6 problems

Graph Complement Conjecture. For any graph $G$,

Let ${\cal A}$ be a sign pattern, and let $mr({\cal A})$ denote its minimum rank over the reals while $mr_{\mathbb Q}({\cal A})$ denotes its minimum rank over the rationals. Ration…

Weak Graph Complement Conjectures. There exist universal constants $b,b_+,b_\nu<2$ such that, for every graph $G$,

Positive semidefinite Graph Complement Conjecture. For any graph $G$,

For a graph $G$ on $n$ vertices, let $S(G)$ be the set of real symmetric matrices whose off-diagonal zero pattern is prescribed by adjacency in $G$. Define $mr(G)$ as the minimum r…

Let $c_2(G)$ denote the minimum cardinality of a subgraph complementation system for a graph $G$, and fix a natural number $k$. A minimal forbidden induced subgraph for the propert…