed polynomial comparisons; there are no fitted parameters and the dependence on α is explicit. The main deficit is not in the computational core but in the missing case distinction in the initial contradictory assumption, which currently leaves the s=1 branch undischarged.

major comments (1)

[§3, s=1 paragraph] The proof by contradiction never assumes G≠K_1∨(K_{n-2}∪K_1). Since the theorem's conclusion is disjunctive, the contrary assumption should be "G contains no {P3,P4,P5}-factor and G≠K_1∨(K_{n-2}∪K_1)". In the s=1 branch the argument correctly derives λ_α(G)≤λ_α(K_1∨(K_{n-2}∪K_1)), with equality if and only if G=K_1∨(K_{n-2}∪K_1). Together with the hypothesis this forces equality and hence G=K_1∨(K_{n-2}∪K_1), which is exactly the allowed exception and is not a contradiction under the stated theorem. The branch is therefore not discharged. A minimal repair is to add G≠G* to the supposition, so that equality contradicts the added condition; alternatively, prove separately that G* itself has a {P3,P4,P5}-factor for n≥25 and use that to contradict the no-factor assumption when G=G*.

minor comments (4)

[§3, Cases 2 and 3] In Case 2 and Case 3 the text says "the quotient matrix of A(G1)", but the displayed matrices B4 and B5 contain α and are quotient matrices of A_α(G1), not of A(G1). The same notation issue occurs for "A(G2)" and "A(G3)" in Subcases 1.1 and 1.3.
[Abstract] The abstract contains the grammatical error "where α be a real number"; it should read "where α is a real number".
[References] In the introduction, reference [28] is cited as "O [28]"; the author name and citation should be given in standard form, for example "S. O [28]".
[§3, s=1 paragraph] Even after the logical repair, the sentence "This is a contradiction" should be replaced by an explicit reference to the added assumption G≠G* or to a proof that G* has a factor, so that the reader can see which premise is contradicted.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is a symbolic extremal comparison against a fixed threshold graph using the external Kano–Lu–Yu criterion; the s=1 branch has a proof gap but no self-referential reduction.

full rationale

The derivation chain is not circular. The only bridge from spectral radius to the existence of a {P3,P4,P5}-factor is Lemma 2.1, an external sufficient condition of Kano, Lu and Yu: if i(G-S) ≤ (2/3)|S| for every S, then G has the required factor. The proof argues by contrapositive, constructs the covering graph G1 = K_s ∨ (K_{n-⌊5s/3⌋−1} ∪ (⌊2s/3⌋+1)K1), and uses monotonicity (Lemma 2.3), equitable quotient matrices (Lemma 2.4), and Cauchy interlacing (Lemma 2.5) to compare λ_α(G1) with the fixed threshold graph G* = K1 ∨ (K_{n−2} ∪ K1). All comparisons are exact symbolic inequalities; no parameter is fitted to the target conclusion and no result of the present authors is load-bearing. The s=1 subcase of Case 1 does contain a genuine logical gap: the proof obtains λ_α(G) ≤ λ_α(G*) with equality iff G=G* and then calls this 'a contradiction', but the theorem's 'unless G=G*' clause means the contrary assumption must also include G≠G* to make that equality contradictory. This is a proof gap (and G* actually has a {P3,P4,P5}-factor for n≥25, so the exception is vacuous), but it is not circularity: the conclusion is not used as an input, and no equation reduces to another by construction. Therefore the circularity score is 0.

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

No free parameters are fitted to data: alpha is a fixed variable in the stated interval, and the threshold graph is determined by n. The proof relies on one domain-specific external lemma (Kano-Lu-Yu) and on standard spectral graph theory facts. No new particles, forces, dimensions, or ad hoc objects are introduced.

assumptions (3)

domain assumption Kano-Lu-Yu sufficient condition (Lemma 2.1): if i(G-S) <= (2/3)|S| for every subset S of V(G), then G contains a {P3,P4,P5}-factor.
External theorem from reference [1], used as the sole criterion linking isolated-vertex counts to the desired factor. The contrapositive drives the entire proof, and no independent proof of this lemma is given.
standard math Standard A_alpha spectral facts: lambda_alpha(K_n)=n-1, strict monotonicity under proper subgraphs, equitable quotient matrix eigenvalue properties, and the Cauchy interlacing theorem (Lemmas 2.2-2.5).
Invoked without proof from references [17], [34], and [35]. These are accepted background results in spectral graph theory.
domain assumption The theorem is scoped to connected simple graphs of order n>=25 with 0<=alpha<2/3.
The proof's case inequalities repeatedly use n>=25 and alpha<2/3, for example to bound quadratic expressions at s=15 or s=16. The theorem does not claim anything outside this range.

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Cite this review

Pith. "Pith review of An A{\alpha}-spectral radius for the existence of {P3, P4, P5}-factors in graphs." pith.science (2026). https://pith.science/paper/23R5PPPC

@misc{pith2026250105029,
  author       = {Pith},
  title        = {Pith review of: An A\alpha-spectral radius for the existence of P3, P4, P5-factors in graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/23R5PPPC}},
  note         = {Machine review of arXiv:2501.05029}
}

abstract

Let $G$ be a connected graph of order $n$ with $n\geq25$. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to an element of $\{P_3,P_4,P_5\}$. Nikiforov introduced the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ [V. Nikiforov, Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107], where $\alpha\in[0,1]$, $D(G)$ denotes the diagonal matrix of vertex degrees of $G$ and $A(G)$ denotes the adjacency matrix of $G$. The largest eigenvalue of $A_{\alpha}(G)$, denoted by $\lambda_{\alpha}(G)$, is called the $A_{\alpha}$-spectral radius of $G$. In this paper, it is proved that $G$ has a $\{P_3,P_4,P_5\}$-factor unless $G=K_1\vee(K_{n-2}\cup K_1)$ if $\lambda_{\alpha}(G)\geq\lambda_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$, where $\alpha$ be a real number with $0\leq\alpha<\frac{2}{3}$.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

M. Kano, H. Lu, Q. Yu, Component factors with large component s in graphs, Applied Mathe- matics Letters 23 (2010) 385–389

work page 2010

[2] [2]

Akiyama, D

J. Akiyama, D. Avis, H. Era, On a {1, 2}-factor of a graph, TRU Math. 16 (1980) 97–102

work page 1980

[3] [3]

Kaneko, A necessary and suﬃcient condition for the existenc e of a path factor every com- ponent of which is a path of length at least two, J

A. Kaneko, A necessary and suﬃcient condition for the existenc e of a path factor every com- ponent of which is a path of length at least two, J. Combin. Theory Se r. B 88 (2003) 195–218

work page 2003

[4] [4]

H. Liu, X. Pan, Independence number and minimum degree for pat h-factor critical uniform graphs, Discrete Applied Mathematics 359 (2024) 153–158

work page 2024

[5] [5]

W. Gao, W. Wang, Y. Chen, Tight bounds for the existence of pat h factors in network vulner- ability parameter settings, Int. J. Intell. Syst. 36(2021)1133–1 158

work page 2021

[6] [6]

G. Dai, Z. Hu, P3-factors in the square of a tree, Graphs and Combinatorics 36 (20 20) 1913– 1925

work page 1913

[7] [7]

S. Zhou, Z. Sun, H. Liu, Some suﬃcient conditions for path-fact or uniform graphs, Aequationes Mathematicae 97(3) (2023) 489–500

work page 2023

[8] [8]

Zhou, Some results on path-factor critical avoidable graphs , Discussiones Mathematicae Graph Theory 43(1) (2023) 233–244

S. Zhou, Some results on path-factor critical avoidable graphs , Discussiones Mathematicae Graph Theory 43(1) (2023) 233–244

work page 2023