立方体类图的正交与酉符号

中文摘要 AI 辅助

一个$d$-正则图的酉符号是一个Hermitian邻接矩阵$M$，其非零元素位于$\{\pm1,\pm i\}$中，并满足$M^2=dI$。受Alon和Zheng关于立方体类图的正交与酉符号工作的启发，我们为生成集$S\subseteq\mathbb Z_2^n$引入了$\Theta$-性质：只要$S$的三个两两不相交的子集具有相同的和，则其中至少有两个子集的大小为偶数。我们证明了每个具有$\Theta$-性质的零自由生成集都能给出一个允许酉符号的立方体类图$Q_S$，并给出了这种符号的显式局部公式。对于Sidon集，$\Theta$-性质也是必要的，从而刻画了允许酉符号的Sidon立方体类图。在这种情况下，恰好有$2^{|S|-n}$个酉符号的切换等价类，并且我们刻画了何时可以选择一个正交的酉符号。$\Theta$-性质允许用$S$的依赖空间$\mathcal D$进行线性代数描述：\\[ |D_1\cap D_2| \equiv |D_1||D_2| \pmod2 \qquad (D_1,D_2\in\mathcal D). \\] 等价地，映射\\[ D\longmapsto \binom{|D|}{2}\pmod2 \\]在$\mathcal D$上是线性的。对于相应的极值集问题（其中允许零），这一表述给出了具有$\Theta$-性质的生成集的尖锐界$|S|\leq 2n+1$。我们给出了对每个$n\geq3$达到此界的初等构造。在附加Sidon条件下，对每个$n\geq10$使用二元自对偶码可以达到相同的极值。对于$3\leq n\leq9$，也确定了精确的Sidon最大值。

英文摘要

A unitary signing of a $d$-regular graph is a Hermitian adjacency matrix $M$ whose nonzero entries lie in $\{\pm1,\pm i\}$ and satisfies $M^2=dI$. Motivated by the work of Alon and Zheng on orthogonal and unitary signings of cube-like graphs, we introduce the $Θ$-property for a generating set $S\subseteq\mathbb Z_2^n$: whenever three pairwise disjoint subsets of $S$ have the same sum, at least two of them have even size. We prove that every zero-free generating set with the $Θ$-property gives a cube-like graph $Q_S$ admitting a unitary signing, and we give an explicit local formula for such a signing. For Sidon sets, the $Θ$-property is also necessary, yielding a characterization of the Sidon cube-like graphs that admit unitary signings. In this setting there are exactly $2^{|S|-n}$ switching-equivalence classes of unitary signings, and we characterize when a unitary signing can be chosen to be orthogonal. The $Θ$-property admits a linear-algebraic description in terms of the dependency space $\mathcal D$ of $S$: \[ |D_1\cap D_2| \equiv |D_1||D_2| \pmod2 \qquad (D_1,D_2\in\mathcal D). \] Equivalently, the map \[ D\longmapsto \binom{|D|}{2}\pmod2 \] is linear on $\mathcal D$. For the corresponding extremal set problem, in which zero is permitted, this formulation yields the sharp bound $|S|\leq 2n+1$ for generating sets with the $Θ$-property. We give elementary constructions attaining this bound for every $n\geq3$. Under the additional Sidon condition, the same extremal value is attained for every $n\geq10$ using binary self-dual codes. The exact Sidon maximum is also determined for $3\leq n\leq9$.

发表机构

• School of Mathematics and Statistics, Xiamen University of Technology（厦门理工学院数学与统计学院）
• Université de Paris, CNRS（巴黎大学，法国国家科学研究中心）

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