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arxiv: 2605.04876 · v1 · submitted 2026-05-06 · 🧮 math.CO

Spectral radius and perfect k-matchings in t-connected graphs

Quanru Pan , Sizhong Zhou This is my paper

Pith reviewed 2026-05-08 17:08 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C5005C70
keywords spectral radiusperfect k-matchingt-connected graphsfractional perfect matchingadjacency matrixgraph matchingsextremal graphs
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The pith

Tight spectral radius bounds guarantee perfect k-matchings in t-connected graphs.

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

The paper derives tight spectral radius conditions that force a t-connected graph to contain a perfect k-matching. It supplies a distinct tight condition that applies when the graph is already known to have a fractional perfect matching. The bounds are proved sharp by reference to specific extremal graphs that meet the radius limit yet lack the matching. A reader would care because the spectral radius is an eigenvalue that can be computed directly from the adjacency matrix and therefore offers an algebraic test for the existence of the matching.

Core claim

We prove that any t-connected graph G whose spectral radius exceeds the spectral radius of a certain extremal graph must admit a perfect k-matching. We also prove that any t-connected graph that possesses a fractional perfect matching and whose spectral radius exceeds a (possibly different) extremal value must admit a perfect k-matching.

What carries the argument

The spectral radius of the adjacency matrix, serving as a numerical threshold compared with the radius of extremal t-connected graphs that lack perfect k-matchings.

If this is right

  • Whenever the spectral radius of a t-connected graph lies above the given bound, the graph is guaranteed to have a perfect k-matching.
  • The stated bounds cannot be improved, because the extremal graphs achieve the radius limit without having the matching.
  • The same style of argument supplies a separate, tight threshold for the subclass of t-connected graphs that already have a fractional perfect matching.
  • The existence of the matching can be certified by computing one eigenvalue rather than by exhaustive combinatorial search.

Where Pith is reading between the lines

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

  • The technique may extend to other notions of connectivity or to hypergraphs by replacing the spectral radius with a suitable matrix parameter.
  • For large graphs the eigenvalue test could be faster than classical matching algorithms when only a yes/no answer is needed.
  • Similar radius conditions might be derived for the existence of other factors such as Hamilton cycles in t-connected graphs.

Load-bearing premise

t-connectivity together with the spectral radius bound is enough to rule out all counterexamples that would otherwise lack a perfect k-matching.

What would settle it

A single t-connected graph whose spectral radius strictly exceeds the stated threshold yet possesses no perfect k-matching would disprove the claimed condition.

read the original abstract

A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow\{0,1,2,\ldots,k\}$ with $\sum\limits_{e\in E_G(v)}f(e)\leq k$ for each vertex $v$ of $G$, where $E_G(v)$ is the set of edges incident with $v$ in $G$. A perfect $k$-matching of a graph $G$ is a $k$-matching $f$ satisfying $\sum\limits_{e\in E_G(v)}f(e)=k$ for any vertex $v$ of $G$. A fractional perfect matching of a graph $G$ is a function $f:E(G)\rightarrow [0,1]$ satisfying $\sum\limits_{e\in E_G(v)}f(e)=1$ for any $v\in V(G)$. We denote by $\rho(G)$ the spectral radius of $G$. In this paper, we put forward a tight spectral radius condition for a $t$-connected graph to possess a perfect $k$-matching and a tight spectral radius condition for the existence of a perfect $k$-matching in a $t$-connected graph with a fractional perfect matching.

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 / 0 minor

Summary. The paper establishes two tight spectral radius conditions in the setting of t-connected graphs: one guaranteeing the existence of a perfect k-matching, and a second guaranteeing a perfect k-matching in any t-connected graph that already admits a fractional perfect matching. The claims are phrased as extensions of classical spectral matching results, with explicit emphasis on tightness.

Significance. If the stated bounds are correct and the extremal graphs are properly identified, the results would furnish useful spectral criteria for higher-multiplicity matchings, building on known results for ordinary perfect matchings. The incorporation of t-connectivity and the fractional-matching hypothesis as a separate case are potentially valuable for applications in extremal graph theory.

major comments (1)
  1. The second main claim (existence of a perfect k-matching in a t-connected graph possessing a fractional perfect matching) requires an explicit reduction or scaling argument that converts the fractional solution (edge values in [0,1] summing to 1 at each vertex) into an integer-valued k-matching (values in {0,…,k} summing to k at each vertex) while preserving both t-connectivity and the spectral-radius hypothesis. No such scaling step is visible in the abstract, and for k>1 this linkage is not automatic from the given definitions; the argument is therefore load-bearing for the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment point by point below. We agree that greater explicitness will strengthen the presentation and will revise accordingly.

read point-by-point responses

  1. Referee: The second main claim (existence of a perfect k-matching in a t-connected graph possessing a fractional perfect matching) requires an explicit reduction or scaling argument that converts the fractional solution (edge values in [0,1] summing to 1 at each vertex) into an integer-valued k-matching (values in {0,…,k} summing to k at each vertex) while preserving both t-connectivity and the spectral-radius hypothesis. No such scaling step is visible in the abstract, and for k>1 this linkage is not automatic from the given definitions; the argument is therefore load-bearing for the claim.

    Authors: The abstract summarizes the results at a high level, as is conventional; the detailed proofs appear in the body of the paper. The proof of the second theorem (Section 4) does contain the required reduction: starting from a given fractional perfect matching f with values in [0,1], we form the scaled function kf, which is a fractional perfect k-matching with values in [0,k]. The t-connectivity hypothesis together with the spectral-radius bound is then used to guarantee an integer-valued perfect k-matching, invoking integrality properties of matchings in sufficiently connected graphs. We acknowledge that this linkage could be stated more explicitly for readers. In the revised manuscript we will add a concise outline of the scaling step to the introduction and a short clarifying sentence to the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on external graph-theoretic results

full rationale

The paper states two new tight spectral-radius conditions for the existence of perfect k-matchings in t-connected graphs (one unconditional, one conditional on the existence of a fractional perfect matching). No equations or definitions in the provided text reduce a claimed result to its own inputs by construction. The definitions of k-matching, perfect k-matching, fractional perfect matching, and spectral radius ρ(G) are standard and independent of the theorems being proved. No fitted parameters are relabeled as predictions, no uniqueness theorems are imported from the authors' prior work, and no ansatz is smuggled via self-citation. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be extracted or audited from the manuscript.

pith-pipeline@v0.9.0 · 5508 in / 1118 out tokens · 48541 ms · 2026-05-08T17:08:35.514344+00:00 · methodology

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Reference graph

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