Table of minimum ranks of graphs of order at most 7 and selected optimal matrices

Details Table of minimum ranks of graphs of order at most 7 and selected optimal matrices Table of minimum ranks of graphs of order at most 7 and selected optimal matrices

2008-12-04

arxiv.org/abs/0812.0870v1

Mathematics

Combinatorics

54 pages, 1 spreadsheet, 1 Sage program

Scientific paper

The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a difficult parameter to compute. However, there are now a number of known reduction techniques and bounds that can be programmed on a computer; we have developed a program using the open-source mathematics software Sage to implement several techniques. We have also established several additional strategies for computation of minimum rank. These techniques have been used to determine the minimum ranks of all graphs of order 7. This paper contains a list of minimum ranks for all graphs of order at most 7. We also present selected optimal matrices.

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