WOW-284的反例、谱障碍与删除稳定性
Counterexamples, Spectral Obstructions, and Deletion Stability for WOW-284
浏览论文内容
中文总结 AI 辅助
该论文反驳了关于连通图最小对偶度与最小距离特征值关系的WOW-284断言，构造了多阶反例，建立相关结构理论、界与删除稳定性，并经Lean内核验证。
中文摘要 AI 辅助
WOW-284断言：每个阶数至少为3、围长至少为5的连通图的最小对偶度不超过其最小距离特征值的相反数。我们用阶数为38、39、40、42和50的精确反例反驳该断言，并建立其失效的结构理论。对于围长至少为5、直径为3的连通k-正则图，我们证明δ*(G)+λ_min(D(G))=2k-2-max_{θ≠k}(θ+1)²，其中θ为非主邻接特征值。我们进一步证明，每个正则严格反例的度数至少为6，直径最多为4，且直径为4时度数至少为10。我们精确求解了相关的单变量非回溯线性规划，包括最优解的刚性。对于直径为3的正则严格反例，最优解产生一个半正定松弛矩阵，其积分余量给出更强的界|V(G)|≤⌊3(k+2)²(k²+3)/(18k+41)⌋，该界源于积分余量的三对一量化定理。松弛矩阵的主子式还能恢复局部环约束，具体而言，正则6度反例的阶数最多为50，且在6度、阶数50的边界处，相关的带符号补图必然不连通。我们确定了Moore图的1个和2个顶点穿孔的距离谱，并建立了统一的删除稳定性界：从Hoffman–Singleton图中删除最多5个顶点后仍为严格反例，而显式的6个顶点删除则不是。所有定理级计算均使用精确算术，Lean 4.31内核在图级别检查了显式的50顶点Hoffman–Singleton反例，在阶数38、39、40、42处检查了有限谱证书，且对所有整数k≥4检查了解析LP最优解与刚性。
英文摘要
WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five does not exceed the negative of its least distance eigenvalue. We refute it with exact counterexamples of orders $38,39,40,42$, and $50$, and develop a structural theory of the failure. For a connected $k$-regular graph of girth at least five and diameter three, we prove $δ^*(G)+λ_{\min}(D(G))=2k-2-\max_{θ\ne k}(θ+1)^2$. Here $θ$ ranges over the nonprincipal adjacency eigenvalues. We further prove that every regular strict counterexample has degree at least six and diameter at most four, while diameter four forces degree at least ten. We solve the associated one-variable nonbacktracking linear program exactly, including optimizer rigidity. For regular strict counterexamples of diameter three, the optimizer yields a positive-semidefinite slack matrix whose integral excess gives the stronger bound $|V(G)|\le\left\lfloor 3(k+2)^2(k^2+3)/(18k+41)\right\rfloor$; this follows from a three-to-one quantization theorem for the integral excess. The slack matrix's principal minors also recover local cycle constraints. In particular, regular degree-six counterexamples have order at most $50$, and at the degree-six, order-$50$ boundary the associated signed complement is necessarily disconnected. We determine the distance spectra of one- and two-vertex punctures of Moore graphs and establish a uniform deletion-stability bound: every deletion of at most five vertices from the Hoffman--Singleton graph remains a strict counterexample, whereas an explicit six-vertex deletion does not. All theorem-level computations use exact arithmetic. Lean 4.31 kernel-checks the explicit $50$-vertex Hoffman--Singleton counterexample at graph level, finite spectral certificates at orders $38,39,40,42$, and the analytic LP optimum and rigidity for every integer $k\ge4$.
