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On the Structure of Bidegreed Graphs with Minimal Spectral Radius

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on-the-structure-of-bidegreed-graphs-with-minimal-spectral-radiusNumber Theorymath.NTmath.SPposed by Francesco Belardorecorded: open · 1 machine check, unexamined

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Statement

Let G' be a graph obtained from a (Δ,1)(\Delta,1) -bidegreed graph G by inserting a bouquet in an edge of a bouquet internal path. Then

(i) if ρ(G)<1+Δ−1, then ρ(G′)>ρ(G);\text{(i) if }\rho(G)<1+\sqrt{\Delta-1}\text{, then }\rho\left(G^{\prime}\right)>\rho(G)\text{;}

(ii) if ρ(G)>1+Δ−1, then ρ(G′)<ρ(G);\text{(ii) if }\rho(G)>1+\sqrt{\Delta-1}\text{, then }\rho\left(G^{\prime}\right)<\rho(G)\text{;}

(iii) if ρ(G)=1+Δ−1, then ρ(G′)=ρ(G).\text{(iii) if }\rho(G)=1+\sqrt{\Delta-1}\text{, then }\rho\left(G^{\prime}\right)=\rho(G)\text{.}

Context

Candidate 1 of the open problems stated in "On the Structure of Bidegreed Graphs with Minimal Spectral Radius", 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 statement: for finite simple connected (Δ,1)(\Delta,1)-bidegreed graphs, ρ(G)\rho(G) is the adjacency spectral radius. The skeleton is the subgraph induced by degree-Δ\Delta vertices. Inserting a bouquet in a skeleton edge uvuv means subdividing uvuv by a new skeleton vertex ww and attaching Δ−2\Delta-2 new leaves to ww. The conjecture claims the three stated monotonicity alternatives relative to 1+Δ−11+\sqrt{\Delta-1}.

    Result: The conjecture is false. A counterexample occurs for Δ=3\Delta=3.

    Let HH be the skeleton with vertices a,b,c,d,e,f,g,h,i,ja,b,c,d,e,f,g,h,i,j and edges

    ab,bc,af,aj,cd,ce,fg,fi,gh.ab,bc,af,aj,cd,ce,fg,fi,gh .

    Form GG by attaching 3−dH(v)3-d_H(v) leaves to each skeleton vertex vv. Then GG is connected and (3,1)(3,1)-bidegreed. The edge abab lies on the bouquet internal path c−b−ac-b-a. Let G′G' be obtained by replacing abab with ak,kbak,kb and attaching one new leaf to kk.

    For a skeleton KK, nonzero eigenvalues tt of its (3,1)(3,1)-completion satisfy

    det⁡ ⁣(t2I−tA(K)−diag⁡(3−dK(v)))=0.\det\!\bigl(t^2I-tA(K)-\operatorname{diag}(3-d_K(v))\bigr)=0.

    For HH,

    DH(t)=t4(t2−2)q(t2),D_H(t)=t^4(t^2-2)q(t^2),

    where

    q(y)=y7−19y6+142y5−532y4+1052y3−1052y2+452y−48.q(y)=y^7-19y^6+142y^5-532y^4+1052y^3-1052y^2+452y-48.

    For the inserted skeleton H′H',

    DH′(t)=t4(t2−2)r(t2),D_{H'}(t)=t^4(t^2-2)r(t^2),

    where

    r(y)=y8−21y7+178y6−784y5+1924y4−2616y3+1840y2−572y+48.r(y)=y^8-21y^7+178y^6-784y^5+1924y^4-2616y^3+1840y^2-572y+48.

    Now

    q(293/50)<0,q(293/50)<0,

    so the largest real zero η\eta of qq satisfies

    η>293/50>3+22=(1+2)2.\eta>293/50>3+2\sqrt2=(1+\sqrt2)^2.

    Thus ρ(G)=η>1+2\rho(G)=\sqrt{\eta}>1+\sqrt2.

    Also

    r(y)=(y−2)q(y)−2(y−4)s(y),r(y)=(y-2)q(y)-2(y-4)s(y),

    with

    s(y)=y5−12y4+48y3−78y2+46y−6.s(y)=y^5-12y^4+48y^3-78y^2+46y-6.

    For y≥293/50y\ge 293/50, s(y)>0s(y)>0 by expanding s(293/50+u)s(293/50+u), whose coefficients are all positive. Hence

    r(η)=−2(η−4)s(η)<0.r(\eta)=-2(\eta-4)s(\eta)<0.

    Since r(y)→+∞r(y)\to+\infty, rr has a real zero ζ>η\zeta>\eta. Therefore G′G' has eigenvalue ζ>η=ρ(G)\sqrt\zeta>\sqrt\eta=\rho(G), so

    ρ(G′)>ρ(G),\rho(G')>\rho(G),

    although ρ(G)>1+2\rho(G)>1+\sqrt2. This contradicts part (ii), which predicts ρ(G′)<ρ(G)\rho(G')<\rho(G).

    Citation: F. Belardo, “On the Structure of Bidegreed Graphs with Minimal Spectral Radius,” Filomat 28(1) (2014), 1–10, Conjecture 2.4. The counterexample above is constructed here.

    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 counterexample attacks the stated Conjecture 2.4(ii). The constructed graph is a connected (3,1)(3,1)-bidegreed graph, the operation described is bouquet insertion on the edge abab of the bouquet internal path c−b−ac-b-a, and the determinant reduction for nonzero adjacency eigenvalues is correct. The displayed polynomials and sign argument show ρ(G)>1+2\rho(G)>1+\sqrt2 but ρ(G′)>ρ(G)\rho(G')>\rho(G), contradicting the conjectured decrease. I found no indication of a prior published counterexample or stronger known resolution.

      Novelty assessment

      TYPE1

      Classification rationale: The accepted result is an explicit finite counterexample to Belardo’s Conjecture 2.4(ii). I found no evidence that this counterexample is already known. However, its mathematical contribution is narrow: it disproves one conjectured monotonicity statement by a constructed graph plus determinant/sign computations, without a broader replacement theorem or new method. It would be suitable as an erratum/short note or part of a larger paper, but likely not as a standalone standard-journal combinatorics paper.

      Literature check: I searched for the exact paper title, “Belardo bidegreed minimal spectral radius,” “Conjecture 2.4 bidegreed,” “bouquet internal path,” “insert/inserting a bouquet,” “(Δ,1)(\Delta,1)-bidegreed spectral radius,” and combinations with “counterexample,” “false,” and the threshold 1+Δ−11+\sqrt{\Delta-1}. I also checked arXiv full-text/title searches; the relevant phrase searches returned no follow-up preprint. The accessible hits point back to the original Filomat/ResearchGate/DOI records or to unrelated bidegreed/spectral graph literature. I found no published paper, preprint, note, forum post, or citation trail recording this counterexample or a stronger disproof.

      Citation: F. Belardo, “On the Structure of Bidegreed Graphs with Minimal Spectral Radius,” Filomat 28(1) (2014), 1–10, Conjecture 2.4, DOI: 10.2298/FIL1401001B.

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