Computing positive semidefinite minimum rank for small graphs

Abstract

The positive semidefinite minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all positive semidefinite real symmetric matrices whose $ij$-th entry (for $i\ne j$) is nonzero whenever $\left\{i,j\right\}$ is an edge in $G$ and is zero otherwise. The computation of this parameter directly is difficult. However, there are a number of known bounding parameters and techniques which can be calculated and performed on a computer. We programmed an implementation of these bounds and techniques in the open-source mathematical software Sage. The program, in conjunction with the orthogonal representation method, establishes the positive semidefinite minimum rank for all graphs of order $7$ or less.