Techniques for determining the minimum rank of a small graph

Abstract

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 ≠ 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.