The Classification of Graphs on $8$ vertices with Coinciding Zero Forcing number and Maximum Nullity
Combinatorics
2025-06-13 v1
Abstract
We study the minimum rank of a (simple, undirected) graph, which is the minimum rank among all matrices in a space determined by the graph. We determine the exact set of graphs on eight vertices for which the nullity of a minimum rank matrix does not coincide with a bound determined by the zero forcing number of a graph. Although our goal was to determine which eight-vertex graphs satisfy maximum nullity equal to the zero forcing number, we also established several additional methods to assist in the computation of minimum rank for general graphs.
Keywords
Cite
@article{arxiv.2506.10726,
title = {The Classification of Graphs on $8$ vertices with Coinciding Zero Forcing number and Maximum Nullity},
author = {Wayne Barrett and Mark Hunnell and John Hutchens and John Sinkovic},
journal= {arXiv preprint arXiv:2506.10726},
year = {2025}
}