#### Vol. 7, No. 5, 2014

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Computing positive semidefinite minimum rank for small graphs

### Steven Osborne and Nathan Warnberg

Vol. 7 (2014), No. 5, 595–609
##### Abstract

The positive semidefinite minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all positive semidefinite real symmetric matrices whose $ij$-th entry (for $i\ne j$) is nonzero whenever $\left\{i,j\right\}$ is an edge in $G$ and is zero otherwise. The computation of this parameter directly is difficult. However, there are a number of known bounding parameters and techniques which can be calculated and performed on a computer. We programmed an implementation of these bounds and techniques in the open-source mathematical software Sage. The program, in conjunction with the orthogonal representation method, establishes the positive semidefinite minimum rank for all graphs of order $7$ or less.

##### Keywords
zero forcing number, maximum nullity, minimum rank, positive semidefinite, zero forcing, graph, matrix
Primary: 05C50
Secondary: 15A03
##### Milestones
Received: 19 July 2011
Revised: 13 December 2011
Accepted: 12 December 2013
Published: 1 August 2014

Communicated by Chi-Kwong Li
##### Authors
 Steven Osborne Department of Mathematics Iowa State University Ames, IA 50011 United States Nathan Warnberg Department of Mathematics Iowa State University Ames, IA 50011 United States