Minimum Rank of Matrices Described by a Graph or Pattern over the Rational, Real and Complex Numbers

Avi Berman, Shmuel Friedland, Leslie Hogben, Uriel G. Rothblum, Bryan Shader
2008 Electronic Journal of Combinatorics  
We use a technique based on matroids to construct two nonzero patterns $Z_1$ and $Z_2$ such that the minimum rank of matrices described by $Z_1$ is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by $Z_2$ is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture by Arav, Hall, Koyucu, Li and Rao about rational realization of minimum rank of sign patterns. Using $Z_1$ and $Z_2$,
more » ... g $Z_1$ and $Z_2$, we construct symmetric patterns, equivalent to graphs $G_1$ and $G_2$, with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.
doi:10.37236/749 fatcat:cybj234mtzazjdp2fnrv5rjpne