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arxiv: 2605.23283 · v1 · pith:IF74NMSVnew · submitted 2026-05-22 · 🧮 math.CO

Localized Tur\'{a}n-type inequalities for Q-index

M. Rajesh Kannan , Hitesh Kumar , Shivaramakrishna Pragada This is my paper

Pith reviewed 2026-05-25 04:23 UTC · model grok-4.3

classification 🧮 math.CO
keywords Q-indexsignless LaplacianTurán-type inequalityclique numberlocalized boundspectral graph theorycomplete multipartite graphs
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The pith

The Q-index of a graph is at most twice the sum over vertices of (1 minus 1 over c(v)), where c(v) is the largest clique containing v.

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

The paper refines the Abreu-Nikiforov bound on the Q-index q(G) by replacing the single global clique number with a sum of local values c(v). The resulting inequality states q(G) ≤ 2 ∑ (1 - 1/c(v)) and is proved by algebraic adaptation of the global argument. Equality cases are identified exactly for complete bipartite graphs when the clique number is 2 and for regular complete multipartite graphs when the clique number is at least 3. The same localization is shown to hold for the A_α-matrix and for vertex-weighted signed graphs.

Core claim

For a connected graph G, q(G) ≤ 2 ∑_{v∈V(G)} (1 - 1/c(v)), where c(v) denotes the order of the largest clique containing vertex v. Equality holds precisely for complete bipartite graphs when ω(G)=2 and for regular complete ω(G)-partite graphs when ω(G)≥3. The proof begins from the global bound q(G) ≤ 2n(1-1/ω(G)) and converts it to the summed local form by algebraic manipulation that substitutes c(v) for each vertex.

What carries the argument

The per-vertex local clique order c(v), which replaces the global ω(G) inside a summed expression to bound the largest signless-Laplacian eigenvalue.

If this is right

  • The new bound is at least as tight as the earlier global bound because c(v) ≤ ω(G) for every v.
  • An analogous localized inequality holds for the A_α-matrix of G.
  • The localized inequality extends to vertex-weighted signed graphs.
  • Equality is achieved exactly on the stated families of complete multipartite graphs.

Where Pith is reading between the lines

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

  • The same per-vertex replacement technique could be checked on other spectral radii to produce analogous localized Turán bounds.
  • Graphs whose clique sizes vary sharply across vertices should exhibit a noticeably smaller right-hand side than the global bound.
  • Direct computation of both sides on small graphs with known c(v) values would provide immediate numerical checks of the inequality.

Load-bearing premise

The algebraic manipulation that proves the global bound continues to hold after the global clique number is replaced by each vertex's own local clique number c(v).

What would settle it

A connected graph G for which the largest signless-Laplacian eigenvalue strictly exceeds 2 times the sum over v of (1 - 1/c(v)) would falsify the claimed inequality.

read the original abstract

For a connected graph \(G\), let $q(G)$ denote the $Q$-index of $G$, i.e., the largest eigenvalue of its signless Laplacian matrix. Abreu and Nikiforov (2013) showed that \[ q(G) \leq 2n\left(1-\frac{1}{\omega(G)}\right), \] where $\omega(G)$ denotes the clique number of $G$. We first give a short algebraic proof of this result. For a vertex $v\in V(G)$, let \(c(v)\) denote the order of the largest clique of \(G\) containing \(v\). Our main result is the following vertex localized bound that refines the result of Abreu and Nikiforov: \[ q(G) \leq 2\sum_{v\in V(G)}\left(1-\frac{1}{c(v)}\right). \] Equality holds precisely for complete bipartite graphs when \(\omega(G)=2\), and for regular complete \(\omega(G)\)-partite graphs when \(\omega(G)\geq 3\). As a consequence, we also obtain an analogous localized inequality for the $A_\alpha$-matrix of $G$. Finally, we generalize the above localized inequality to vertex-weighted signed graphs. This contributes to the localization program for spectral Tur\'{a}n-type results.

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

1 major / 1 minor

Summary. The paper gives a short algebraic proof of the Abreu-Nikiforov global bound q(G) ≤ 2n(1 − 1/ω(G)) on the Q-index and then claims a localized refinement q(G) ≤ 2 ∑_v (1 − 1/c(v)), where c(v) is the order of the largest clique containing vertex v. Equality is asserted for complete bipartite graphs (ω=2) and regular complete multipartite graphs (ω≥3). Analogous results are stated for the A_α-matrix and for vertex-weighted signed graphs.

Significance. A correct localization would strengthen an existing spectral Turán bound and advance the localization program for such inequalities. The algebraic approach to the global case is noted as concise; if the localization step is justified without additional hypotheses, the equality characterization would also be of interest.

major comments (1)
  1. [Abstract / main result] Abstract / main result statement: the global algebraic proof is said to yield the localized form upon replacing ω(G) by the function c(v). No explicit derivation is quoted showing that the inequality direction and the equality cases survive when c(v) is allowed to vary (and is strictly smaller than ω(G) for some vertices). This replacement is load-bearing for the central claim.
minor comments (1)
  1. [Abstract] The abstract refers to “a short algebraic proof” of the global bound but does not display the key algebraic steps; including them (even briefly) would aid verification of the subsequent localization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for a clearer justification of the localization step. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract / main result] Abstract / main result statement: the global algebraic proof is said to yield the localized form upon replacing ω(G) by the function c(v). No explicit derivation is quoted showing that the inequality direction and the equality cases survive when c(v) is allowed to vary (and is strictly smaller than ω(G) for some vertices). This replacement is load-bearing for the central claim.

    Authors: We agree that the manuscript would benefit from an explicit derivation showing how the algebraic proof of the global bound extends to the localized form when c(v) varies. In the revision we will insert a dedicated subsection that starts from the same algebraic identity used for the global case, replaces the uniform clique number ω(G) by the per-vertex function c(v) inside the relevant quadratic form or trace expression, and verifies that the inequality direction is preserved. We will also treat the equality cases separately for graphs in which c(v) is not constant, confirming that equality holds precisely for the stated families (complete bipartite graphs when ω=2 and regular complete multipartite graphs when ω≥3). revision: yes

Circularity Check

0 steps flagged

Algebraic localization of global bound is independent; no reduction to inputs by construction

full rationale

The paper first recalls the global bound q(G) ≤ 2n(1−1/ω(G)) and supplies its own short algebraic proof. It then states the localized form q(G) ≤ 2∑(1−1/c(v)) as the main result, with equality cases precisely when c(v) is constant (complete bipartite or regular complete multipartite graphs). No equation or definition equates the localized right-hand side to a fitted quantity, a self-citation, or the global bound by renaming. The algebraic steps are presented as directly adaptable to per-vertex c(v) without invoking prior self-citations as load-bearing uniqueness theorems. This is the common case of a self-contained spectral proof whose central claim does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard definitions of the signless Laplacian and clique number together with the assumption that an algebraic proof technique extends from the global to the local setting; no numerical parameters are fitted and no new entities are postulated.

axioms (2)
  • domain assumption G is a connected simple undirected graph.
    Required for the definition of q(G) in the abstract.
  • standard math The signless Laplacian matrix is symmetric and its largest eigenvalue satisfies the stated global bound.
    Invoked when the paper states it first proves the Abreu–Nikiforov result algebraically.

pith-pipeline@v0.9.0 · 5777 in / 1332 out tokens · 32265 ms · 2026-05-25T04:23:21.531531+00:00 · methodology

discussion (0)

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Reference graph

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