Ela the Minimum Rank Problem over Finite Fields

The problem of finding mr(F,G) and describing Gk(F ) has recently attracted considerable attention, particularly for the case in which F = R (see [29, 17, 26, 25, 27, 13, 33, 5, 9, 22, 2, 11, 6, 7, 10, 18, 4]). The minimum rank problem over R is a sub-problem of a much more general problem, the inverse eigenvalue problem for symmetric matrices: given a family of real numbers, find every symmetric matrix that has the family as its eigenvalues. More particularly, the minimum rank problem is a sub-problem of the inverse eigenvalue problem for graphs, which fixes a zero/nonzero pattern for the symmetric matrices considered in the inverse eigenvalue problem. The minimum rank problem can also be thought of in this way: given a fixed pattern of off-diagonal zeros, what is the smallest rank that a symmetric matrix having that pattern can achieve?