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On the A_α-spectra of graphs

Analysis · math.CO · math.SP · posed by Huiqiu Lin, Jie Xue, Jinlong Shu · open

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Statement

Let G be a connected graph of order n. If 12<α<1\frac{1}{2}<\alpha<1 , then λn(Aα(G))≥λn(Aα(K1,n−1)),\lambda_{n}\left(A_{\alpha}(G)\right)\geq \lambda_{n}\left(A_{\alpha}\left(K_{1,n-1}\right)\right),the equality holds if and only if G≅K1,n−1G \cong K_{1,n-1} .

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  • On the A_α-spectra of graphs
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: For finite simple connected undirected graphs GG of order nn, define

    Aα(G)=αD(G)+(1−α)A(G),12<α<1,A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),\qquad \tfrac12<\alpha<1,

    and let λn\lambda_n denote the smallest eigenvalue. The conjecture is:

    λn(Aα(G))≥λn(Aα(K1,n−1)),\lambda_n(A_\alpha(G))\ge \lambda_n(A_\alpha(K_{1,n-1})),

    with equality iff G≅K1,n−1G\cong K_{1,n-1}. For n=1n=1 this is trivial; assume n≥2n\ge2.

    Result: The conjecture is true.

    Put β=1−α\beta=1-\alpha. For the star Sn=K1,n−1S_n=K_{1,n-1}, its least eigenvalue is the smaller root

    r=αn−α2(n−2)2+4β2(n−1)2r=\frac{\alpha n-\sqrt{\alpha^2(n-2)^2+4\beta^2(n-1)}}2

    of

    (α−r)(α(n−1)−r)=β2(n−1).(\alpha-r)(\alpha(n-1)-r)=\beta^2(n-1).

    Thus 0<r<α0<r<\alpha. Set

    a=α−r,h=rn−1.a=\alpha-r,\qquad h=\frac r{n-1}.

    The star equation gives

    h=α−β2a.h=\alpha-\frac{\beta^2}{a}.

    First consider a tree TT. Since TT is bipartite, Aα(T)A_\alpha(T) is orthogonally similar to

    MT=αD(T)−βA(T)=βL(T)+(2α−1)D(T).M_T=\alpha D(T)-\beta A(T)=\beta L(T)+(2\alpha-1)D(T).

    We prove MT−rI⪰0M_T-rI\succeq0.

    Root TT at a vertex ρ\rho. In the usual leaf-elimination Schur complement process, define pivots fvf_v. For a non-root vertex vv, with children uu,

    fv=α(∣C(v)∣+1)−r−∑u∈C(v)β2fu.f_v=\alpha(|C(v)|+1)-r-\sum_{u\in C(v)}\frac{\beta^2}{f_u}.

    Let eve_v be the number of edges in the rooted subtree below vv. We claim

    fv≥a+evh>0.f_v\ge a+e_vh>0.

    This is clear for leaves. If the children have subtree-edge counts eue_u, then using induction it is enough to note that for 0≤e≤n−20\le e\le n-2,

    α−β2a+eh−(e+1)h=h2e(n−2−e)a+eh≥0.\alpha-\frac{\beta^2}{a+eh}-(e+1)h =\frac{h^2e(n-2-e)}{a+eh}\ge0.

    Hence

    fv≥a+∑u∈C(v)(eu+1)h=a+evh.f_v\ge a+\sum_{u\in C(v)}(e_u+1)h=a+e_vh.

    For the root,

    fρ=α∣C(ρ)∣−r−∑u∈C(ρ)β2fu≥∑u∈C(ρ)(eu+1)h−r=(n−1)h−r=0.f_\rho=\alpha|C(\rho)|-r-\sum_{u\in C(\rho)}\frac{\beta^2}{f_u} \ge \sum_{u\in C(\rho)}(e_u+1)h-r =(n-1)h-r=0.

    Thus all Schur pivots are positive except possibly the final one, so MT−rI⪰0M_T-rI\succeq0. Therefore

    λn(Aα(T))=λmin⁡(MT)≥r.\lambda_n(A_\alpha(T))=\lambda_{\min}(M_T)\ge r.

    Equality in the displayed inequality occurs only when every child-subtree size ee is 00 or n−2n-2. This forces the rooted tree to be a star: either the root is adjacent to all other vertices, or the root has a single child adjacent to all remaining vertices. Hence for trees,

    λn(Aα(T))=r⟺T≅K1,n−1.\lambda_n(A_\alpha(T))=r \quad\Longleftrightarrow\quad T\cong K_{1,n-1}.

    Now let GG be connected and let TT be a spanning tree. For each added edge uvuv,

    Aα(G)−Aα(T)A_\alpha(G)-A_\alpha(T)

    has on {u,v}\{u,v\} the block (αββα)\begin{pmatrix}\alpha&\beta\\ \beta&\alpha\end{pmatrix}, whose eigenvalues are 11 and 2α−1>02\alpha-1>0. Hence

    Aα(G)−Aα(T)⪰0,A_\alpha(G)-A_\alpha(T)\succeq0,

    so by the variational principle,

    λn(Aα(G))≥λn(Aα(T))≥r=λn(Aα(K1,n−1)).\lambda_n(A_\alpha(G))\ge \lambda_n(A_\alpha(T))\ge r =\lambda_n(A_\alpha(K_{1,n-1})).

    If equality holds, then TT must be a star. If GG had any additional edge, it would join two leaves of that star. The least star eigenvalue rr is simple, with eigenvector nonzero and equal on all leaves. The positive definite edge contribution from an added leaf-leaf edge gives strictly positive quadratic form on this eigenvector, contradicting equality. Therefore no extra edge exists and G≅K1,n−1G\cong K_{1,n-1}.

    Thus the conjecture holds exactly as stated for n≥2n\ge2, with the n=1n=1 case trivial.

    Citation: No prior resolution is used here. The conjecture is from Lin, Xue, and Shu, “On the AαA_\alpha-spectra of graphs,” Linear Algebra Appl. 556 (2018), 210–219; arXiv:1709.00182.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE2

    PASS

    The proof attacks exactly Conjecture 3.2. The star eigenvalue formula is correct, the Schur-complement pivot induction for trees is valid and proves Aα(T)A_\alpha(T) has least eigenvalue at least the star value. Edge addition is positive semidefinite for α>1/2\alpha>1/2, so the extension from spanning trees to all connected graphs is sound. The equality case is also justified: equality forces the spanning tree to be a star, and any added leaf-leaf edge strictly increases the Rayleigh quotient on the unique least star eigenvector. I found no fatal gap in the argument.

    Novelty assessment

    TYPE2

    Classification rationale: This appears to be a genuine resolution of a published conjecture in spectral graph theory. The argument is short and elementary, but not a routine corollary of a standard theorem; it gives the exact extremal graph and equality case. The topic is specialized, so it is not top-journal level, but it plausibly supports a short standalone note in a standard linear algebra / spectral graph theory journal.

    Literature check: I checked the original arXiv/LAA paper, metadata and citation records, and searched for the conjecture/result using variants of “smallest/least AαA_\alpha-eigenvalue,” “AαA_\alpha-spectra,” “Conjecture 3.2,” and the star K1,n−1K_{1,n-1}. I did not find an existing proof of this conjecture or a stronger theorem implying it. The original paper still lists it as a conjecture.

    Citation: Huiqiu Lin, Jie Xue, Jinlong Shu, “On the AαA_\alpha-spectra of graphs,” Linear Algebra and its Applications 556 (2018), 210–219; arXiv:1709.00182.

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