Cooling Number

Author

Eric W. Weisstein
February 8, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/CoolingNumber.nb.

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Source

Figure

g1=With[{p=PathGraph[Range@4,VertexSize->Large]},
Tree[
Labeled[p,Style[Subscript[Style["P",Italic],4],Directive[24,Black,FontFamily->"Times"]],Left],
{
Tree[HighlightGraph[p,{Style[1,Blue]}],{
Tree[HighlightGraph[p,{Style[1|2,Hue[.5]],Style[3,Blue]}],{HighlightGraph[p,{Style[1|2|3|4,Hue[.5]]}]}],
Tree[HighlightGraph[p,{Style[1|2,Hue[.5]],Style[4,Blue]}],{HighlightGraph[p,{Style[1|2|3|4,Hue[.5]]}]}]
}],
Tree[HighlightGraph[p,{Style[2,Blue]}],{
HighlightGraph[p,{Style[1|2|3,Hue[.5]],Style[4,Blue]}]
}
]},
ImageSize->800,
TreeLayout->Left]
]

g2=With[{p=CompleteGraph[{2,3},VertexSize->Large],blue=Blue,aqua=Hue[.5]},
Tree[
Labeled[p,Style[Subscript[Style["K",Italic],2,3],Directive[24,Black,FontFamily->"Times"]],Left],
{
Tree[HighlightGraph[p,{Style[1,blue]}],{
Tree[HighlightGraph[p,{Style[1|3|4|5,aqua],Style[2,blue]}],{HighlightGraph[p,{Style[1|2|3|4|5,aqua]}]}]
}],
Tree[HighlightGraph[p,{Style[3,blue]}],{
Tree[HighlightGraph[p,{Style[1|2|3,aqua],Style[4,blue]}],{HighlightGraph[p,{Style[1|2|3|4|5,aqua]}]}],
Tree[HighlightGraph[p,{Style[1|2|3,aqua],Style[5,blue]}],{HighlightGraph[p,{Style[1|2|3|4|5,aqua]}]}]
}],
Tree[HighlightGraph[p,{Style[4,blue]}],{
Tree[HighlightGraph[p,{Style[1|2|4,aqua],Style[3,blue]}],{HighlightGraph[p,{Style[1|2|3|4|5,aqua]}]}]
}]
},
ImageSize->1000,
TreeLayout->Left]
]

GraphicsRow[{g1,g2},ImageSize->2000,Dividers->All]

Counts

Computation

Do[Print[{n,(Put[CoolingNumber[#]&/@Graphs[n],"CoolingNumber_"<>If[n<10,"0",""]<>ToString[n]<>".m"])//Timing}],{n,9}]

All

GraphData

A369279

Flatten[all=Table[With[{t=Tally[Sort[GraphData[#,"CoolingNumber"]&/@GraphData[n]]]},If[FreeQ[t,{1,_}],Prepend[t,{1,0}],t]][[All,-1]],{n,7}]]

all//Column

n-node files

Flatten[all=Table[With[{t=Tally[Sort[Get[FileNameJoin[{$GraphDataDirectory,"Nodes","CoolingNumber","CoolingNumber_"<>If[n<10,"0",""]<>ToString[n]<>".m"}]]]]},If[FreeQ[t,{1,_}],Prepend[t,{1,0}],t]][[All,-1]],{n,10}]]

all//Column

Length/@all

Connected

A369280

GraphData

Flatten[connected=Table[With[{t=Tally[Sort[GraphData[#,"CoolingNumber"]&/@GraphData["Connected",n]]]},If[FreeQ[t,{1,_}],Prepend[t,{1,0}],t]][[All,-1]],{n,7}]]

connected//Column

Table[1+Floor[n/2],{n,7}]

n-node files

Flatten[connected=Table[With[{t=Tally[Sort[Get[FileNameJoin[{$GraphDataDirectory,"Nodes","CoolingNumber","CoolingNumber_"<>If[n<10,"0",""]<>ToString[n]<>".m"}]][[ConnectedGraphIndices[n]]]]]},If[FreeQ[t,{1,_}],Prepend[t,{1,0}],t]][[All,-1]],{n,10}]]

connected//Column

Length/@connected

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

GraphDataCountTable

GraphDataCountTable["CoolingNumber",{"Antiprism"->057354,"Cycle"->002264,"Flower"->004526,"FoldedCube"->004526,"Ladder"->004523,"Path"->004526,"Star"->040000,"Web"->004526}]

Parametrized families

<<MathWorld`Graphs`

Alkane

Inferred for n > 8:

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

Do[Print[{gn={"Alkane",n},{0,CoolingNumber[GraphData[gn]]//Timing}}],{n,10}]

AlternatingGroupGraph

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

Do[Print[{gn=GraphData[{"AlternatingGroupGraph",n},"#"],CoolingNumber[GraphData@gn]//Timing}],{n,2,6}]

Andrasfai

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

Do[Print[{gn={"Andrasfai",n},{0,CoolingNumber[GraphData[gn]]//Timing}}],{n,20}]

Antelope n×n

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

Do[Print[{gn={"Antelope",{n,n}},CoolingNumber[GraphData@gn,Debug->True,MaxMemoryUsed[]/10^9.]//Timing}],{n,20}]

Antiprism

A057354

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

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

gf=(x^3)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2&&n∈Integers]

Table[Floor[2(n+4)/5],{n,3,25}]

Do[Print[{gn={"Antiprism",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,25}]

Apollonian

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

Do[Print[{gn={"Apollonian",n},CoolingNumber[GraphData@gn]//Timing}],{n,5}]

Barbell

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

BlackBishop n×n

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

Do[Print[{gn={"BlackBishop",{n,n}},CoolingNumber[GraphData@gn]//Timing}],{n,20}]

Bishop n×n

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

Do[Print[{gn={"Bishop",{n,n}},CoolingNumber[GraphData@gn]//Timing}],{n,20}]

Book

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

Do[Print[{gn={"Book",n},CoolingNumber[GraphData@gn]//Timing}],{n,20}]

Bruhat

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

Do[Print[{gn={"Bruhat",n},CoolingNumber[GraphData@gn]//Timing}],{n,6}]

Camel n×n

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

Length@DeleteMissing@seq

Do[Print[{gn={"Camel",{n,n}},CoolingNumber[GraphData@gn,Debug->True]//Timing}],{n,20}]

Centipede

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

FindSequenceFunction[seq,n]

CocktailParty

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

Complete

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

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

FindSequenceFunction[seq,n]

CompleteBipartite m×n

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

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"CompleteBipartite",{m,n}},"CoolingNumber"],
Piecewise[{{2,Max[m,n]<3}},3]
},{n,10},{m,n}],1],{_,x_,x_}]

CompleteBipartite n×n

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

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

CompleteTripartite k×m×n

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

DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"CoolingNumber"],
Piecewise[{{2,Max[k,m,n]<3}},3]
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CompleteTripartite n×n×n

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

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

CrossedPrism

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

Do[Print[{gn={"CrossedPrism",2n},CoolingNumber[GraphData@gn]//Timing}],{n,2,12}]

Crown

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

CubeConnectedCycle

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

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

Do[Print[{gn={"CubeConnectedCycle",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,8}]

Cycle

A002264

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

https://arxiv.org/pdf/2401.03496.pdf

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

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

Table[(1/9) (9+3 n-2 (Sqrt[3]) Sin[((2 n π)/3)]),{n,3,30}]

Table[Ceiling[((n+2)/3)],{n,3,30}]

Table[Floor[((n+4)/3)],{n,3,30}]

Table[CoolingNumber[CycleGraph[n]],{n,3,30}]//Timing

CycleComplement

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

DiagonalIntersection

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

Do[Print[{gn={"DiagonalIntersection",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,14}]

Dipyramid

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

DorogovtsevGoltsevMendes

Inferred for n > 5:

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

Do[Print[{gn={"DorogovtsevGoltsevMendes",n},CoolingNumber[GraphData@gn]//Timing}],{n,0,7}]

Do[Print[{gn={"DorogovtsevGoltsevMendes",n},CoolingNumber[DorogovtsevGoltsevMendesGraph[n],Debug->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,0,7}]

DoubleCone

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

Empty

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

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

Table[CoolingNumber[Graph[Range[n],{}]],{n,10}]//Timing

FibonacciCube

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

Do[Print[{gn={"FibonacciCube",n},CoolingNumber[GraphData@gn]//Timing}],{n,15}]

Fiveleaper n×n

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

Do[Print[{gn={"Fiveleaper",{n,n}},CoolingNumber[GraphData@gn,Debug->True]//Timing}],{n,20}]

Flower

A004526

Inferred for n > 18:

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

Table[2+Ceiling[n/2],{n,5,20}]

Do[Print[{gn={"Flower",n},CoolingNumber[FlowerGraph[n],Debug->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,5,20}]

FoldedCube

A004526

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

Table[Floor[n/2]+1,{n,2,11}]

Do[Print[{gn={"FoldedCube",n},CoolingNumber[FoldedCubeGraph[n]]//Timing}],{n,2,11}]

Gear

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

Do[Print[{gn={"Gear",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,20}]

Giraffe n×n

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

Length@DeleteMissing@seq

Do[Print[{gn={"Giraffe",{n,n}},CoolingNumber[GraphData@gn,Debug->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,20}]

{"Giraffe", {8, 8}} needs 138+ GB

Do[Print[{gn={"Giraffe",{n,n}},CoolingNumber[GiraffeGraph[n,n],Debug->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,20}]

Do[Print[{gn={"Giraffe",{n,n}},CoolingNumber[GiraffeGraph[n,n],Debug->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,9,10}]

Goldberg

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

Do[Print[{gn={"Goldberg",n},CoolingNumber[GoldbergGraph[n],Debug->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,3,20}]

Grid m×n

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

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

Grid n×n

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

https://arxiv.org/pdf/2401.03496.pdf

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

Cf. A001651

Floor[(3 Range[20]-1)/2]

Table[2n-2Ceiling[Log2[n+3]]+Interval[{0,2}],{n,20}]

Do[Print[{gn={"Grid",{n,n}},CoolingNumber[GraphData[gn],"UseSymmetry"->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,10}]//Timing

Do[Print[{gn={"Grid",{n,n}},CoolingNumber[GraphData[gn],"UseSymmetry"->True,Debug->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,8,8}];

Grid k×m×n

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

Sort@Flatten[Table[{
{k,m,n},
GraphData[{"Grid",{k,m,n}},"CoolingNumber"],
XXX
},{n,5},{m,n},{k,m}],2]

Grid n×n×n

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

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

Table[Floor[((((n+1))^2)/3)],{n,10}]

Do[Print[{gn={"Grid",{n,n,n}},CoolingNumber[GraphData[gn],"UseSymmetry"->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,3}];

HalvedCube

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

Do[Print[{gn={"HalvedCube",n},CoolingNumber[GraphData@gn]//Timing}],{n,10}]

Hanoi

seq=Table[GraphData[{"Hanoi",n},"CoolingNumber"],{n,5}]

Do[Print[{gn={"Hanoi",n},CoolingNumber[GraphData@gn]//Timing}],{n,5}]

Helm

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

Do[Print[{gn={"Helm",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,20}]

Hypercube

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

Inferred for n > 8:

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

Length@DeleteMissing@seq-1

Do[Print[{gn={"Hypercube",n},CoolingNumber[GraphData@gn]//Timing}],{n,0,10}]

Do[Print[{gn={"Hypercube",n},CoolingNumber[GraphData[gn],"UseSymmetry"->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,10}]//Timing

Do[Print[{gn={"Hypercube",n},CoolingNumber[GraphData[gn],"UseSymmetry"->True,Debug->True]//Timing,MaxMemoryUsed[]/10^9.}],{n,9,10}]//Timing

Keller

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

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

Do[Print[{gn={"Keller",n},CoolingNumber[GraphData@gn]//Timing}],{n,5}]

King n×n

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

Do[Print[{gn={"King",{n,n}},CoolingNumber[GraphData@gn]//Timing}],{n,20}]

Knight m×n

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

Knight n×n

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

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

Length@DeleteMissing@seq

FindSequenceFunction[Rest@DeleteMissing@seq,n-1]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,z]//Head//Timing

Do[Print[{gn={"Knight",{n,n}},CoolingNumber[GraphData@gn]//Timing}],{n,20}]

Ladder

A004523

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

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

Table[Floor[2(n+2)/3],{n,20}]

Do[Print[{gn={"Ladder",n},CoolingNumber[GraphData@gn]//Timing}],{n,20}]

LadderRung

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

Do[Print[{gn={"LadderRung",n},CoolingNumber[GraphData@gn]//Timing}],{n,20}]

LucasCube

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

Length@DeleteMissing@seq

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

Simplify[SeriesCoefficient[gf,{x,0,n}],n>0]

Do[Print[{gn={"LucasCube",n},CoolingNumber[GraphData@gn]//Timing}],{n,15}]

MengerSponge

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

MoebiusLadder

A288156

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

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

Table[Floor[(2n+9)/5],{n,3,25}]

Do[Print[{gn={"MoebiusLadder",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,25}]

Mycielski

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

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

Do[Print[{gn={"Mycielski",n},CoolingNumber[GraphData@gn]//Timing}],{n,11}]

Odd

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

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

Do[Print[{gn={"Odd",n},CoolingNumber[GraphData@gn]//Timing}],{n,7}]

Pan

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

Path

A004526

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

https://arxiv.org/pdf/2401.03496.pdf

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

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

Table[(1/4) (3+(((-1))^n)+2 n),{n,20}]

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

Table[CoolingNumber[PathGraph[Range@n]],{n,20}]//Timing

PathComplement

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

Pell

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

Length@DeleteMissing@seq

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

Do[Print[{gn={"Pell",n},CoolingNumber[GraphData@gn]//Timing}],{n,9}]

Prism

A288156

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

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

gf=(x^3)FindGeneratingFunction[DeleteMissing@seq,x]

Table[Floor[(2n+9)/5],{n,3,25}]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n∈Integers&&n>2]

Do[Print[{gn={"Prism",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,25}]

Queen n×n

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

Do[Print[{gn={"Queen",{n,n}},CoolingNumber[GraphData@gn]//Timing}],{n,20}]

Rook n×n

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

RookComplement n×n

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

SierpinskiCarpet

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

SierpinskiGasket

seq=Table[GraphData[{"SierpinskiGasket",n},"CoolingNumber"],{n,5}]

SierpinskiTetrahedron

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

Spoke n×n

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

Do[Print[{gn={"Spoke",{n,n}},CoolingNumber[GraphData@gn]//Timing}],{n,5}]

StackedBook

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

StackedPrism

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

Star

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

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

Do[Print[{gn={"Star",n},CoolingNumber[GraphData[gn]]//Timing}],{n,25}]

Sun

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

Do[Print[{gn={"Sun",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,20}]

Sunlet

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

Do[Print[{gn={"Sunlet",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,20}]

TorusGrid n×n

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

Do[Print[{gn={"TorusGrid",{n,n}},CoolingNumber[GraphData@gn]//Timing}],{n,3,20}]

Transposition

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

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

Do[Print[{gn={"Transposition",n},CoolingNumber[GraphData@gn]//Timing}],{n,6}]

Trapezohedron

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

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

Do[Print[{gn={"Trapezohedron",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,20}]

Triangular

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

TriangularGrid

seq=Table[GraphData[{"TriangularGrid",n},"CoolingNumber"],{n,0,10}]

Do[Print[{gn={"TriangularGrid",n},CoolingNumber[GraphData@gn]//Timing}],{n,0,10}]

TriangularHoneycombAcuteKnight

Inferred for n > 10:

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

Table[Piecewise[{{1,n==1},{3,n==2},{4,n==3}},2n-3],{n,20}]

TriangularHoneycombBishop

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

Do[Print[{gn=GraphData[{"TriangularHoneycombBishop",n},"#"],CoolingNumber[GraphData@gn]//Timing}],{n,20}]

TriangularHoneycombObtuseKnight

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

Length@DeleteMissing@seq

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

Do[Print[{gn=GraphData[{"TriangularHoneycombObtuseKnight",n},"#"],CoolingNumber[GraphData@gn]//Timing,MaxMemoryUsed[]/10^9.}],{n,20}]

TriangularHoneycombQueen

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

Do[Print[{gn=GraphData[{"TriangularHoneycombQueen",n},"#"],CoolingNumber[GraphData@gn]//Timing}],{n,20}]

TriangularHoneycombRook

Inferred for n > 10:

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

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

TriangularSnake

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

Web

A004526

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

Table[3+Floor[n/2],{n,3,20}]==seq

Do[Print[{gn={"Web",n},CoolingNumber[GraphData@gn]//Timing}],{n,3,20}]

Wheel

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

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

Do[Print[{gn={"Wheel",n},CoolingNumber[GraphData[gn]]//Timing}],{n,4,20}]

WheelComplement

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

WhiteBishop n×n

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

Do[Print[{gn={"WhiteBishop",{n,n}},CoolingNumber[GraphData@gn]//Timing}],{n,2,20}]

Zebra n×n

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

Length@DeleteMissing@seq

Do[Print[{gn=GraphData[{"Zebra",{n,n}},"#"],CoolingNumber[GraphData@gn]//Timing,MaxMemoryUsed[]/10^9.}],{n,20}]

Algorithm

{"Path", 4}

GraphData[{"Path",4},{"Diameter","BurningNumber","CoolingNumber"}]

BurningNumber[PathGraph[Range@4],Debug->All,MaxSteps->5]

CoolingNumber[PathGraph[Range@4],Debug->All]

{"CompleteBipartite", {2, 3}}

GraphData[{"CompleteBipartite",{2,3}},{"Diameter","BurningNumber","CoolingNumber"}]

BurningNumber[GraphData[{"CompleteBipartite",{2,3}}],Debug->All]

SymmetricallyDistinctVertices@GraphData[{"CompleteBipartite",{2,3}}]

CoolingNumber[GraphData[{"CompleteBipartite",{2,3}}],"UnitSymmetry"->True,Debug->True]

Checks

DeleteCases[Monitor[{gn=#,GraphData[#,"CoolingNumber"],CoolingNumber[GraphData[#],"UseSymmetry"->False]}&/@GraphData[;;10],gn],{_,x_,x_}]//Timing

DeleteCases[Monitor[{gn=#,GraphData[#,"CoolingNumber"],CoolingNumber[GraphData[#],"UseSymmetry"->False]}&/@GraphData[;;12],gn],{_,x_,x_}]//Timing

DeleteCases[Monitor[{gn=#,GraphData[#,"CoolingNumber"],CoolingNumber[GraphData[#],"UseSymmetry"->True]}&/@GraphData[;;10],gn],{_,x_,x_}]//Timing

DeleteCases[Monitor[{gn=#,GraphData[#,"CoolingNumber"],CoolingNumber[GraphData[#],"UseSymmetry"->True]}&/@GraphData[;;12],gn],{_,x_,x_}]//Timing