Minimum Rank

Author

Eric W. Weisstein
February 13, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/MinimumRank.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/MinimumRank.html.

©2026 Wolfram Research, Inc. except for portions noted otherwise

Sources

https://arxiv.org/pdf/2506.10726

Determining the minimum rank /maximum nullity of a graph has been studied extensively since [15]. For most graphs mrpGqpequivalently, MpGqqis difficult to compute.

An exact algorithm for calculating the minimum rank of a graph is known, but the number of variables makes the computation cost prohibitive. In [11], a purely combinatorial bound on the maximum nullity was proven:

[15] Peter M Nylen. Minimum-rank matrices with prescribed graph. Linear Algebra and its Applications, 248:303–316, 1996.

[11] AIM Minimum Rank-Special Graphs Work Group et al. Zero forcing sets and the minimum rank of graphs. Linear algebra and its applications, 428(7):1628–1648, 2008.

[8] Laura DeLoss, Jason Grout, Leslie Hogben, Tracy McKay, Jason Smith, and Geoff Tims. Table of minimum ranks of graphs of order at most 7 and selected optimal matrices. arXiv preprint arXiv:0812.0870, 2008.

https://arxiv.org/pdf/0812.0870

/Users/eww/Downloads/0812.0870/data/minrank-without-precomputed.csv

https://aimath.org/WWN/matrixspectrum/catalog2.html

https://aimath.org/pastworkshops/catalog1.html#E

Catalogs

DeLoss et al. 2008 (all graphs on <= 7 vertices)

bug(470928): Import[XXX.csv", "Tabular"] mishandles trailing whitespace

DeLossCatalog=ImportString[StringReplace[Import["/Users/eww/Documents/Mathematica/Paclets/GraphData/Collections/MinimumRank/minrank-without-precomputed.csv","String"],Whitespace~~"\n"->"\n"],"Tabular","HeaderLines"->1]

Iconize[DeLossCatalog]

ColumnTypes[DeLossCatalog]

DeleteCases[
	With[
		{
			atlas = #,
			order = #2,
			size = #3,
			connected = #4,
			tree = #5,
			gn = GraphData[{"Atlas", #}, "#"]
		},
		{
			atlas, gn,
			order, GraphData[gn, "V"],
			size, GraphData[gn, "M"],
			connected, GraphData[gn, "Connected"],
			tree, GraphData[gn, "Tree"]
		}
	] & @@@ Values @ FromTabular[
		DeLossCatalog[[All, {"Atlas number", "Order", "Size", "Connected", "Tree"}]],
		"Rows"
	],
	{_, _, n_, n_, m_, m_, c_, c_, t_, t_}
]

DeLossCatalogLookup[g_Graph,prop_]:=Module[
{rows=Rule[Lookup[#,"Atlas number"],#]&/@FromTabular[DeLossCatalog,"Rows"]},
Lookup[Lookup[rows,AtlasIndex[g]],prop]
]

DeLossCatalogLookup[GraphData["NetGraph"],"Minimum Rank"]

MinimumRank[GraphData["NetGraph"]]//Timing

GraphData["NetGraph","MinimumRank"]

MaximumNullity[GraphData["NetGraph"]]//Timing

GraphData["NetGraph","MaximumNullity"]

DeLossCatalogLookup[GraphData["NetGraph"],"Zero forcing LB"]

GraphData["NetGraph","ZeroForcingNumber"]

Graph catalog: Spectra of small graphs

http://admin.aimath.org/resources/graph-invariants/minimumrankoffamilies/

Raw

The catalog contains many data errors.  The only one fixed in the below is the incorrect use of "EhZw" for {6, 135}

https://aimath.org/pastworkshops/catalog1.html#E

data=Import["https://aimath.org/pastworkshops/catalog1.html#E","Data"];

catalog=Drop[data,3];

CatalogRowAssociation[l_List]:=Module[
{
g6=FirstCase[l,{"graph6 :",g6_}:>g6]
},
(* catalog incorrectly uses "EhZw" for {6,135} *)
g6=Replace[g6,"EhZw"->ExportString[GraphData[{6,135}],"Graph6","Header"->False]];
<|
"StandardName"->RecognizeGraph[ImportString[g6,"Graph6"]],
"Name"->If[StringQ[l[[1]]],StringTrim@StringReplace[l[[1]],StartOfString~~",":>""],Missing["NotAvailable"]],
"Graph6"->g6,
"Graph"->ImportString[g6,"Graph6"],
"VertexCount"->FirstCase[l,{"Order :",ord_}:>ord],
"MinimumRank"->FirstCase[l,{"mr :",mr_}:>mr],
"MaximumNullity"->FirstCase[l,{"M :",M_}:>M],
"FieldIndependent"->FirstCase[l,{"field independent :",fi_}:>Switch[fi,"yes",True,"no",False,_,Missing["NotAvailable"]]],
"ColinDeVeriereTypeParameter"->FirstCase[l,{"ξ :",xi_}:>xi],
"CliqueNumber"->FirstCase[l,{"ω :",omega_}:>omega],
"MinimumVertexDegree"->FirstCase[l,{"tw :",tw_}:>tw],
"Treewdith"->FirstCase[l,{"tw :",d_}:>d],
"CliqueCoveringNumber"->FirstCase[l,{"cc :",cc_}:>cc],
"Diameter"->FirstCase[l,{"Diameter :",d_}:>d],
"MaximumInducedPathLength"->FirstCase[l,{"maxinducedpath :",maxinducedpath_}:>maxinducedpath],
"ZeroForcingNumber"->FirstCase[l,{"Z :",Z_}:>Z],
"IndependenceNumber"->FirstCase[l,{"α :",alpha_}:>alpha],
"GName"->FirstCase[l,{"G name :",gname_}:>gname]
|>
]

Catalog=Tabular[CatalogRowAssociation/@catalog]

Find and correct Duplicated graphs

Select[Split[Sort[Catalog[[All,1]]]],Length[#]>1&]

Cases[Catalog,HoldPattern[{6,96}->_]]

"EhZw"->ExportString[GraphData[{6,135}],"Graph6","Header"->False]]

Corrected

Pictures are actually wrong; they're of the *same* graph

G180:

v={{127,39},{176,127},{152,212},{51,134},{106,107},{46,11}};
ee={e[6,1,2,3,5,6],e[1,4,2,5,1],{4,5}};
G180=ScaleGraph[#,AspectRatio->Automatic]&@Graph[Range@Length@v,UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@G180

Lookup[Catalog,Key@{6,96}]

MinimumRank[GraphData[{6,96}]]//Timing

GraphData[{6,96},{"VertexCount","CliqueNumber","CliqueCoveringNumber","Diameter","MinimumRank","MaximumNullity"}]

v={{78,21},{188,166},{154,91},{77,122},{108,199},{9,196}};
ee={e[4,1,3,2,5,6,4,2],e[3,5,4,3]};
G181=ScaleGraph[#,AspectRatio->Automatic]&@Graph[Range@Length@v,UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@G181

Lookup[Catalog,Key@{6,135}]

MinimumRank[GraphData[{6,135}]]//Timing

GraphData[{6,135},{"VertexCount","CliqueNumber","CliqueCoveringNumber","Diameter","MinimumRank","MaximumNullity"}]

Code

AIMathWorkshopMinimumRankCatalog=

AIMathWorkshopMinimumRankCatalogLookup[gn_,prop__]:=With[{key=Lookup[AIMathWorkshopMinimumRankCatalog,Key@gn]},If[ListQ[key],Lookup[key,prop],key]]

Counts by Vertex

Tally[GraphData[#,"V"]&/@Catalog[[All,1]]]//Sort

Length/@Table[GraphData["Connected",n],{n,6}]

Table[Complement[GraphData["Connected",n],Select[Catalog[[All,1]],GraphData[#,"V"]==n&]],{n,2,6}]

GraphData[{6,96},"Diameter"]

GraphData[{6,135},"Diameter"]

Select[Catalog[[All,1]],GraphData[#,"V"]>6&]

CliqueCoveringNumber

DeleteCases[{std=Lookup[#,"StandardName"],GraphData[std,"CliqueCoveringNumber"],Lookup[#,"CliqueCoveringNumber"]}&/@Catalog,{_,x_,x_}]

CliqueNumber

DeleteCases[{std=Lookup[#,"StandardName"],GraphData[std,"CliqueNumber"],Lookup[#,"CliqueNumber"]}&/@Catalog,{_,x_,x_}]

Select[Catalog,!FreeQ[#,{6,111}]&]

CliqueNumber[GraphData[{6,111}]]

FindAllCliques[GraphData[{6,111}]]

Diameter

DeleteCases[{std=Lookup[#,"StandardName"],GraphData[std,"Diameter"],Lookup[#,"Diameter"]}&/@Catalog,{_,x_,x_}]

Select[Catalog,!FreeQ[#,{6,96}]&]

IsomorphicGraphQ[ImportString["EjVW","Graph6"],ImportString["EhZw","Graph6"]]

IndependenceNumber

DeleteCases[{std=Lookup[#,"StandardName"],GraphData[std,"IndependenceNumber"],Lookup[#,"IndependenceNumber"]}&/@Catalog,{_,x_,x_}]

DeleteCases[%,{_,_,_Missing}]

IndependenceNumber[GraphData["HammerGraph"]]

IndependenceNumber[GraphData["K2,3"]]

MaximumNullity

DeleteCases[
{#,AIMathWorkshopMinimumRankCatalogLookup[#,"MaximumNullity"],GraphData[#,"MaximumNullity"]}&/@GraphData["Connected",2;;6],
{_,x_,x_}]//Column

MinimumRank

DeleteCases[
{#,AIMathWorkshopMinimumRankCatalogLookup[#,"MinimumRank"],GraphData[#,"MinimumRank"]}&/@GraphData["Connected",2;;6],
{_,x_,x_}]//Column

VertexCount

DeleteCases[{std=Lookup[#,"StandardName"],GraphData[std,"V"],Lookup[#,"VertexCount"]}&/@Catalog,{_,x_,x_}]

ZeroForcingNumber

DeleteCases[
{#,AIMathWorkshopMinimumRankCatalogLookup[#,"ZeroForcingNumber"],GraphData[#,"ZeroForcingNumber"]}&/@GraphData["Connected",2;;6],
{_,x_,x_}]//Column

Z is wrong since this is a tree, the catalog is wrong since MN must equal Z

Lookup[AIMathWorkshopMinimumRankCatalog,"HGraph"]

mr == n

0

Empty (edgeless)

Select[GraphData[All],GraphData[#,"MinimumRank"]===0&]

1

(Subscript[K, k])+(n-k) (Subscript[K, k]) for k>=2

Select[GraphData[All],GraphData[#,"MinimumRank"]===1&]

<=2 and <= 8 vertices

https://arxiv.org/pdf/2506.10726

[6] Wayne Barrett, Hein Van Der Holst, and Raphael Loewy. Graphs whose minimal rank is two. Electronic Journal of Linear Algebra, 11:258–280, 2004.

https://journals.uwyo.edu/index.php/ela/article/view/275/275

gs=GraphData/@{"P4","CampstoolGraph","DartGraph","P3+P2",{"LadderRung",3},{"CompleteTripartite",{3,3,3}}}

ForbiddenInducedSubgraphFree=Monitor[
Select[
GraphData[All],
NoneTrue[gs,Function[g,IsomorphicInducedSubgraphQ[g,GraphData[gn=#]]]]&
],
gn];//Timing

Length[ForbiddenInducedSubgraphFree]

ForbiddenInducedSubgraphFree

l=Cases[{#,GraphData[#,"MinimumRank"]}&/@ForbiddenInducedSubgraphFree,{gn_,_Missing}:>{gn,{0,2}}]

<=2 alternate characterization

https://journals.uwyo.edu/index.php/ela/article/view/275/275

SmallMinimumRank[g_Graph,opts___?OptionQ]:=Module[
{
hcc,cc,ind,complete,
debug=Lookup[{opts},Debug,False]
},
Which[
EdgeCount[g]==0,
	(* mr = 0 *)
	0,
And[
AllTrue[cc=ConnectedGraphComponents[g],CompleteGraphQ],
MatchQ[VertexCount/@cc,{_?(GreaterThan[1]),1...}]
],
	1,
hcc=ConnectedGraphComponents[GraphComplement[g]];
If[debug,Print["Complement connected components: ",hcc]];
ind=CompleteKPartiteIndices/@hcc;
If[debug,Print["CompleteKPartite indices: ",ind]];
And[
FreeQ[ind,{}],
MatchQ[ind,{({1..}|{_,_})..}],
Count[
Cases[ind,one:{1..}:>Length[one]],
_?(GreaterEqualThan[3])]<=2
],
	2,
True,
	Missing["NotAvailable"]
]
]

DeleteCases[
{
#,
GraphData[#,"MinimumRank"],
SmallMinimumRank[GraphData[#]]
}&/@GraphData["Connected",;;10],
{_,x_,x_}|{_,_,_Missing}
]

GraphDataCountTable

table=GraphDataCountTable["MinimumRank"]

DateString[]

TextGrid[OEISUpdates[table],Dividers->All]

Parametrized families

<<MathWorld`Graphs`

Alkane

seq=Table[GraphData[{"Alkane",n},"MinimumRank"],{n,10}]

AlternatingGroupGraph

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimumRank"],{n,2,6}]

Andrasfai

seq=Table[GraphData[{"Andrasfai",n},"MinimumRank"],{n,20}]

Antelope n×n

seq=Table[GraphData[{"Antelope",{n,n}},"MinimumRank"],{n,20}]

Antiprism

Entity["Graph",{"Antiprism",n}]["MinimumRank"]

AIM Minimum Rank—Special Graphs Work Group.  "Zero forcing sets and the minimum rank of graphs."  Linear Algebra and its Applications 428, 1628–1648, 2008.  https://draganstevanovic.wordpress.com/wp-content/uploads/2012/09/published-version2.pdf?utm_source=chatgpt.com

seq=Table[GraphData[{"Antiprism",n},"MinimumRank"],{n,3,25}]

Apollonian

seq=Table[GraphData[{"Apollonian",n},"MinimumRank"],{n,5}]

Barbell

seq=Table[GraphData[{"Barbell",n},"MinimumRank"],{n,3,20}]

Bishop n×n

seq=Table[GraphData[{"Bishop",{n,n}},"MinimumRank"],{n,20}]

BlackBishop n×n

seq=Table[GraphData[{"BlackBishop",{n,n}},"MinimumRank"],{n,20}]

Book

seq=Table[GraphData[{"Book",n},"MinimumRank"],{n,20}]

Bruhat

seq=Table[GraphData[{"Bruhat",n},"MinimumRank"],{n,6}]

Camel n×n

seq=Table[GraphData[{"Camel",{n,n}},"MinimumRank"],{n,20}]

Cameron

seq=Table[GraphData[{"Cameron",n},"MinimumRank"],{n,20}]

Centipede

seq=Table[GraphData[{"Centipede",n},"MinimumRank"],{n,10}]

FullSimplify[FindSequenceFunction[seq,n]//ExpandAll,n∈Integers]

CocktailParty

seq=Table[GraphData[{"CocktailParty",n},"MinimumRank"],{n,20}]

Complete

Entity["Graph",{"Complete",n}]["MinimumRank"]

seq=Table[GraphData[{"Complete",n},"MinimumRank"],{n,30}]

https://aimath.org/WWN/matrixspectrum/catalog2.html

CompleteBipartite n×n

Entity["Graph",{"CompleteBipartite",{n,n}}]["MinimumRank"]

seq=Table[GraphData[{"CompleteBipartite",{n,n}},"MinimumRank"],{n,20}]

Table[Piecewise[{{1,n==1}},2],{n,20}]

CompleteBipartite m×n

Entity["Graph",{"CompleteBipartite",{m,n}}]["MinimumRank"]

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"CompleteBipartite",{m,n}},"MinimumRank"],
Piecewise[{{1,m==n==1}},2]
},{n,10},{m,n}],1],{_,x_,x_}]

CompleteKPartite

https://aimath.org/WWN/matrixspectrum/catalog2.html

http://admin.aimath.org/resources/graph-invariants/minimumrankoffamilies/#/compmult

CompleteMultipartiteMinimumRank[parts_List]:=Module[{ns,k,n1,n3},If[!VectorQ[parts,IntegerQ[#]&&#>=1&],Return[Missing["InvalidPartsSizes"]]];
ns=Reverse@Sort[parts];
k=Length[ns];
n1=ns[[1]];
n3=If[k>=3,ns[[3]],0];
Which[k==1,0,n1==1,1,k>=3&&n3>=3,3,True,2]]

CompleteMultipartiteMaximumNullity[parts_List]:=Module[{mr},mr=CompleteMultipartiteMinimumRank[parts];
If[IntegerQ[mr],Total[parts]-mr,mr]]

new=DeleteCases[
{#,parts=GraphData[#,"NotationRules","CompleteKPartite"],CompleteMultipartiteMaximumNullity[parts],GraphData[#,"MaximumNullity"]}&/@GraphData["CompleteKPartite"],
{_,_,x_,x_}]

CompleteTripartite n×n×n

Entity["Graph",{"CompleteTripartite",{n,n,n}}]["MinimumRank"]

seq=Table[GraphData[{"CompleteTripartite",{n,n,n}},"MinimumRank"],{n,10}]

CompleteTripartite k×m×n

Entity["Graph",{"CompleteTripartite",{k,m,n}}]["MinimumRank"]

DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"MinimumRank"],
Piecewise[{{1,Max[k,m,n]==1},{3,Min[k,m,n]>=3}},2]
},{n,10},{m,n},{k,m}],2],
{_,x_,x_}]

CossidentePenttila

seq=Table[GraphData[{"CossidentePenttila",n},"MinimumRank"],{n,10}]

CrossedPrism

seq=Table[GraphData[{"CrossedPrism",2n},"MinimumRank"],{n,2,12}]

Crown

seq=Table[GraphData[{"Crown",n},"MinimumRank"],{n,20}]

CubeConnectedCycle

Entity["Graph",{"CubeConnectedCycle",n}]["MinimumRank"]

seq=Table[GraphData[{"CubeConnectedCycle",n},"MinimumRank"],{n,3,8}]

Cycle

Entity["Graph",{"Cycle",n}]["MinimumRank"]

seq=Table[GraphData[{"Cycle",n},"MinimumRank"],{n,3,30}]

Table[n-2,{n,3,20}]

CycleComplement

seq=Table[GraphData[{"CycleComplement",n},"MinimumRank"],{n,3,20}]

https://aimath.org/WWN/matrixspectrum/catalog2.html

DiagonalIntersection

seq=Table[GraphData[{"DiagonalIntersection",n},"MinimumRank"],{n,3,14}]

Dipyramid

seq=Table[GraphData[{"Dipyramid",n},"MinimumRank"],{n,3,20}]

DorogovtsevGoltsevMendes

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumRank"],{n,0,7}]

DoubleCone

seq=Table[GraphData[{"DoubleCone",n},"MinimumRank"],{n,3,20}]

Empty

Entity["Graph",{"Empty",n}]["MinimumRank"]

seq=Table[GraphData[{"Empty",n},"MinimumRank"],{n,30}]

ErectedPolygon

http://admin.aimath.org/resources/graph-invariants/minimumrankoffamilies/#/pathstrongprod

FibonacciCube

seq=Table[GraphData[{"FibonacciCube",n},"MinimumRank"],{n,15}]

Fiveleaper n×n

seq=Table[GraphData[{"Fiveleaper",{n,n}},"MinimumRank"],{n,20}]

Flower

seq=Table[GraphData[{"Flower",n},"MinimumRank"],{n,5,20}]

FoldedCube

seq=Table[GraphData[{"FoldedCube",n},"MinimumRank"],{n,2,10}]

Gear

seq=Table[GraphData[{"Gear",n},"MinimumRank"],{n,3,20}]

Giraffe n×n

seq=Table[GraphData[{"Giraffe",{n,n}},"MinimumRank"],{n,20}]

Goldberg

seq=Table[GraphData[{"Goldberg",n},"MinimumRank"],{n,3,20}]

Grid n×n

Entity["Graph",{"Grid",{n,n}}]["MinimumRank"]

seq=Table[GraphData[{"Grid",{n,n}},"MinimumRank"],{n,20}]

Table[n(n-1),{n,20}]

https://aimath.org/WWN/matrixspectrum/catalog2.html

Grid m×n

Entity["Graph",{"Grid",{m,n}}]["MinimumRank"]

https://aimath.org/WWN/matrixspectrum/catalog2.html

Sort@DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"Grid",{m,n}},"MinimumRank"],
m n-Min[m,n]
},{n,10},{m,n}],1],{_,x_,x_}]

Grid n×n×n

Entity["Graph",{"Grid",{n,n,n}}]["MinimumRank"]

seq=Table[GraphData[{"Grid",{n,n,n}},"MinimumRank"],{n,10}]

Grid k×m×n

Entity["Graph",{"Grid",{k,m,n}}]["MinimumRank"]

Sort@DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"Grid",{k,m,n}},"MinimumRank"],
XXX
},{n,5},{m,n},{k,m}],2],{_,x_,x_}]

HalvedCube

seq=Table[GraphData[{"HalvedCube",n},"MinimumRank"],{n,10}]

Hanoi

seq=Table[GraphData[{"Hanoi",n},"MinimumRank"],{n,6}]

Helm

seq=Table[GraphData[{"Helm",n},"MinimumRank"],{n,3,20}]

Hypercube

Entity["Graph",{"Hypercube",n}]["MinimumRank"]

seq=Table[GraphData[{"Hypercube",n},"MinimumRank"],{n,0,10}]

Table[Piecewise[{{0,n==0}},(2^(n-1))],{n,0,10}]

KC

https://aimath.org/WWN/matrixspectrum/catalog2.html

DeleteCases[
With[{gn=#,t=#2,s=#3},{gn,(s-2)t,GraphData[gn,"MinimumRank"]}]&@@@DeleteCases[
{#,Sequence@@GraphData[#,"NotationRules","KC"]}&/@GraphData["KC"],
{_,_,_?(LessThan[4])}
],
{_,x_,x_}]

Keller

Entity["Graph",{"Keller",n}]["MinimumRank"]

seq=Table[GraphData[{"Keller",n},"MinimumRank"],{n,5}]

King m×n

https://aimath.org/WWN/matrixspectrum/catalog2.html

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"King",{m,n}},"MinimumRank"],
Piecewise[{{Max[m,n]-1,Min[m,n]==1}},(m-1)(n-1)]
},{n,10},{m,n}],1],{_,x_,x_}]

King n×n

seq=Table[GraphData[{"King",{n,n}},"MinimumRank"],{n,20}]

FindSequenceFunction[seq,n]

Knight n×n

Entity["Graph",{"Knight",{n,n}}]["MinimumRank"]

seq=Table[GraphData[{"Knight",{n,n}},"MinimumRank"],{n,20}]

Knight m×n

Entity["Graph",{"Knight",{m,n}}]["MinimumRank"]

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"Knight",{m,n}},"MinimumRank"],
XXX
},{n,10},{m,n}],1],{_,x_,x_}]

KP

https://aimath.org/WWN/matrixspectrum/catalog2.html

DeleteCases[
With[{gn=#,s=#2,t=#3},{gn,s(t-1),GraphData[gn,"MinimumRank"]}]&@@@DeleteCases[
{#,Sequence@@GraphData[#,"NotationRules","KP"]}&/@GraphData["KP"],
{_,_,1}
],
{_,x_,x_}]

Ladder

Entity["Graph",{"Ladder",n}]["MinimumRank"]

seq=Table[GraphData[{"Ladder",n},"MinimumRank"],{n,20}]

Table[Piecewise[{{1,n==1}},2(n-1)],{n,20}]

LadderRung

seq=Table[GraphData[{"LadderRung",n},"MinimumRank"],{n,20}]

LindgrenSousselier

seq=Table[GraphData[{"LindgrenSousselier",n},"MinimumRank"],{n,11}]

LucasCube

seq=Table[GraphData[{"LucasCube",n},"MinimumRank"],{n,15}]

MengerSponge

seq=Table[GraphData[{"MengerSponge",n},"MinimumRank"],{n,3}]

MoebiusLadder

Entity["Graph",{"MoebiusLadder",n}]["MinimumRank"]

seq=Table[GraphData[{"MoebiusLadder",n},"MinimumRank"],{n,3,25}]

Table[2(n-2),{n,3,25}]

https://aimath.org/WWN/matrixspectrum/catalog2.html

Mycielski

Entity["Graph",{"Mycielski",n}]["MinimumRank"]

seq=Table[GraphData[{"Mycielski",n},"MinimumRank"],{n,11}]

Necklace

seq=Table[GraphData[{"Necklace",n},"MinimumRank"],{n,10}]

Table[3n-2,{n,10}]

http://admin.aimath.org/resources/graph-invariants/minimumrankoffamilies/#/neck

https://aimath.org/WWN/matrixspectrum/catalog2.html

[DG+09] L. DeAlba, J. Grout, L. Hogben, R. Mikkelson, and K. Rasmussen. Universally optimal matrices and field independence of the minimum rank of a graph. Elec. J. Lin. Alg., 18: 403–419, 2009.

Odd

Entity["Graph",{"Odd",n}]["MinimumRank"]

seq=Table[GraphData[{"Odd",n},"MinimumRank"],{n,7}]

Paley

seq=Table[GraphData[{"Paley",n},"MinimumRank"],{n,XXX}]

Pan

seq=Table[GraphData[{"Pan",n},"MinimumRank"],{n,3,20}]

Path

Entity["Graph",{"Path",n}]["MinimumRank"]

https://aimath.org/WWN/matrixspectrum/catalog2.html

seq=Table[GraphData[{"Path",n},"MinimumRank"],{n,20}]

PathComplement

seq=Table[GraphData[{"PathComplement",n},"MinimumRank"],{n,20}]

Peisert

seq=Table[GraphData[{"Peisert",n},"MinimumRank"],{n,XXX}]

Pell

seq=Table[GraphData[{"Pell",n},"MinimumRank"],{n,9}]

Pineapple

new=With[{gn=#,m=#2[[1]],k=#2[[2]]},{gn,3,GraphData[gn,"MinimumRank"]}]&@@@DeleteCases[
{#,GraphData[#,"NotationRules","Pineapple"]}&/@GraphData["Pineapple"],
{_,{_,1}}
]

DeleteCases[new,{_,x_,x_}]

https://aimath.org/WWN/matrixspectrum/catalog2.html

http://admin.aimath.org/resources/graph-invariants/minimumrankoffamilies/#/crciclo

http://admin.aimath.org/resources/graph-invariants/minimumrankoffamilies/view/_id/5303cb962d62b7385f0be323

[DG+09] L. DeAlba, J. Grout, L. Hogben, R. Mikkelson, and K. Rasmussen. Universally optimal matrices and field independence of the minimum rank of a graph. Elec. J. Lin. Alg., 18: 403–419, 2009.

PlummerToft

seq=Table[GraphData[{"PlummerToft",n},"MinimumRank"],{n,20}]

Prism

Entity["Graph",{"Prism",n}]["MinimumRank"]

seq=Table[GraphData[{"Prism",n},"MinimumRank"],{n,3,25}]

Table[Piecewise[{{3,n==3}},2(n-2)],{n,3,25}]

Queen n×n

seq=Table[GraphData[{"Queen",{n,n}},"MinimumRank"],{n,20}]

Rook m×n

https://aimath.org/WWN/matrixspectrum/catalog2.html

DeleteCases[With[{gn=#,s=#2,t=#3},{gn,s+t-2,GraphData[gn,"MinimumRank"]}]&@@@DeleteCases[
{#,Sequence@@GraphData[#,"NotationRules","Rook"]}&/@GraphData["Rook"],
{_,1,_}],
{_,x_,x_}]

Rook n×n

seq=Table[GraphData[{"Rook",{n,n}},"MinimumRank"],{n,20}]

FindSequenceFunction[seq,n]

RookComplement n×n

seq=Table[GraphData[{"RookComplement",{n,n}},"MinimumRank"],{n,20}]

SierpinskiCarpet

seq=Table[GraphData[{"SierpinskiCarpet",n},"MinimumRank"],{n,4}]

SierpinskiGasket

seq=Table[GraphData[{"SierpinskiGasket",n},"MinimumRank"],{n,7}]

SierpinskiTetrahedron

seq=Table[GraphData[{"SierpinskiTetrahedron",n},"MinimumRank"],{n,6}]

Spoke n×n

seq=Table[GraphData[{"Spoke",{n,n}},"MinimumRank"],{n,5}]

StackedBook m×n

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"StackedBook",{m,n}},"MinimumRank"],
XXX
},{n,10},{m,n}],1],{_,x_,x_}]

StackedBook n×n

seq=Table[GraphData[{"StackedBook",{n,n}},"MinimumRank"],{n,10}]

StackedPrism m×n

https://aimath.org/WWN/matrixspectrum/catalog2.html

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"StackedPrism",{m,n}},"MinimumRank"],
m n-Min[m,2n]
},{m,3,8},{n,5}],1],{_,x_,x_}]

StackedPrism n×n

seq=Table[GraphData[{"StackedPrism",{n,n}},"MinimumRank"],{n,3,5}]

Star

Entity["Graph",{"Star",n}]["MinimumRank"]

seq=Table[GraphData[{"Star",n},"MinimumRank"],{n,25}]

Table[Piecewise[{{0,n==1},{1,n==2}},2],{n,25}]

Sun

seq=Table[GraphData[{"Sun",n},"MinimumRank"],{n,3,20}]

Sunlet

seq=Table[GraphData[{"Sunlet",n},"MinimumRank"],{n,3,20}]

https://aimath.org/WWN/matrixspectrum/catalog2.html

Tetrahedral

seq=Table[GraphData[{"Tetrahedral",n},"MinimumRank"],{n,6,10}]

TorusGrid m×n

http://admin.aimath.org/resources/graph-invariants/minimumrankoffamilies/#/pathstrongprod

DeleteCases[With[{gn=#,s=#2,t=#3},{gn,If[t==s&&OddQ[s],s(t-2)+1,s(t-2)],GraphData[gn,"MinimumRank"]}]&@@@(
{#,Sequence@@GraphData[#,"NotationRules","TorusGrid"]}&/@GraphData["TorusGrid"]),
{_,x_,x_}]

TorusGrid n×n

seq=Table[GraphData[{"TorusGrid",{n,n}},"MinimumRank"],{n,3,20}]

FullSimplify[FindSequenceFunction[seq,n],n∈Integers]

Transposition

Entity["Graph",{"Transposition",n}]["MinimumRank"]

seq=Table[GraphData[{"Transposition",n},"MinimumRank"],{n,6}]

Trapezohedron

Entity["Graph",{"Trapezohedron",n}]["MinimumRank"]

seq=Table[GraphData[{"Trapezohedron",n},"MinimumRank"],{n,3,20}]

Tree

https://aimath.org/WWN/matrixspectrum/catalog2.html

Triangular

seq=Table[GraphData[{"Triangular",n},"MinimumRank"],{n,2,20}]

TriangularGrid

seq=Table[GraphData[{"TriangularGrid",n},"MinimumRank"],{n,0,20}]

Simplify[(1/2)(n+1)(n+2)-(n+1)]

Table[n(n+1)/2,{n,0,20}]

Table[Binomial[n+1,2],{n,0,20}]

https://aimath.org/WWN/matrixspectrum/catalog2.html

mth supertriangle = {"TriangularGrid", m-1}

GraphData[{"TriangularGrid",2}]

Table[GraphData[{"TriangularGrid",n},"V"],{n,10}]

Table[(1/2)(n+1)(n+2),{n,10}]

Table[GraphData[{"TriangularGrid",n},"Diameter"],{n,10}]

Table[n,{n,10}]

TriangularHoneycombAcuteKnight

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MinimumRank"],{n,20}]

TriangularHoneycombBishop

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MinimumRank"],{n,20}]

TriangularHoneycombObtuseKnight

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"MinimumRank"],{n,20}]

TriangularHoneycombQueen

seq=Table[GraphData[{"TriangularHoneycombQueen",n},"MinimumRank"],{n,20}]

TriangularHoneycombRook

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MinimumRank"],{n,20}]

TriangularSnake

seq=Table[GraphData[{"TriangularSnake",n},"MinimumRank"],{n,1,20,2}]

Unicyclic

https://aimath.org/WWN/matrixspectrum/catalog2.html

Web

seq=Table[GraphData[{"Web",n},"MinimumRank"],{n,3,20}]

Wheel

Entity["Graph",{"Wheel",n}]["MinimumRank"]

seq=Table[GraphData[{"Wheel",n},"MinimumRank"],{n,4,20}]

WheelComplement

seq=Table[GraphData[{"WheelComplement",n},"MinimumRank"],{n,4,20}]

WhiteBishop n×n

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimumRank"],{n,2,20}]

WhiteCamel n×n

seq=Table[GraphData[{"Bishop",{n,n}},"MinimumRank"],{n,{5}}]

Zebra n×n

seq=Table[GraphData[{"Zebra",{n,n}},"MinimumRank"],{n,20}]