On the Minimum Rank Among Positive Semidefinite Matrices with a Given Graph

Let $\mathcal{P}(G)$ be the set of all positive semidefinite matrices whose graph is $G$, and $\operatorname{msr}(G)$ be the minimum rank of all matrices in $\mathcal{P}(G)$. Upper and lower bounds for $\operatorname{msr}(G)$ are given and used to determine $\operatorname{msr}(G)$ for some well-known graphs, including chordal graphs, and for all simple graphs on less than seven vertices.