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arxiv: 2604.05869 · v1 · submitted 2026-04-07 · 🧮 math.CO

Distance spectral radius and perfect matchings in graphs with given fractional property

Sizhong Zhou This is my paper

Pith reviewed 2026-05-10 19:36 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C5005C70
keywords distance spectral radiusperfect matchingfractional perfect matchingk-connected graphextremal graphjoin graph
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The pith

A k-connected graph of even order with a fractional perfect matching has a perfect matching if its distance spectral radius is at most that of a specific join graph, unless the graph is exactly that join.

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

The paper proves a spectral condition for perfect matchings. Take any k-connected graph G on an even number n of vertices that admits a fractional perfect matching. When n is at least 8k+6 and the distance spectral radius of G is no larger than the radius of the graph obtained by joining a complete graph K_k to the disjoint union of k isolated vertices, one triangle, and one larger complete graph, then G must contain a perfect matching. The sole exception is when G itself is precisely that join graph. The result therefore converts an upper bound on the largest distance-matrix eigenvalue into a guarantee of a perfect matching, once connectivity and the fractional condition are already satisfied.

Core claim

Let G be a k-connected graph of even order n with a fractional perfect matching, where k is a positive integer. We denote by μ(G) the distance spectral radius of G. In this paper, we prove that if n≥8k+6 and μ(G)≤μ(K_k∨(kK_1∪K_3∪K_{n-2k-3})), then G contains a perfect matching unless G=K_k∨(kK_1∪K_3∪K_{n-2k-3}).

What carries the argument

The distance spectral radius μ(G), the largest eigenvalue of the distance matrix, whose upper bound forces the graph to be either the named extremal join or to possess a perfect matching.

Load-bearing premise

The order must be at least 8k+6 and the distance spectral radius must not exceed the value for that specific join graph; smaller orders or larger radii remove the guarantee.

What would settle it

A k-connected even-order graph with a fractional perfect matching, n at least 8k+6, whose distance spectral radius is at most that of the extremal join yet which lacks a perfect matching and is not identical to the join.

read the original abstract

A matching in a graph $G$ is a set of independent edges in $G$. A perfect matching in a graph $G$ is a matching which saturates all the vertices of $G$. A fractional perfect matching in a graph $G$ is a function $h:E(G)\rightarrow [0,1]$ such that $\sum\limits_{e\in E_G(v)}h(e)=1$ for every $v\in V(G)$, where $E_G(v)$ is the set of edges incident to $v$ in $G$. Clearly, the existence of a fractional perfect matching in a graph is a necessary condition for the graph to possess a perfect matching. Let $G$ be a $k$-connected graph of even order $n$ with a fractional perfect matching, where $k$ is a positive integer. We denote by $\mu(G)$ the distance spectral radius of $G$. In this paper, we prove that if $n\geq8k+6$ and $\mu(G)\leq\mu(K_k\vee(kK_1\cup K_3\cup K_{n-2k-3}))$, then $G$ contains a perfect matching unless $G=K_k\vee(kK_1\cup K_3\cup K_{n-2k-3})$.

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 manuscript proves that if G is a k-connected graph of even order n with a fractional perfect matching, and if n ≥ 8k+6 and the distance spectral radius μ(G) ≤ μ(K_k ∨ (kK_1 ∪ K_3 ∪ K_{n-2k-3})), then G has a perfect matching unless G is isomorphic to the extremal join graph H = K_k ∨ (kK_1 ∪ K_3 ∪ K_{n-2k-3}).

Significance. If correct, the result gives a precise spectral extremal condition for perfect matchings under k-connectivity and the existence of a fractional perfect matching. It strengthens the literature on distance-spectral conditions for matchings by identifying an explicit extremal graph and using the fractional-matching hypothesis as a natural weakening of the conclusion.

major comments (1)
  1. [Main result (abstract statement)] The size threshold n ≥ 8k+6 is load-bearing for the central claim (see the statement in the abstract). The proof must show that every graph satisfying the connectivity and fractional-matching hypotheses but lacking a perfect matching has strictly larger distance spectral radius than H once n reaches this value; if the eigenvalue comparison fails to be strict for some barrier configuration (different odd-set partitions or component sizes) at this n, the implication does not hold for all k.
minor comments (1)
  1. [Abstract] The notation K_k ∨ (kK_1 ∪ K_3 ∪ K_{n-2k-3}) is compact but would benefit from an explicit small-k example or diagram showing the distance matrix structure of H.

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 major comment point by point below.

read point-by-point responses

  1. Referee: The size threshold n ≥ 8k+6 is load-bearing for the central claim (see the statement in the abstract). The proof must show that every graph satisfying the connectivity and fractional-matching hypotheses but lacking a perfect matching has strictly larger distance spectral radius than H once n reaches this value; if the eigenvalue comparison fails to be strict for some barrier configuration (different odd-set partitions or component sizes) at this n, the implication does not hold for all k.

    Authors: Our proof of the main theorem (Theorem 1.1) does establish the required strict inequality for every admissible graph at n ≥ 8k+6. After invoking Tutte's theorem, we let S be a minimum Tutte set with |S| = k (guaranteed by k-connectivity). The fractional perfect matching hypothesis restricts the possible deficiencies, so the odd components of G − S must be analyzed in two exhaustive cases: (i) the extremal partition consisting of k isolated vertices, one triangle, and one large clique of order n − 2k − 3 (joined to the K_k), and (ii) all other partitions with different numbers or sizes of odd components. In Lemmas 3.3–3.6 we compare the distance matrices directly via the Perron vector of the extremal graph H. For any non-extremal partition the sum of distances from the large component to the rest of the graph is strictly larger once n ≥ 8k+6; the resulting quadratic form is therefore strictly larger than the corresponding form for H, yielding μ(G) > μ(H). The threshold 8k+6 is the smallest integer that makes the inequality strict uniformly in k, as verified by direct computation for the boundary value and by monotonicity for larger n. Consequently the implication holds for all positive integers k. revision: no

Circularity Check

0 steps flagged

No circularity; standard extremal graph theorem proof

full rationale

The paper states and proves a conditional implication: for k-connected even-order graphs with a fractional perfect matching, an upper bound on the distance spectral radius μ(G) forces a perfect matching (except for one explicit extremal join graph) once n ≥ 8k+6. The abstract and description contain no fitted parameters, no self-referential definitions of the spectral radius or matching property, and no load-bearing self-citations that reduce the central claim to prior work by the same author. The proof is expected to combine Tutte-type barrier analysis with distance-matrix eigenvalue comparisons; these steps are independent of the target conclusion and rely on external graph-theoretic facts rather than re-deriving the input hypotheses. No step equates the conclusion to a renaming or construction of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies exclusively on standard definitions from graph theory and spectral graph theory; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract statement.

axioms (1)
  • standard math Standard definitions of k-connectivity, fractional perfect matching, distance matrix, and its spectral radius (largest eigenvalue).
    These background notions are invoked to state the hypotheses and conclusion.

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Reference graph

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