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Perturbation of Fiedler vector: interest for graph measures and shape analysis

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perturbation-of-fiedler-vector-interest-for-graph-measures-and-shape-2Spectral Theorymath.COmath.SPposed by Julien Lefevre, Justine Fraize, David Germanaudrecorded: open · 1 machine check, unexamined

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Statement

Clearly a(v) \le \bar{a}(v) and we conjecture that a(v) = \bar{a}(v) based on empirical observations.

Context

Candidate 2 of the open problems stated in "Perturbation of Fiedler vector: interest for graph measures and shape analysis", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: Reconstructed conjecture: for a finite connected simple graph G=(V,E)G=(V,E), a vertex vv, and the graph GxG_x obtained by adding a new pendant vertex pp adjacent only to vv with edge-weight x>0x>0, choose the Fiedler vector Φx\Phi_x of the weighted Laplacian of GxG_x with the sign for which Φx(p)>0\Phi_x(p)>0 for small xx. Define

    a(v)=sup⁡{a>0: Φx(p)=max⁡iΦx(i) for every 0<x≤a},a(v)=\sup\{a>0:\ \Phi_x(p)=\max_i\Phi_x(i)\text{ for every }0<x\le a\},

    and

    aˉ(v)=sup⁡{x>0: Φx(p)=max⁡iΦx(i)}.\bar a(v)=\sup\{x>0:\ \Phi_x(p)=\max_i\Phi_x(i)\}.

    The conjecture is that a(v)=aˉ(v)a(v)=\bar a(v). This is the natural formalization of the paper’s definitions around its equation defining a(v)a(v) and its later definition of aˉ(v)\bar a(v).

    Result: The conjecture is false.

    Take GG on vertices 0,…,90,\dots,9 with edges

    (0,3),(0,5),(1,2),(1,3),(1,7),(1,8),(1,9),(2,7),(3,4),(4,6),(4,7),(4,9),(5,7),(6,9),\begin{aligned} &(0,3),(0,5),(1,2),(1,3),(1,7),(1,8),(1,9),(2,7),\\ &(3,4),(4,6),(4,7),(4,9),(5,7),(6,9), \end{aligned}

    and perturb at v=4v=4. Let p=10p=10 be the added pendant vertex.

    For λ<λ2(G)\lambda<\lambda_2(G), let r(λ)r(\lambda) solve

    (L−λI)r=λe4.(L-\lambda I)r=\lambda e_4.

    Then (r(λ),1)(r(\lambda),1) is a Fiedler eigenvector of the augmented graph for

    x=λ1−r4(λ).x=\frac{\lambda}{1-r_4(\lambda)}.

    A Sturm check for

    χL(t)=t(t9−28t8+332t7−2180t6+8697t5−21748t4+33900t3−31604t2+15944t−3320)\chi_L(t)=t\bigl(t^9-28t^8+332t^7-2180t^6+8697t^5-21748t^4 +33900t^3-31604t^2+15944t-3320\bigr)

    gives λ2(G)>77/100\lambda_2(G)>77/100.

    At λ1=19/25\lambda_1=19/25,

    x1=168954454714587151448225975,x_1=\frac{16895445471458}{7151448225975},

    and

    r6=925178531325889233972182>1.r_6=\frac{925178531325}{889233972182}>1.

    Thus the new vertex pp, whose coordinate is 11, is not maximal.

    At λ2=77/100\lambda_2=77/100,

    x2=704736249366627541203457078222166600>x1,x_2=\frac{704736249366627541}{203457078222166600}>x_1,

    and

    r=1100676607052375363(89274952888530000−83657533855410000−6841680880047000037629483175361900782963284479370377217870887753000097102304797421900−495140969168100−36372840806700000041139506452891900),r=\frac1{100676607052375363} \begin{pmatrix} 89274952888530000\\ -83657533855410000\\ -68416808800470000\\ 37629483175361900\\ 78296328447937037\\ 72178708877530000\\ 97102304797421900\\ -495140969168100\\ -363728408067000000\\ 41139506452891900 \end{pmatrix},

    so every old coordinate is <1<1. Hence pp is the strict maximum at x2x_2.

    Therefore the property holds again after failing:

    a(v)≤x1<x2≤aˉ(v),a(v)\le x_1<x_2\le \bar a(v),

    so a(v)≠aˉ(v)a(v)\ne \bar a(v).

    Citation: No known literature resolution used. This is an explicit counterexample to the conjecture from Lefevre–Fraize–Germanaud, “Perturbation of Fiedler vector: interest for graph measures and shape analysis,” arXiv:2306.04327.

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The solution gives a valid explicit counterexample to the stated a(v)=aˉ(v)a(v)=\bar a(v) conjecture. The resolvent equation correctly parameterizes the Fiedler eigenvector of the pendant-edge perturbation for λ<λ2(G)\lambda<\lambda_2(G), and the stated Sturm/characteristic-polynomial check ensures the two chosen λ\lambda's are in that range. At the smaller weight x1x_1, an old vertex coordinate exceeds the pendant coordinate 11, so the property has failed; at the larger weight x2x_2, all old coordinates are <1<1, so the pendant vertex is again the strict maximum. Hence a(v)≤x1<x2≤aˉ(v)a(v)\le x_1<x_2\le\bar a(v), disproving equality.

      I found no existing prior resolution in available citation/search checks.

      Novelty assessment

      TYPE1

      Classification rationale: This appears to be a genuinely new explicit counterexample, but the target conjecture is recent, specialized, and empirically motivated in an applied/spectral-graph paper. The resolution is a finite computational/algebraic example rather than a broad theorem or structural explanation. On its own it is likely a useful note or erratum-level observation, not a standalone standard combinatorics-journal paper.

      Literature check: I found no prior resolution. Searches for the exact paper title locate only the original arXiv/HAL/Springer/ResearchGate/DBLP/alphaXiv records. Semantic Scholar lists three citations to the 2023 paper, but they concern unrelated PCA/shape-analysis applications and do not address the graph-measure conjecture. Targeted searches for “Fiedler vector” with “pendant vertex/pendant edge,” “a(v), \bar a(v),” and related extremal-Fiedler-vector terminology found only background literature on Fiedler-vector extrema, graph elongation, and edge perturbation sensitivity, not this pendant-weight monotonicity question or a counterexample. A later related paper by Noschese–Reichel on Fiedler eigenpair sensitivity discusses edge-weight perturbations but not this conjecture.

      Citation: Original conjecture: Julien Lefevre, Justine Fraize, David Germanaud, “Perturbation of Fiedler vector: interest for graph measures and shape analysis,” arXiv:2306.04327; Springer GSI 2023, DOI 10.1007/978-3-031-38299-4_61.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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