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arxiv: 2501.03806 · v1 · submitted 2025-01-07 · 🧮 math.CO

Bounds on A_α-eigenvalues using graph invariants

Jo\~ao Domingos Gomes da Silva Junior , Carla Silva Oliveira , Liliana Manuela Gaspar Cerveira da Costa This is my paper

Pith reviewed 2026-05-23 06:21 UTC · model grok-4.3

classification 🧮 math.CO
keywords A_alpha matrixeigenvalue boundsgraph invariantsadjacency matrixsignless Laplacianspectral graph theoryconvex combination
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The pith

Bounds on the largest and smallest eigenvalues of the A_α-matrix are derived from graph invariants.

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

The paper derives several inequalities that bound the largest and smallest eigenvalues of the A_α-matrix using standard graph invariants such as the number of vertices, number of edges, and degree information. The A_α-matrix is presented as a unifying object that interpolates between the adjacency matrix and the signless Laplacian matrix via a convex combination parameter α. A reader interested in spectral graph theory would care because these bounds apply to every graph and every α in [0,1], offering estimates that do not require direct computation of the full spectrum. The work extends known spectral results by expressing them in a single parameterized form.

Core claim

The authors present some bounds for the largest and smallest eigenvalue of the A_α-matrix involving invariants associated to graphs. The A_α-matrix is introduced as the linear convex combination (1−α)A + αD of the adjacency matrix A and the degree diagonal matrix D, serving as a structure that combines properties of both the adjacency matrix and the signless Laplacian matrix.

What carries the argument

The A_α-matrix defined by the convex combination (1-α)A + αD, with eigenvalue bounds expressed directly in terms of graph invariants such as order, size, and degrees.

If this is right

  • The bounds recover known results for adjacency eigenvalues when α equals 0 and for signless Laplacian eigenvalues when α equals 1.
  • The inequalities supply estimates for spectral radius and smallest eigenvalue using only the order n, size m, and maximum degree Δ.
  • The same inequalities apply uniformly across the entire family of A_α-matrices for fixed graph invariants.
  • Direct comparison of graphs becomes possible via their shared invariants without computing full spectra.

Where Pith is reading between the lines

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

  • The bounds may prove sharpest on regular or nearly regular graphs, where degree variation is minimal.
  • Similar bounding techniques could be applied to other parameterized matrix families such as weighted or signed versions.
  • Numerical checks on small graphs would reveal whether the new inequalities improve upon existing separate bounds for A and Q.

Load-bearing premise

The stated bounds are valid for every graph under the standard definition of the A_α matrix as the convex combination (1-α)A + αD.

What would settle it

A concrete graph G together with a specific α in [0,1] for which one of the stated eigenvalue bounds fails to hold.

Figures

Figures reproduced from arXiv: 2501.03806 by Carla Silva Oliveira, Jo\~ao Domingos Gomes da Silva Junior, Liliana Manuela Gaspar Cerveira da Costa.

Figure 1
Figure 1. Figure 1: Examples of graphs to illustrate the concepts of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

In 2017, Nikiforov introduced the concept of the $A_{\alpha}$-matrix, as a linear convex combination of the adjacency matrix and the degree diagonal matrix of a graph. This matrix has attracted increasing attention in recent years, as it serves as a unifying structure that combines the adjacency matrix and the signless Laplacian matrix. In this paper, we present some bounds for the largest and smallest eigenvalue of $A_{\alpha}$-matrix involving invariants associated to graphs.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript presents several bounds on the largest and smallest eigenvalues of the A_α matrix (defined as the convex combination (1-α)A + αD) expressed in terms of standard graph invariants such as order, size, maximum/minimum degree, and related quantities.

Significance. If the derivations are correct, the bounds add to the body of inequalities in A_α-spectral graph theory, which unifies the adjacency and signless-Laplacian spectra; they may be useful for extremal problems and graph characterization when the invariants are easy to compute.

minor comments (3)
  1. The abstract states that bounds are presented but does not name the specific invariants or indicate whether the bounds are sharp; adding one sentence listing the main invariants would improve readability.
  2. Include at least one small example graph (e.g., a cycle or complete graph) with explicit numerical verification of each stated bound to allow readers to check the inequalities directly.
  3. Ensure that the introduction cites the most recent papers on A_α-eigenvalues (post-2020) so that the novelty of the new bounds relative to existing literature is clear.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work on bounds for the A_α-eigenvalues and for recommending minor revision. The referee's summary accurately captures the content of the manuscript.

Circularity Check

0 steps flagged

No circularity: bounds derived from standard linear algebra without reduction to inputs

full rationale

The paper derives bounds on the largest and smallest A_α-eigenvalues from graph invariants using the standard convex combination definition of A_α introduced by Nikiforov. No load-bearing step reduces a claimed bound to a fitted parameter, self-definition, or self-citation chain; the abstract and structure indicate conventional spectral graph theory inequalities that remain independent of the target results. This is the expected non-circular outcome for such bounding arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of the A_alpha matrix and basic facts from linear algebra about eigenvalues of symmetric matrices; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The A_α matrix equals (1-α)A + αD, where A is the adjacency matrix and D the diagonal degree matrix.
    Invoked directly in the abstract as the unifying structure introduced by Nikiforov.
  • standard math Eigenvalues of real symmetric matrices are real and can be bounded using trace, norms, or other matrix invariants.
    Implicit background assumption required for any eigenvalue bound.

pith-pipeline@v0.9.0 · 5609 in / 1250 out tokens · 30473 ms · 2026-05-23T06:21:22.037907+00:00 · methodology

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Reference graph

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