Sharp upper bounds on the $A_\alpha$-spectral radius of graphs

Zheng-Jiang Xia; Zhen-Mu Hong; Zhi Qiao

arxiv: 2606.00843 · v1 · pith:FL6BTRXDnew · submitted 2026-05-30 · 🧮 math.CO

Sharp upper bounds on the A_α-spectral radius of graphs

Zhen-Mu Hong , Zheng-Jiang Xia , Zhi Qiao This is my paper

Pith reviewed 2026-06-28 18:02 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C50

keywords A_alpha spectral radiusupper boundsvertex deletionspectral radiussignless Laplacianadjacency matrixgraph eigenvalues

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The pith

The A_α-spectral radius satisfies sharp upper bounds on its drop after vertex deletion, extending and tightening prior spectral inequalities.

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

The paper derives upper bounds on how much the A_α-spectral radius decreases when a vertex is removed from a simple graph. It extends the known squared-radius inequality to a family parameterized by k that matches the Sun-Das bound at k=2 but is strictly smaller otherwise when degree differs from the radius. A short proof is supplied for the direct difference bound on ρ_α itself. The work also unifies earlier separate inequalities for the adjacency spectral radius and the signless Laplacian spectral radius into statements using the single A_α family.

Core claim

For any simple graph G and non-isolated vertex v of degree d_v, the inequality ρ_α(G) − ρ_α(G−v) ≤ α + (1−α)² d_v / (ρ_α(G) − α d_v) holds. As a corollary, ρ²(G) − ρ²(G−v) ≤ 2d_v −1 + (k−2)(d_v/ρ(G)−1) for every k in [0, d_v+1]; this recovers the Sun-Das bound exactly at k=2 and improves it whenever k≠2 and d_v ≠ ρ(G). The same framework yields a unified generalization of the Hong-Shu-Fang bound for ρ(G) and Nikiforov's bound for q(G).

What carries the argument

The A_α-matrix A_α(G) = α D(G) + (1−α) A(G) whose largest eigenvalue ρ_α(G) interpolates between the adjacency spectral radius at α=0 and half the signless Laplacian radius at α=1/2.

If this is right

The parameterized family recovers the Sun-Das squared-radius bound exactly when k=2.
Choosing k≠2 yields a strictly tighter upper bound whenever d_v ≠ ρ(G).
The direct A_α difference bound recovers known cases at the endpoint values α=0 and α=1/2.
A single statement now covers both the Hong-Shu-Fang inequality for the adjacency radius and Nikiforov's inequality for the signless Laplacian radius.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The k-parameterization offers a continuum of bounds that could be optimized over specific graph classes or degree sequences.
The same matrix-family approach may extend to other convex combinations of graph matrices beyond the A_α line.
The short self-contained proofs suggest the underlying matrix inequalities are robust enough to apply uniformly for all α in [0,1].

Load-bearing premise

The proofs assume the largest eigenvalue is simple with a positive eigenvector so that Perron-Frobenius properties and standard variational or matrix inequalities can be applied directly.

What would settle it

A concrete simple graph G, non-isolated vertex v, and value α where ρ_α(G) − ρ_α(G−v) exceeds α + (1−α)² d_v / (ρ_α(G) − α d_v) would disprove the central inequality.

read the original abstract

Let $G$ be a simple graph with degree diagonal matrix $D(G)$ and adjacency matrix $A(G)$. The signless Laplacian matrix of $G$ is defined as $Q(G)=D(G)+A(G)$. For a real number $\alpha \in [0, 1]$, Nikiforov (2017) proposed the $A_\alpha$-matrix of a graph $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$. The $A_\alpha$-spectral radius of $G$, denoted by $\rho_\alpha(G)$, is the largest eigenvalue of $A_\alpha(G)$, where $\rho_0(G)=\rho(G)$ is the spectral radius of $A(G)$ and $2\rho_{\frac{1}{2}}(G)=q(G)$ is the spectral radius of $Q(G)$. Sun and Das (2020) proved that for any non-isolated vertex $v$ of degree $d_v$, $\rho^2(G)-\rho^2(G-v) \leq 2 d_v-1$, which confirmed the conjecture originally posed by Guo, Wang, and Li (2019). Recently, Liu and Ning (2026) provided a short and self-contained proof of this inequality. In this paper, we establish the corresponding result for $\rho_\alpha(G)$. As a corollary, for every $k\in [0,d_v+1]$, we have $$ \rho^2(G)- \rho^2(G-v) \leq 2d_v-1 +(k-2)\left(\frac{d_v}{\rho(G)}-1\right). $$ This inequality coincides with that of Sun and Das when $k=2$, and is strictly sharper than theirs whenever $k\neq 2$ and $d_v\neq \rho(G)$. We also give a short proof of the inequality $\rho_{\alpha}(G)-\rho_{\alpha}(G-v)\leq \alpha +\frac{(1-\alpha)^2d_v}{\rho_{\alpha}(G)-\alpha d_v}$, which is obtained by Wang and She (2022). Moreover, we obtain a unified generalization of Hong, Shu and Fang's inequality for $\rho(G)$ and Nikiforov's inequality for $q(G)$ in terms of $\rho_\alpha(G)$.

Editorial analysis

A structured set of objections, weighed in public.

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.

Desk Editor's Note private letter to a colleague

The paper adds a tunable k-parameter that can tighten the Sun-Das bound on ρ²(G)−ρ²(G−v) for some graphs and unifies a few older inequalities under the A_α matrix.

read the letter

The main takeaway is that this work extends the Sun-Das bound on the drop in squared spectral radius after vertex removal by introducing a free parameter k in [0, d_v+1]. The resulting family recovers the original inequality at k=2 and is claimed to be strictly sharper otherwise when d_v differs from ρ(G). They also supply a short proof of the A_α difference bound that Wang-She had and fold Hong-Shu-Fang and Nikiforov into one statement via ρ_α.

What is actually new is the k-dependent corollary and the explicit unification across α values. The paper does well by stating the special cases clearly and by working from standard matrix inequalities rather than ad-hoc constructions.

The soft spots are limited. The improvement is not uniform; for many graphs the best k may not yield a noticeable gain, and the abstract leaves the exact first step of the derivation implicit. Since the arguments rest on Perron-Frobenius for the nonnegative matrix A_α and on Rayleigh quotients, they inherit the usual conditions for connected graphs. No internal contradictions appear in the stated claims.

This is for readers already working on spectral radius bounds or A_α matrices. Someone who follows Sun-Das or Nikiforov will see the direct extensions and can check the corollaries against known graphs. The paper shows clear engagement with the literature and produces falsifiable statements, so it deserves a serious referee even if the practical sharpening turns out modest.

Referee Report

0 major / 1 minor

Summary. The paper establishes sharp upper bounds on the A_α-spectral radius ρ_α(G) of a simple graph G. It proves the inequality ρ_α(G) − ρ_α(G − v) ≤ α + (1 − α)² d_v / (ρ_α(G) − α d_v) for a non-isolated vertex v of degree d_v, gives a short proof of a result previously obtained by Wang and She, derives the parameterized corollary ρ²(G) − ρ²(G − v) ≤ 2 d_v − 1 + (k − 2)(d_v / ρ(G) − 1) for k ∈ [0, d_v + 1] that sharpens the Sun-Das bound when k ≠ 2 and d_v ≠ ρ(G), and obtains a unified generalization of the Hong-Shu-Fang inequality for ρ(G) and Nikiforov's inequality for q(G) in the A_α setting.

Significance. If the derivations hold, the work supplies a natural extension of existing spectral-radius bounds to the one-parameter family A_α(G) = α D(G) + (1 − α) A(G), recovers the classical cases α = 0 and α = 1/2 as special instances, and supplies a strictly sharper family of inequalities for the adjacency spectral radius via the free parameter k. The short proofs and unification are useful contributions to the literature on A_α-spectral radii.

minor comments (1)

The abstract states the corollary for every k ∈ [0, d_v + 1], but the manuscript should explicitly verify that the inequality remains valid (or is vacuously true) at the endpoints k = 0 and k = d_v + 1 when d_v ≠ ρ(G).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its contributions to the A_α-spectral radius literature, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivations use standard matrix inequalities

full rationale

The paper derives upper bounds on ρ_α(G) by extending known results for ρ(G) and q(G) via the definition A_α(G) = αD(G) + (1-α)A(G) and standard applications of the Rayleigh quotient and Perron-Frobenius theorem for nonnegative matrices. The claimed inequalities are obtained from eigenvector equations and variational characterizations without defining any quantity in terms of itself or renaming fitted parameters as predictions. Self-citations (e.g., to Hong-Shu-Fang) are used only for unification of prior independent inequalities and are not load-bearing for the new proofs. The derivations remain self-contained against external matrix theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard linear-algebra facts about symmetric matrices and the variational characterization of the largest eigenvalue; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math The largest eigenvalue of a real symmetric matrix is characterized by the Rayleigh quotient and satisfies standard interlacing or perturbation inequalities.
Invoked implicitly to relate ρ_α(G) and ρ_α(G−v).

pith-pipeline@v0.9.1-grok · 5971 in / 1350 out tokens · 25621 ms · 2026-06-28T18:02:43.296047+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages

[1]

Utilizing matrix theory, Jin, Zhang and Zhang [5] obtained several upper bounds of the spectral radius of symmetric matrices, which yielded a new proof for the above equality

confirmed this conjecture using the eigenvector corresponding toρ(G), and characterized all connected graphs for which this bound is attained. Utilizing matrix theory, Jin, Zhang and Zhang [5] obtained several upper bounds of the spectral radius of symmetric matrices, which yielded a new proof for the above equality. Recently, Liu and Ning [6] provided a ...
[2]

Guo, Z.W

J.M. Guo, Z.W. Wang, X. Li, Sharp upper bounds of the spectral radius of a graph, Discrete Math. 342 (2019) 2559–2563. doi:10.1016/j.disc.2019.05.017

work page doi:10.1016/j.disc.2019.05.017 2019
[3]

Hong, A bound on the spectral radius of graphs, Linear Algebra Appl

Y. Hong, A bound on the spectral radius of graphs, Linear Algebra Appl. 108 (1988) 135–139. doi:10.1016/0024-3795(88)90183-8

work page doi:10.1016/0024-3795(88)90183-8 1988
[4]

Y. Hong, J. Shu, and K. Fang, A sharp upper bound of the spectral radius of graphs, J. Combin. Theory, Ser. B, 81 (2001) 177–183. doi: 10.1016/S0012-365X(99)90280-7

work page doi:10.1016/s0012-365x(99)90280-7 2001
[5]

Huang, H

X. Huang, H. Lin, J. Xue, The Nordhaus–Gaddum type inequalities ofA α-matrix, Appl. Math. Comput. 365 (2020) 124716. doi:10.1016/j.amc.2019.124716

work page doi:10.1016/j.amc.2019.124716 2020
[6]

Y.L. Jin, J. Zhang, X.D. Zhang, Upper bounds of spectral radius of symmetric matrices and graphs, Linear Algebra Appl. 682 (2024) 152–163. doi:10.1016/j.laa.2023.11.008

work page doi:10.1016/j.laa.2023.11.008 2024
[7]

Liu and B

L. Liu and B. Ning, A short proof of an inequality on spectral radius perturbation, arXiv:2603.29335v1 11

arXiv
[8]

C. Li, H. Wang, P.V. Mieghem, Bounds for the spectral radius of a graph when nodes are removed, Linear Algebra Appl. 437 (2012) 319–323. doi:10.1016/j.laa.2012.02.023

work page doi:10.1016/j.laa.2012.02.023 2012
[9]

Mieghem, D

P.V. Mieghem, D. Stevanovi´ c, F.A. Kuipers, C. Li, R. Bovenkamp, D. Liu, H. Wang, Decreasing the spectral radius of a graph by link removals, Phys. Rev. E 84 (1) (2011) 016101. doi:10.1103/PhysRevE.84.016101

work page doi:10.1103/physreve.84.016101 2011
[10]

Nikiforov, Some inequalities for the largest eigenvalue of a graph, Combin

V. Nikiforov, Some inequalities for the largest eigenvalue of a graph, Combin. Probab. Comput. 11 (2002) 179–189. doi:10.1017/S0963548301004928

work page doi:10.1017/s0963548301004928 2002
[11]

Nikiforov, Maxima of theQ-index: degenerate graphs, Electron

V. Nikiforov, Maxima of theQ-index: degenerate graphs, Electron. J. Linear Algebra, 27 (2014) 250–257. doi:10.13001/1081-3810.1616

work page doi:10.13001/1081-3810.1616 2014
[12]

1, 81–107

V. Nikiforov, Merging theA-andQ-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81–107. doi:10.2298/aadm1701081n

work page doi:10.2298/aadm1701081n 2017
[13]

Sun, K.C

S. Sun, K.C. Das, A conjecture on the spectral radius of graphs, Linear Algebra Appl. 588 (2020) 74–80. doi:10.1016/j.laa.2019.11.028

work page doi:10.1016/j.laa.2019.11.028 2020
[14]

C. Wang, T. She, A bound for theA α-spectral radius of a connected graph after vertex deletion, RAIRO-Oper. Res. 56 (2022) 3635–3642. doi: 10.1051/ro/2022176

work page doi:10.1051/ro/2022176 2022
[15]

Zhou and H.H

B. Zhou and H.H. Cho, Remarks on spectral radius and Laplacian eigenvalues of a graph, Czechoslovak Math. J. 55 (2005) 781–790. doi:10.1007/s10587-005-0064-3 12

work page doi:10.1007/s10587-005-0064-3 2005

[1] [1]

Utilizing matrix theory, Jin, Zhang and Zhang [5] obtained several upper bounds of the spectral radius of symmetric matrices, which yielded a new proof for the above equality

confirmed this conjecture using the eigenvector corresponding toρ(G), and characterized all connected graphs for which this bound is attained. Utilizing matrix theory, Jin, Zhang and Zhang [5] obtained several upper bounds of the spectral radius of symmetric matrices, which yielded a new proof for the above equality. Recently, Liu and Ning [6] provided a ...

[2] [2]

Guo, Z.W

J.M. Guo, Z.W. Wang, X. Li, Sharp upper bounds of the spectral radius of a graph, Discrete Math. 342 (2019) 2559–2563. doi:10.1016/j.disc.2019.05.017

work page doi:10.1016/j.disc.2019.05.017 2019

[3] [3]

Hong, A bound on the spectral radius of graphs, Linear Algebra Appl

Y. Hong, A bound on the spectral radius of graphs, Linear Algebra Appl. 108 (1988) 135–139. doi:10.1016/0024-3795(88)90183-8

work page doi:10.1016/0024-3795(88)90183-8 1988

[4] [4]

Y. Hong, J. Shu, and K. Fang, A sharp upper bound of the spectral radius of graphs, J. Combin. Theory, Ser. B, 81 (2001) 177–183. doi: 10.1016/S0012-365X(99)90280-7

work page doi:10.1016/s0012-365x(99)90280-7 2001

[5] [5]

Huang, H

X. Huang, H. Lin, J. Xue, The Nordhaus–Gaddum type inequalities ofA α-matrix, Appl. Math. Comput. 365 (2020) 124716. doi:10.1016/j.amc.2019.124716

work page doi:10.1016/j.amc.2019.124716 2020

[6] [6]

Y.L. Jin, J. Zhang, X.D. Zhang, Upper bounds of spectral radius of symmetric matrices and graphs, Linear Algebra Appl. 682 (2024) 152–163. doi:10.1016/j.laa.2023.11.008

work page doi:10.1016/j.laa.2023.11.008 2024

[7] [7]

Liu and B

L. Liu and B. Ning, A short proof of an inequality on spectral radius perturbation, arXiv:2603.29335v1 11

arXiv

[8] [8]

C. Li, H. Wang, P.V. Mieghem, Bounds for the spectral radius of a graph when nodes are removed, Linear Algebra Appl. 437 (2012) 319–323. doi:10.1016/j.laa.2012.02.023

work page doi:10.1016/j.laa.2012.02.023 2012

[9] [9]

Mieghem, D

P.V. Mieghem, D. Stevanovi´ c, F.A. Kuipers, C. Li, R. Bovenkamp, D. Liu, H. Wang, Decreasing the spectral radius of a graph by link removals, Phys. Rev. E 84 (1) (2011) 016101. doi:10.1103/PhysRevE.84.016101

work page doi:10.1103/physreve.84.016101 2011

[10] [10]

Nikiforov, Some inequalities for the largest eigenvalue of a graph, Combin

V. Nikiforov, Some inequalities for the largest eigenvalue of a graph, Combin. Probab. Comput. 11 (2002) 179–189. doi:10.1017/S0963548301004928

work page doi:10.1017/s0963548301004928 2002

[11] [11]

Nikiforov, Maxima of theQ-index: degenerate graphs, Electron

V. Nikiforov, Maxima of theQ-index: degenerate graphs, Electron. J. Linear Algebra, 27 (2014) 250–257. doi:10.13001/1081-3810.1616

work page doi:10.13001/1081-3810.1616 2014

[12] [12]

1, 81–107

V. Nikiforov, Merging theA-andQ-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81–107. doi:10.2298/aadm1701081n

work page doi:10.2298/aadm1701081n 2017

[13] [13]

Sun, K.C

S. Sun, K.C. Das, A conjecture on the spectral radius of graphs, Linear Algebra Appl. 588 (2020) 74–80. doi:10.1016/j.laa.2019.11.028

work page doi:10.1016/j.laa.2019.11.028 2020

[14] [14]

C. Wang, T. She, A bound for theA α-spectral radius of a connected graph after vertex deletion, RAIRO-Oper. Res. 56 (2022) 3635–3642. doi: 10.1051/ro/2022176

work page doi:10.1051/ro/2022176 2022

[15] [15]

Zhou and H.H

B. Zhou and H.H. Cho, Remarks on spectral radius and Laplacian eigenvalues of a graph, Czechoslovak Math. J. 55 (2005) 781–790. doi:10.1007/s10587-005-0064-3 12

work page doi:10.1007/s10587-005-0064-3 2005