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Subdivision and Graph Eigenvalues

Algebra · math.RT · math.SP · posed by Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada, Hanmeng Zhan · open

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Statement

We have max⁡tmHt(2,∞)=max⁡tmGt(2,∞)\max_t m_{H_t}(2, \infty) = \max_t m_{G_t}(2, \infty) and max⁡tmHt(−∞,−2)=max⁡tmGt(−∞,−2).\max_t m_{H_t}(-\infty, -2) = \max_t m_{G_t}(-\infty, -2).

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  • Subdivision and Graph Eigenvalues
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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 a finite simple graph GG, a fixed set S⊆E(G)S\subseteq E(G), and t≥1t\ge1, let Gt=Gt(S)G_t=G_t(S) be obtained by replacing every edge of SS by a path of length tt. Let Ht=Ht(S)H_t=H_t(S) be obtained from G2t+1G_{2t+1} by deleting the middle edge in each stretched SS-path. For an interval II, let mX(I)m_X(I) be the number of adjacency eigenvalues of XX in II, counted with multiplicity.

    Conjecture 21 asks whether

    max⁡t≥1mHt(2,∞)=max⁡t≥1mGt(2,∞)\max_{t\ge1}m_{H_t}(2,\infty)=\max_{t\ge1}m_{G_t}(2,\infty)

    and

    max⁡t≥1mHt(−∞,−2)=max⁡t≥1mGt(−∞,−2).\max_{t\ge1}m_{H_t}(-\infty,-2)=\max_{t\ge1}m_{G_t}(-\infty,-2).

    Result: The conjecture is true.

    Let A0A_0 be the adjacency matrix of the spanning subgraph (V(G),E(G)∖S)(V(G),E(G)\setminus S), let ASA_S be the adjacency matrix of (V(G),S)(V(G),S), and let D=diag⁡(dS(v))D=\operatorname{diag}(d_S(v)). Define

    B=2I−A0−D,C=−2I−A0+D.B=2I-A_0-D,\qquad C=-2I-A_0+D.

    Then in fact

    max⁡tmGt(2,∞)=max⁡tmHt(2,∞)=n−(B),\max_t m_{G_t}(2,\infty)=\max_t m_{H_t}(2,\infty)=n_-(B),

    and

    max⁡tmGt(−∞,−2)=max⁡tmHt(−∞,−2)=n+(C),\max_t m_{G_t}(-\infty,-2)=\max_t m_{H_t}(-\infty,-2)=n_+(C),

    where n−n_- and n+n_+ denote negative and positive inertia.

    Proof. Let p0(x)=1p_0(x)=1, p1(x)=xp_1(x)=x, pk(x)=xpk−1(x)−pk−2(x)p_k(x)=xp_{k-1}(x)-p_{k-2}(x), the characteristic polynomial of a path on kk vertices, and set p−1=0p_{-1}=0. Then

    pk(2)=k+1,pk(−2)=(−1)k(k+1).p_k(2)=k+1,\qquad p_k(-2)=(-1)^k(k+1).

    By Schur complementation on the internal path vertices, for ∣λ∣>2|\lambda|>2,

    FG,t(λ)=λI−A0−pt−2(λ)pt−1(λ)D−1pt−1(λ)ASF_{G,t}(\lambda)=\lambda I-A_0-\frac{p_{t-2}(\lambda)}{p_{t-1}(\lambda)}D-\frac1{p_{t-1}(\lambda)}A_S

    detects the eigenvalues of GtG_t outside [−2,2][-2,2], while

    FH,t(λ)=λI−A0−pt−1(λ)pt(λ)DF_{H,t}(\lambda)=\lambda I-A_0-\frac{p_{t-1}(\lambda)}{p_t(\lambda)}D

    detects those of HtH_t. Since the eliminated path blocks have spectrum in (−2,2)(-2,2), the Schur complements are well-defined at λ=±2\lambda=\pm2. Also

    F′(λ)=I+R(λ)2F'(\lambda)=I+R(\lambda)^2

    in Schur-complement form, hence is positive definite; therefore the number of roots above 22 is n−(F(2))n_-(F(2)), and the number of roots below −2-2 is n+(F(−2))n_+(F(-2)).

    Evaluating at 22 gives

    mGt(2,∞)=n− ⁣(B+1t(D−AS)),m_{G_t}(2,\infty)=n_-\!\left(B+\frac1t(D-A_S)\right), mHt(2,∞)=n− ⁣(B+1t+1D).m_{H_t}(2,\infty)=n_-\!\left(B+\frac1{t+1}D\right).

    Both D−ASD-A_S and DD are positive semidefinite. Hence each negative inertia is at most n−(B)n_-(B), and for all sufficiently large tt, by continuity of eigenvalues, the negative inertia equals n−(B)n_-(B). Thus the positive-side maxima are both n−(B)n_-(B).

    Similarly, evaluating at −2-2,

    mGt(−∞,−2)=n+ ⁣(C−1t(D+(−1)t−1AS)),m_{G_t}(-\infty,-2) = n_+\!\left(C-\frac1t\bigl(D+(-1)^{t-1}A_S\bigr)\right), mHt(−∞,−2)=n+ ⁣(C−1t+1D).m_{H_t}(-\infty,-2) = n_+\!\left(C-\frac1{t+1}D\right).

    Here D−ASD-A_S, D+ASD+A_S, and DD are positive semidefinite. Thus these matrices are all ≤C\le C in Loewner order, so their positive inertia is at most n+(C)n_+(C). For sufficiently large tt, the perturbations have arbitrarily small norm, so all positive eigenvalues of CC remain positive. Hence both negative-side maxima are n+(C)n_+(C).

    This proves both asserted equalities.

    Citation: Conjecture source: Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada, Hanmeng Zhan, “Subdivision and Graph Eigenvalues,” arXiv:2303.10488, Conjecture 21. No prior resolution is used here.

  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 the correct Conjecture 21 and is mathematically sound. The Schur-complement formulas for GtG_t and HtH_t outside [−2,2][-2,2] are correct, the monotonicity F′(λ)≻0F'(\lambda)\succ0 justifies converting root counts to inertia at ±2\pm2, and the evaluations at ±2\pm2 give the stated perturbations of BB and CC. Since DD, D−ASD-A_S, and D+ASD+A_S are positive semidefinite, the inertia bounds and eventual equality for large tt follow. This proves the claimed maxima equalities.

    Novelty assessment

    TYPE2

    Classification rationale: The result appears genuinely new and gives a clean exact inertia formula, not just the conjectured equality. It resolves Conjecture 21 of Kumar–Mohar–Pragada–Zhan and also implies stabilization of the negative-side sequence from their Conjecture 20. The proof is short and based on standard Schur-complement/inertia ideas, so this is not a top-journal-level advance, but the exact formula and resolution of explicit published conjectures should support a short standalone note in a spectral graph theory / linear algebra journal.

    Literature check: I found the conjecture still open in the final published version of “Subdivision and graph eigenvalues,” Linear Algebra and its Applications 710 (2025), where Conjectures 20 and 21 are stated at the end of Section 6. Searches for the exact title, Conjecture 21, the DOI, “eigenvalue interval multiplicity,” and formula-specific terms such as mHtm_{H_t}, mGtm_{G_t}, and subdivision eigenvalues outside [−2,2][-2,2] did not reveal any prior solution or stronger theorem. OpenAlex records a few citations to the published paper, but accessible search results and repository/forum searches did not show a cited work resolving this conjecture.

    Citation: Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada, Hanmeng Zhan, “Subdivision and graph eigenvalues,” Linear Algebra and its Applications 710 (2025), 336–355, DOI: 10.1016/j.laa.2025.01.044; arXiv:2303.10488. Conjecture 21 is the source conjecture.

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