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arxiv: 2606.00691 · v1 · pith:DM5X4GKWnew · submitted 2026-05-30 · 🧮 math.CO

Sharp A_α-Spectral Conditions for Odd [1,b]-Factors When α>1/2

Silin Huang This is my paper

Pith reviewed 2026-06-28 18:27 UTC · model grok-4.3

classification 🧮 math.CO
keywords A_α-spectral radiusodd [1,b]-factorextremal graphspectral conditionobstruction graphcrossing threshold
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The pith

For α > 1/2 the A_α-spectral radius of graphs without an odd [1,b]-factor is maximized by either G1 or Gk, with the maximum switching at a single crossing value α*.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper resolves the extremal problem for the A_α-spectral radius among all even-order graphs that lack an odd [1,b]-factor when α lies between 1/2 and 1. Prior work had reduced the question to comparing the radii of a specific family of obstruction graphs Gs. The authors show that, for every such α, the largest radius in the family is attained only at the two endpoint graphs G1 and Gk. They further prove that these two radii cross exactly once at a unique α*(n,b) that depends on the order n and the parameter b, so the identity of the unique maximizer changes exactly once as α increases through the interval.

Core claim

For every α ∈ (1/2,1) the equality max_{1≤s≤k} ρ_α(G_s) = max{ρ_α(G_1), ρ_α(G_k)} holds; moreover, for each fixed odd b ≥ 3 and all even n ≥ N_b there exists a unique α* = α*(n,b) ∈ (1/2,1) at which ρ_α(G_1) = ρ_α(G_k), with the consequence that G_1 alone maximizes the radius when 1/2 < α < α*, both graphs maximize it at α = α*, and G_k alone maximizes it when α* < α < 1.

What carries the argument

The one-parameter family of obstruction graphs G_s = K_s ∇ (K_{n−(b+1)s−1} ∪ (bs+1)K_1) for s = 1 to k = ⌊(n−2)/(b+1)⌋, together with the A_α-matrix whose spectral radius is compared across the family.

Load-bearing premise

That the global maximum among all obstruction graphs occurs only at the endpoints s=1 and s=k, and that the two endpoint radii cross exactly once for all sufficiently large even n.

What would settle it

For a concrete odd b ≥ 3 and even n ≥ N_b, compute the functions ρ_α(G_1) and ρ_α(G_k) and check whether they intersect at more than one point in (1/2,1) or whether some intermediate G_s exceeds both endpoints for some α.

Figures

Figures reproduced from arXiv: 2606.00691 by Silin Huang.

Figure 1
Figure 1. Figure 1: The sign pattern of Dn(c) in the endpoint comparison. The plotted points are computed from the quotient matrices for b = 3 and n = N3 = 58. Equivalently, Kb = ( 14, b ∈ {3, 5}, 2b + 3, b ≥ 7, b odd, Nb =    58, b = 3, 86, b = 5, 2b 2 + 5b + 5, b ≥ 7, b odd. (88) Note that n ≥ Nb is equivalent to k ≥ Kb. Lemma 5.1. Let b ≥ 3 be odd and let n ≥ Nb be even. Then Dn(c) < 0 (0 < c ≤ b + 1). (89) Proof. I… view at source ↗
read the original abstract

We solve, for all sufficiently large even orders, the problem proposed by Chen et al. on sharp $A_\alpha$-spectral conditions for the existence of odd $[1,b]$-factors when $\alpha>1/2$. Chen et al. showed that every connected graph of even order $n$ with no odd $[1,b]$-factor has $A_\alpha$-spectral radius at most $\max_{1\le s\le k}\rho_\alpha(G_s)$, where $G_s=K_s\nabla\left(K_{n-(b+1)s-1}\cup(bs+1)K_1\right)$ and $k=\lfloor(n-2)/(b+1)\rfloor$. Thus the problem reduces to finding the graph with the largest $A_\alpha$-spectral radius among these obstruction graphs. We prove that, for every $\alpha\in(1/2,1)$, $\max_{1\le s\le k}\rho_\alpha(G_s)=\max\{\rho_\alpha(G_1),\rho_\alpha(G_k)\}$. Moreover, for each fixed odd $b\ge 3$ and every even $n\ge N_b=(b+1)\max\{2b+3,14\}+2$, there exists a unique $\alpha=\alpha_\ast(n,b)\in(1/2,1)$ at which $\rho_\alpha(G_1)=\rho_\alpha(G_k)$. Consequently, $G_1$ is the unique extremal graph for $1/2<\alpha<\alpha_\ast(n,b)$, both $G_1$ and $G_k$ are extremal at $\alpha=\alpha_\ast(n,b)$, and $G_k$ is the unique extremal graph for $\alpha_\ast(n,b)<\alpha<1$. This gives the exact $A_\alpha$-spectral threshold, together with the sharp exceptional graphs, for odd $[1,b]$-factors when $\alpha>1/2$ and $n\ge N_b$.

Editorial analysis

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Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper solves the extremal problem posed by Chen et al. for A_α-spectral conditions guaranteeing an odd [1,b]-factor when α>1/2. It proves that among the obstruction graphs G_s = K_s ∇ (K_{n-(b+1)s-1} ∪ (bs+1)K_1) for s=1 to k=⌊(n-2)/(b+1)⌋, the maximum A_α-spectral radius is attained only at the endpoints s=1 or s=k for every α∈(1/2,1). For each fixed odd b≥3 and even n≥N_b=(b+1)max{2b+3,14}+2, there is a unique crossing value α_*(n,b)∈(1/2,1) at which ρ_α(G_1)=ρ_α(G_k), so that G_1 is uniquely extremal below α_*, both are extremal at α_*, and G_k is uniquely extremal above α_*. This yields the exact spectral threshold together with the sharp exceptional graphs.

Significance. The result completes the A_α-spectral characterization of odd [1,b]-factors for α>1/2 and all sufficiently large even orders, supplying the precise threshold function and the two families of extremal graphs. The argument rests on explicit characteristic equations for the radii ρ_α(G_s), sign analysis near the endpoints α=1/2+ and α=1−, and a monotonicity argument establishing exactly one sign change; these steps are internal to the manuscript and do not rely on fitted parameters.

minor comments (3)
  1. §1, line after (1.3): the notation N_b is introduced without an immediate parenthetical reminder of its explicit form; repeating the formula (b+1)max{2b+3,14}+2 would improve readability for readers who consult only the introduction.
  2. §3, after Lemma 3.2: the auxiliary function whose monotonicity implies uniqueness of α_* is defined only implicitly via the difference of the two characteristic polynomials; an explicit one-line formula for this function would make the crossing argument easier to follow.
  3. Table 1 (if present) or the numerical checks in §5: the reported values of α_*(n,b) for small b should include the corresponding n-range to confirm they satisfy n≥N_b.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the thorough and positive report, which accurately captures the main contributions of the paper. We are pleased that the referee recommends acceptance.

Circularity Check

0 steps flagged

Derivation self-contained with no circular steps

full rationale

The paper takes the family of obstruction graphs {G_s} and the bound max ρ_α(G_s) as given from the external Chen et al. result (different authors), then derives the endpoint-maximum property and the unique crossing α_* by explicit characteristic equations for ρ_α(G_s), sign analysis of the difference near α=1/2+ and α=1−, and monotonicity of an auxiliary function. These steps are independent of the input data and do not reduce any claimed prediction or uniqueness statement back to a fitted quantity or self-citation chain defined by the same paper. No self-definitional, fitted-input, or ansatz-smuggling patterns appear.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior theorem of Chen et al. that every connected even-order graph without an odd [1,b]-factor has A_α-radius at most max ρ_α(G_s), together with standard facts about the spectral radius of the A_α-matrix (non-negative matrix theory, Perron-Frobenius) and the assumption that n is large enough for the crossing analysis to hold. No free parameters or invented entities appear.

axioms (2)
  • standard math The A_α-matrix is a convex combination of the adjacency and degree matrices; its spectral radius is a continuous, strictly increasing function of α on (0,1) for connected graphs.
    Invoked implicitly when comparing ρ_α(G_s) across different α.
  • domain assumption The family of obstruction graphs G_s constructed by Chen et al. exhausts all minimal connected graphs without an odd [1,b]-factor.
    Taken directly from the cited prior result; the present paper only compares radii inside that family.

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Reference graph

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