The Classification of Graphs on $8$ vertices with Coinciding Zero Forcing number and Maximum Nullity

TL;DR

This paper classifies all eight-vertex graphs where the nullity of the minimum rank matrix equals the zero forcing number, providing new methods for computing minimum rank in graph theory.

Contribution

It precisely identifies graphs on eight vertices with coinciding nullity and zero forcing number, and introduces methods for minimum rank computation.

Findings

Identified all 8-vertex graphs with matching nullity and zero forcing number.

Developed new techniques for minimum rank calculation.

Enhanced understanding of graph nullity and zero forcing relationships.

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.