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Ordering Trees by Their ABC Spectral Radii

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ordering-trees-by-their-abc-spectral-radiiSpectral Theorymath.COmath.SPposed by Wenshui Lin, Zhangyong Yan, Peifang Fu, Jia-Bao Liurecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

If GG is a connected graph of order n≥4n \ge 4 and ρABC(G)≤2\rho_{ABC}(G) \le \sqrt{2}, then G∈{Pn,Cn,S4}G \in \{P_n, C_n, S_4\}.

Context

Candidate 1 of the open problems stated in "Ordering Trees by Their ABC Spectral Radii", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
    AI involvement
    ai discovered
    — the result was found by a model.
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    GPT-5.5 xhigh

    NEW

    Problem: Reconstructed statement: for every finite simple connected graph GG of order n≥4n\ge 4, with ABC matrix

    (AABC)uv={d(u)+d(v)−2d(u)d(v)uv∈E(G),0otherwise,(A_{ABC})_{uv}=\begin{cases} \sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}} & uv\in E(G),\\ 0&\text{otherwise}, \end{cases}

    if ρABC(G)≤2\rho_{ABC}(G)\le \sqrt2, then GG is isomorphic to PnP_n, CnC_n, or S4=K1,3S_4=K_{1,3}.

    Result: The conjecture is false.

    Let GG have vertices {p,q,c,r,s}\{p,q,c,r,s\} and edges

    pc, qc, cr, rs.pc,\ qc,\ cr,\ rs .

    This is the tree obtained from S4=K1,3S_4=K_{1,3} by subdividing one pendant edge once. Its degrees are

    d(c)=3,d(r)=2,d(p)=d(q)=d(s)=1.d(c)=3,\quad d(r)=2,\quad d(p)=d(q)=d(s)=1.

    Thus the ABC weights are

    w(pc)=w(qc)=3+1−23=23,w(cr)=w(rs)=12.w(pc)=w(qc)=\sqrt{\frac{3+1-2}{3}}=\sqrt{\frac23},\qquad w(cr)=w(rs)=\sqrt{\frac{1}{2}}.

    In the vertex order (p,q,c,r,s)(p,q,c,r,s), set

    x=(13,13,1,23,13)T>0.x=\left(\frac1{\sqrt3},\frac1{\sqrt3},1,\frac23,\frac13\right)^T>0.

    A direct computation gives

    AABC(G)x=2 x.A_{ABC}(G)x=\sqrt2\,x.

    Since GG is connected, AABC(G)A_{ABC}(G) is nonnegative and irreducible, so by Perron-Frobenius the positive eigenvalue with positive eigenvector is the spectral radius. Hence

    ρABC(G)=2.\rho_{ABC}(G)=\sqrt2.

    But GG has order 55, so it is not S4S_4; and its maximum degree is 33, so it is neither P5P_5 nor C5C_5. Therefore G∉{Pn,Cn,S4}G\notin\{P_n,C_n,S_4\}, while ρABC(G)≤2\rho_{ABC}(G)\le\sqrt2.

    Citation: No external resolution is used; the counterexample is constructed and verified above.

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    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 proposed counterexample is valid. The 5-vertex tree with edges pc,qc,cr,rspc,qc,cr,rs has ABC weights 2/3,2/3,1/2,1/2\sqrt{2/3},\sqrt{2/3},1/\sqrt2,1/\sqrt2. The displayed positive vector is correctly checked to satisfy

      AABCx=2 x,A_{ABC}x=\sqrt2\,x,

      so by Perron–Frobenius its ABC spectral radius is 2\sqrt2. The graph is connected of order 55, but is not P5P_5, not C5C_5, and not S4S_4. Hence it rigorously disproves the stated conjecture.

      Novelty assessment

      TYPE1

      Classification rationale: A valid 5-vertex counterexample to the conjecture, but it is a very small explicit computation with no new method or broader classification. Even if genuinely new, it would be suitable only as an erratum/short remark unless expanded into a corrected theorem.

      Literature check: I found no accessible prior paper/note giving this counterexample or a stronger corrected classification. arXiv searches for “ABC spectral radius”, “ABC matrix spectral radius”, “Conjecture 4.1 ABC spectral radius”, and related terms returned only the original graph papers and a later hypergraph paper, not a resolution of this conjecture. I also checked the original arXiv record and related earlier ABC spectral-radius work; nothing there records this counterexample.

      Citation: Original conjecture: W. Lin, Z. Yan, P. Fu, J.-B. Liu, “Ordering Trees by Their ABC Spectral Radii,” arXiv:2008.00689.

      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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