Zero forcing is a combinatorial game played on a graph with a goal of turning all of the vertices of the graph black while having to use as few unforced moves as possible. This leads to a parameter known as the zero forcing number which can be used t
o give an upper bound for the maximum nullity of a matrix associated with the graph. We introduce a new variation on the zero forcing game which can be used to give an upper bound for the maximum nullity of a matrix associated with a graph that has $q$ negative eigenvalues. This gives some limits to the number of positive eigenvalues that such a graph can have and so can be used to form lower bounds for the inertia set of a graph.
The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i eq j$) is nonzero whenever ${i,j}$ is an edge in $G$ and is zero otherwise. Minimum rank is a difficult par
ameter to compute. However, there are now a number of known reduction techniques and bounds that can be programmed on a computer; we have developed a program using the open-source mathematics software Sage to implement several techniques. We have also established several additional strategies for computation of minimum rank. These techniques have been used to determine the minimum ranks of all graphs of order 7. This paper contains a list of minimum ranks for all graphs of order at most 7. We also present selected optimal matrices.
