秩三投影与路径补图的最小重数二分划
Rank-Three Projections and Minimal Multiplicity Bipartitions of Path Complements
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中文总结 AI 辅助
本研究针对路径补图的最小特征值重数参数MB(G)问题，通过构造秩三正交投影结合局部障碍证明了n≥6时MB(\overline{P_n})=3，解决了此前未决的可被3整除的n≥9阶情形，并补充了小阶例外情况的证明。
中文摘要 AI 辅助
对于存在恰含两个不同特征值的实对称实现的图\boldsymbol{G}，\boldsymbol{MB(G)}是所有此类实现中两个特征值重数里较小值的最小值。Adm、Fallat、Meagher、Nasserasr、Plosker和Yang提出了顶点数至少为8的路径补图的该参数求解问题。我们通过证明\boldsymbol{MB(\overline{P_n})=3\quad (n\ge 6)}，完全解答了他们的问题。特别地，这解决了此前未解决的、可被3整除的\boldsymbol{n\ge 9}阶情形。该证明是精确且构造性的：我们构造了\boldsymbol{\mathbb{R}^3}中的6个向量，它们的内积恰在相邻索引时为零，其秩一外积构成\boldsymbol{\mathbb{S}^3}的一组基，且存在严格正的Parseval缩放。随后，一个基础吸收引理允许为该向量链的任意有限忠实正交扩展添加小的正权重，同时修正6个原始权重以保持Parseval恒等式。得到的Gram矩阵是\boldsymbol{\mathcal{S}(\overline{P_n})}中的一个秩三正交投影。一个局部二维正交障碍给出了匹配的下界。为完整起见，我们包含了例外小阶情形的自包含证明：\boldsymbol{MB(\overline{P_3})=1}，而\boldsymbol{q(\overline{P_4})=4}且\boldsymbol{q(\overline{P_5})=3}。
英文摘要
For a graph \(G\) admitting a real symmetric realization with exactly two distinct eigenvalues, \(MB(G)\) is the minimum, over all such realizations, of the smaller of the two eigenvalue multiplicities. Adm, Fallat, Meagher, Nasserasr, Plosker, and Yang asked for this parameter for the complement of a path on at least eight vertices. We answer their question completely by proving $$ MB(\overline{P_n})=3 \qquad (n\ge 6). $$ In particular, this resolves the previously unresolved orders \(n\ge 9\) divisible by three. The proof is exact and constructive. We exhibit six vectors in \(\mathbb{R}^3\) whose mutual inner products vanish exactly for consecutive indices, whose rank-one outer products form a basis of \(\mathbb{S}^3\), and which admit a strictly positive Parseval scaling. An elementary absorption lemma then permits any finite faithful orthogonal extension of this vector chain to be added with small positive weights while the six original weights are corrected to retain the Parseval identity. The resulting Gram matrix is a rank-three orthogonal projection in \(\mathcal{S}(\overline{P_n})\). A local two-dimensional orthogonality obstruction gives the matching lower bound. For completeness, we include self-contained proofs of the exceptional small orders: \(MB(\overline{P_3})=1\), whereas \(q(\overline{P_4})=4\) and \(q(\overline{P_5})=3\).
