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arXiv · 2506.10726

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

Wayne Barrett·Mark Hunnell·John Hutchens·John Sinkovic
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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.

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Wayne Barrett, Mark Hunnell, John Hutchens, John Sinkovic. 2025-06-12. The Classification of Graphs on $8$ vertices with Coinciding Zero Forcing number and Maximum Nullity. https://arxiv.org/abs/2506.10726

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