Complementary Vanishing Graphs

Given a graph $G$ with vertices $\{v_1,\dots ,v_n\}$, we define $\mathcal{S}(G)$ to be the set of symmetric matrices $A=[a_{i,j}]$ such that for $i\mathrm{\ne }j$ we have $a_{i,j}\mathrm{\ne }0$ if and only if $v_i{v}_j\in E(G)$. Motivated by the Graph Complement Conjecture, we say that a graph $G$ is complementary vanishing if there exist matrices $A\in \mathcal{S}(G)$ and $B\in \mathcal{S}(\bar{G})$ such that $AB=O$. We provide combinatorial conditions for when a graph is or is not complementary vanishing, and we characterize which graphs are complementary vanishing in terms of certain minimal complementary vanishing graphs. In addition to this, we determine which graphs on at most $8$ vertices are complementary vanishing.