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The minimum rank of a simple graph $G$ is defined to be the smallest possiblerank over all symmetric real matrices whose $ij$th entry (for $i\neq j$) isnonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. Minimumrank is a difficult parameter to compute. However, there are now a number ofknown reduction techniques and bounds that can be programmed on a computer; wehave developed a program using the open-source mathematics software Sage toimplement several techniques. We have also established several additionalstrategies for computation of minimum rank. These techniques have been used todetermine the minimum ranks of all graphs of order 7. This paper contains alist of minimum ranks for all graphs of order at most 7. We also presentselected optimal matrices.

Table of minimum ranks of graphs of order at most 7 and selected optimal matrices