10 problems

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…

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

Let $G$ be a graph, let $A_X(G)$ denote its adjacency matrix with vertex variables, let $\operatorname{mr}_\mathbb{R}(G)$ be the minimum rank of $G$ over $\mathbb{R}$, and let…

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 $G$,

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…

Let $G$ be a graph. The real minimum-rank conjecture. … For algebraically closed fields, the analogous inequality follows from the Weak Nullstellensatz, but the relation between mi…

Let $G$ be a graph, let $delta(G)$ denote its minimum degree, let $M(G)$ denote its maximum nullity over matrices associated with $G$, and let $v(G)$ denote its number of vertices.…

Let $G$ be a graph with minimum degree $\delta(G)$, maximum nullity $\operatorname{M}(G)$, minimum rank $\operatorname{mr}(G)$, and order $\operatorname{ord}(G)$. Delta conjecture.…