AI 中文总结
该研究确定克内泽尔图$\operatorname{KG}(n,k)$在无限域上补图的对称最小秩，给出公式$\operatorname{mr}^{\mathbb F}(I(n,k)) = n - 2k + 2$，解决了AIM研讨会关于图描述矩阵族谱的一个开放问题。
AI 中文摘要
我们确定了克内泽尔图$\operatorname{KG}(n,k)$在每个无限域上补图的对称最小秩。具体而言，若$I(n,k)$是$[n]$的$k$子集上的图，其中两个顶点在相交时相邻，那么对于每个无限域$\mathbb F$以及所有满足$2\leq k\leq n/2$的整数$n,k$，有$\operatorname{mr}^{\mathbb F}(I(n,k)) = n - 2k + 2$。特别地，在实数域上，这解决了AIM研讨会关于图描述的矩阵族谱的开放问题报告中提出的一个问题。
英文摘要
Let $J_{\ge s}(n,k)$ be the graph whose vertices are the $k$-subsets of $[n]$, with two distinct vertices adjacent whenever their intersection has size at least $s$. Equivalently, $J_{\ge s}(n,k)$ is the complement of a threshold Kneser graph. We determine both the symmetric minimum rank over an arbitrary infinite field and the real positive semidefinite minimum rank of this family. Specifically, for $k\ge2$, $1\le s\le k-1$, and $n\ge2k-s$, we prove $$ \operatorname{mr}^{\mathbb F}\left(J_{\ge s}(n,k)\right) = \binom{n-2(k-s)}{s} $$ for every infinite field $\mathbb F$, and $$ \operatorname{mr}_{+}^{\mathbb R}\left(J_{\ge s}(n,k)\right) = \binom{n-2(k-s)}{s}. $$ The lower bound follows from a diagonal submatrix indexed by two carefully chosen families of $k$-subsets. For the upper bound, we construct a symmetric matrix using an exterior power of a bilinear form, a Lagrange interpolation identity, and a generic nonvanishing argument. Over $\mathbb R$, an interlacing choice of parameters makes the bilinear form positive definite and yields a positive semidefinite matrix attaining the required upper bound. As consequences, we answer a question from an American Institute of Mathematics workshop, determine the real faithful orthogonality dimension of all graphs $J_{\ge s}(n,k)$ in the stated range, and recover the known minimum-rank formula for Johnson graphs.
Comments12 pages. v2: Substantially revised and expanded. The previous result for the case $s=1$ is extended to all thresholds $1\le s\le k-1$, and the exact real positive semidefinite minimum rank is also determined. Title changed
