See also
Maximum Nullity,
Null Space,
Nullity,
Rank,
Rank-Nullity Theorem,
Zero
Forcing Number
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References
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 Mathematics 3, 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.
Cite this as:
Weisstein, Eric W. "Minimum Rank." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MinimumRank.html
be a field,
be a simple undirected graph on 
vertices, and let 
be the set of all 
symmetric matrices
whose nonzero off-diagonal elements occur in exactly the positions corresponding
to the edges of 
.
Then the minimum rank 
is the minimum of the rank over all matrices
in 
.
. The most common case of 
is commonly assumed unless otherwise stated, in whihc case
is simply omitted from the notation.
by the rank-nullity
theorem
