The zero forcing number is a graph parameter that is closely related to the maximum nullity and that bounds it from above:
(AIM 2008, Barrett et al. 2025).
Results from the American Institute of Mathematics (2008) showed that for any graph on 6 or fewer vertices. DeLoss et
al. (2008) subsequently computed and bounds on for all 7-vertex graphs, and the resulting table establishes
that
for all graphs on 7 vertices (Barrett et al. 2025). Barrett et al. (2025)
subsequently showed that for all graphs on 8 vertices with the exception of
a set of 7 graphs denoted to and which are illustrated above. As it turns out, is the graph complement
of the 3-Plummer-Toft graph.
AIM Minimum Rank--Special Graphs Work Group. "Zero Forcing Sets and the Minimum Rank of Graphs." Lin. Alg. Appl.428,
1628-1648, 2008.Almodovar, E.; DeLoss, L.; Hogben, L.; Hogenson, K.;
Murphy, K.; Peters, T.; and Ramírez, C. A. "Minimum Rank, Maximum Nullity
and Zero Forcing Number for Selected Graph Families." Involve: A Journal
of Mathematics3, 371-392, 2010.American Institute of Mathematics.
"Graph Catalog: Families of Graphs." https://aimath.org/WWN/matrixspectrum/catalog2.html.American
Institute of Mathematics. "AIM Minimum Rank Graph Catalog." http://admin.aimath.org/resources/graph-invariants/minimumrankoffamilies/#/super.American
Institute of Mathematics. "AIM Minimum Rank Graph Catalog References."
https://web.archive.org/web/20160207084159/http://orion.math.iastate.edu/lhogben/AIMmrgraphcatrefs.pdf.Barrett,
W.; Hunnell, M.; Hutchens, J.; and Sinkovic, J. "The Classification of Graphs
on 8 vertices with Coinciding Zero Forcing number and Maximum Nullity." 12 Jun
2025. https://arxiv.org/abs/2506.10726.Barrett,
W.; van der Holst, H.; and Loewy, R. "Graphs Whose Minimal Rank is Two."
Elec. J. Lin. Alg.11, 258-280, 2004.DeLoss, L.; Grout,
J.; Hogben, L.; McKay, T.; Smith, J.; and Tims, J. "Table of Minimum Ranks of
Graphs of Order at Most 7 and Selected Optimal Matrices." 4 Dec 2008. https://arxiv.org/abs/0812.0870.Fallat,
S. M. and Hogben, L. "Minimum Rank, Maximum Nullity, and Zero Forcing Number
of Graphs." Ch. 46 in Handbook of Linear Algebra, 2nd ed. Boca Raton,
FL: CRC Press, 2014.Nylen, P. M. "Minimum-Rank Matrices With
Prescribed Graph." Lin. Alg. Appl.248, 303-316, 1996.
is a graph parameter that is closely related to the maximum nullity 
and that bounds it from above:
for any graph on 6 or fewer vertices. DeLoss et
al. (2008) subsequently computed 
and bounds on 
for all 7-vertex graphs, and the resulting table establishes
that 
for all graphs on 7 vertices (Barrett et al. 2025). Barrett et al. (2025)
subsequently showed that 
for all graphs on 8 vertices with the exception of
a set of 7 graphs denoted 
to 
and which are illustrated above. As it turns out, 
is the graph complement
of the 3-Plummer-Toft graph.
