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

TL;DR

This paper determines the minimum ranks of all graphs with up to 7 vertices using computational techniques and presents selected optimal matrices, advancing understanding of this complex graph parameter.

Contribution

The paper provides a complete list of minimum ranks for all graphs of order at most 7 and introduces new computational strategies for this challenging problem.

Findings

Minimum ranks for all graphs of order up to 7 are determined.

New computational techniques and strategies are developed and implemented.

Selected optimal matrices for these graphs are presented.

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 $i\neq j$) 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.