Minimum rank of complements of Kneser graphs

Abstract

We determine the symmetric minimum rank of the complement of the Kneser graph $\operatorname{KG}(n,k)$ over every infinite field. More precisely, if $I(n,k)$ is the graph on the $k$-subsets of $[n]$ in which two vertices are adjacent exactly when they intersect, then $$ \operatorname{mr}^{\mathbb F}(I(n,k))=n-2k+2 $$ for every infinite field $\mathbb F$ and all integers $n,k$ with $2\le k\le n/2$. In particular, over the real numbers, this settles a question posed in the AIM workshop open questions report on spectra of families of matrices described by graphs.