Variants on the minimum rank problem: A survey II

Shaun Fallat, Leslie Hogben

Published 2011-02-25, updated 2014-10-08Version 2

The minimum rank problem for a (simple) graph $G$ is to determine the smallest possible rank over all real symmetric matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. This paper surveys the many developments on the (standard) minimum rank problem and its variants since the survey paper \cite{FH}. In particular, positive semidefinite minimum rank, zero forcing parameters, and minimum rank problems for patterns are discussed.