  The minimum rank of a graph G is the minimum rank over all real symmetric
matrices whose off-diagonal sparsity pattern is the same as that of the
adjacency matrix of G. In this note we present the first exact algorithm for
the minimum rank of an arbitrary graph G. In particular, we use the notion of
determinantal rank to transform the minimum rank problem into a system of
polynomial equations that can be solved by computational tools from algebraic
geometry and commutative algebra. We provide computational results, explore
possibilities for improvement, and discuss how the algorithm can be extended to
other problems such as finding the minimum positive semidefinite rank of a
graph.
