Graph Tetrahedron

Author

Eric W. Weisstein
December 22, 2025

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

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

©2025 Wolfram Research, Inc. except for portions noted otherwise

GraphDataCountTable

table=GraphDataCountTable["TetrahedronCount",Join[
{"Apollonian"->003462,"Barbell"->034827,"Bishop"->067056,"BlackBishop"->391766,"CocktailParty"->130810,"Complete"->000332,"CycleComplement"->005582,"HalvedCube"->391778,"Keller"->390018,"King"->000290,"PlummerToft"->001477,"Queen"->388998,"Rook"->391978,"RookComplement"->179059,"SierpinskiTetrahedron"->000302,"Sun"->000332,"Triangular"->174002,"TriangularHoneycombBishop"->277935,"TriangularHoneycombQueen"->253942,"TriangularHoneycombRook"->000389,"Wheel"->000007,"WheelComplement"->005582},
Thread[{"Antiprism","CompleteBipartite","CompleteTripartite","CubeConnecetdCycle","Cycle","Empty",{"Grid",2},{"Grid",3},"Hypercube","Knight","Ladder","MoebiusLadder","Mycielski","Odd","Path","Prism","Star","Trapezohedron"}->000004]]]

DateString[]

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

Parametrized families

<<MathWorld`Graphs`

Alkane

A000004

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

AlternatingGroupGraph

A000004

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

Andrasfai

A000004

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

Antelope n×n

A000004

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

Antiprism

A000004

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

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

Apollonian

A003462

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

FindSequenceFunction[seq,n]

Barbell

A034827

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

FindSequenceFunction[seq,n-2]//Factor

Table[2Binomial[n,4],{n,3,20}]

Bishop n×n

A067056

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

FindSequenceFunction[seq,n]//Factor

BlackBishop n×n

A391766

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

FindSequenceFunction[seq,n]

FindLinearRecurrence[Drop[seq,4]]//Head//Timing

(gf=x FindGeneratingFunction[seq,x,ValidationLength->2])//Timing

OEISGeneratingFunctionString[gf,x]

seriesc=Simplify[SeriesCoefficient[(((x^4) (1+8 x+4 (x^2)+14 (x^3)+3 (x^4)+2 (x^5)))/((((1-x))^6) (((1+x))^4))),{x,0,n}],n>0]//Factor

StringReplace[ToString[seriesc,InputForm]," "->""]

CoefficientList[Series[(((x^3) (1+8 x+4 (x^2)+14 (x^3)+3 (x^4)+2 (x^5)))/((((1-x))^6) (((1+x))^4))),{x,0,20}],x]

seq=Table[(1/240) (3-4 n+(n^2)) (-5+5 (((-1))^n)+22 n-10 (((-1))^n) n-14 (n^2)+4 (n^3)),{n,40}]

seq=Table[((n^2)-4n+3) (4 (n^3)-14 (n^2)+22 n-5+5(-1)^n(1-2n))/240,{n,40}]

StringLength[ToString@seq]

coef=FindLinearRecurrence[seq]

Take[seq,Length@coef]

LinearRecurrence[{2,3,-8,-2,12,-2,-8,3,2,-1},{0,0,0,1,10,27,90,182,412,714},20]

OEISRecurrenceIndexString[coef]

OEISRecurrenceString[coef]

Book

A000004

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

Bruhat

A000004

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

Butterfly

Camel n×n

A000004

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

Cameron

A000004

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

Centipede

A000004

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

CocktailParty

A130810

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

FindSequenceFunction[seq,n]

Table[16Binomial[n,4],{n,20}]

Complete

A000332

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

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

FindSequenceFunction[seq,n]

Table[Binomial[n,4],{n,10}]

Binomial[Range[10],4]

CompleteBipartite n×n

A000004

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

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

CompleteBipartite m×n

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

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"CompleteBipartite",{m,n}},"TetrahedronCount"],
0
},{n,10},{m,n}],1],{_,x_,x_}]

CompleteTripartite n×n×n

A000004

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

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

CompleteTripartite k×m×n

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

DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"TetrahedronCount"],
0
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CossidentePenttila

Lookup[GraphData["CossidentePenttila","NotationRules"],"CossidentePenttila"]//Sort

seq=Table[GraphData[{"CossidentePenttila",n},"TetrahedronCount"],{n,3,7,2}]

CrossedPrism

A000004

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

Crown

A000004

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

CubeConnectedCycle

A000004

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

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

Cycle

A000004

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

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

CycleComplement

A005582

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

seq=Table[n(n-5)(n-6)(n-7)/24,{n,3,20}]

seq=Table[n*(n+1)*(n+2)*(n+7)/24,{n,0,30}]

FindLinearRecurrence[seq]

LinearRecurrence[{5,-10,10,-5,1},{2,9,25,55,105},{0,20}]

gf=FindGeneratingFunction[seq,x]

CoefficientList[Series[(((x-2) x)/(((x-1))^5)),{x,0,20}],x]

Table[Binomial[n+3,n-1]+Binomial[n+2,n-1],{n,0,20}]

Table[Binomial[n+2,4]+2Binomial[n+2,3],{n,0,20}]

DiagonalIntersection

A000004

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

Dipyramid

A000004

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

DorogovtsevGoltsevMendes

A000004

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

DoubleCone

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

Empty

A000004

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

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

FibonacciCube

A000004

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

Fiveleaper n×n

A000004

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

Flower

A000004

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

FoldedCube

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

Gear

A000004

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

GeneralizedPetersen

GraphData["GeneralizedPetersen","TetrahedronCount"]

Giraffe n×n

A000004

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

Goldberg

A000004

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

Grid n×n

A000004

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

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

Grid m×n

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

Sort@DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"Grid",{m,n}},"TetrahedronCount"],
0
},{n,10},{m,n}],1],{_,x_,x_}]

Grid n×n×n

A000004

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

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

Grid k×m×n

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

Sort@DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"Grid",{k,m,n}},"TetrahedronCount"],
0
},{n,5},{m,n},{k,m}],2],{_,x_,x_}]

HalvedCube

A391778

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

FindSequenceFunction[seq,n]//FullSimplify

seq=Table[n(n-1)(n-2)^2 2^(n-4)/3,{n,28}]

StringLength[ToString@seq]

coef=FindLinearRecurrence[seq]

OEISRecurrenceIndexString[coef]

OEISRecurrenceString[coef]

LinearRecurrence[{10,-40,80,-80,32},{0,0,1,16,120,640},20]

gf=x FindGeneratingFunction[seq,x]

OEISGeneratingFunctionString[gf,x]

CoefficientList[Series[(((x^2)(1+6 x))/(((1-2 x))^5)),{x,0,20}],x]

Hanoi

A000004

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

Harary

GraphData["Harary","TetrahedronCount"]

Helm

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

Hypercube

A000004

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

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

KC

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"KC",{m,n}},"TetrahedronCount"],
XXX
},{n,3,10},{m,10}],1],{_,x_,x_}]

Keller

A390018

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

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

FindSequenceFunction[seq,n]

Do[Print[{gn={"Keller",n},With[{sizes=IGCliqueSizeCounts[GraphData[gn],{4}]},If[sizes==={},0,sizes[[-1]]]]//Timing}],{n,5}]

King n×n

A000290

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

FindSequenceFunction[seq,n]

King m×n

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"King",{m,n}},"TetrahedronCount"],
(m-1)(n-1)
},{n,10},{m,n}],1],{_,x_,x_}]

Knight n×n

A000004

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

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

Knight m×n

A000004

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

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"Knight",{m,n}},"TetrahedronCount"],
0
},{n,10},{m,n}],1],{_,x_,x_}]

KP

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

Ladder

A000004

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

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

LadderRung

A000004

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

LindgrenSousselier

A000004

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

LucasCube

A000004

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

MengerSponge

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

MoebiusLadder

A000004

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

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

Mycielski

A000004

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

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

Odd

A000004

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

Paley

GraphData["Paley","TetrahedronCount"]

Pan

A000004

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

Path

A000290

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

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

PathComplement

A000332

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

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

Table[Binomial[n-3,4],{n,20}]

Peisert

GraphData["Peisert","TetrahedronCount"]

Pell

A000004

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

PlummerToft

A001477

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

FindSequenceFunction[seq,n]

Prism

A000004

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

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

Queen n×n

A388998

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

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

OEISGeneratingFunctionString[gf,x]

CoefficientList[Series[((x (1+14 x+8 (x^3)+19 (x^4)+14 (x^5)))/((((-1+x))^6) (((1+x))^2))),{x,0,20}],x]

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

Factor[(7 (n^4)-45(n^3)+215(n^2)-375n+183)]

seq=Table[n (7 (n^4)-45 (n^3)+215 (n^2)-375 n+183-15 (((-1))^n))/60,{n,36}]

StringLength[ToString@seq]

StringReplace[ToString[n (7 (n^4)-45 (n^3)+215 (n^2)-375 n+183-15 (((-1))^n))/60,InputForm]," "->""]

coef=FindLinearRecurrence[seq]

Take[seq,Length@coef]

LinearRecurrence[{4,-4,-4,10,-4,-4,4,-1},{0,1,18,68,204,501,1106,2208},20]

OEISRecurrenceIndexString[coef]

OEISRecurrenceString[coef]

Rook n×n

A391978

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

FindSequenceFunction[seq,n]

seq=Table[2n Binomial[n,4],{n,37}]

StringLength[ToString@seq]

gf=x FindGeneratingFunction[seq,x]

OEISGeneratingFunctionString[gf,x]

CoefficientList[Series[((2 (x^3) (4+x))/(((-1+x))^6)),{x,0,20}],x]

coef=FindLinearRecurrence[seq]

LinearRecurrence[{6,-15,20,-15,6,-1},{0,0,0,8,50,180},20]

OEISRecurrenceIndexString[coef]

OEISRecurrenceString[coef]

RookComplement n×n

A179059

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

FindSequenceFunction[seq,n]

Table[4!Binomial[n,4]^2,{n,20}]

seq=4!Binomial[Range[20],4]^2

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[-((24 (x^3)(1+16 x+36 (x^2)+16 (x^3)+(x^4)))/(((-1+x))^9)),{x,0,20}],x]

SierpinskiCarpet

A000004

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

SierpinskiGasket

A000004

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

SierpinskiTetrahedron

A000302

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

FindSequenceFunction[seq,n]

Spoke n×n

A000004

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

StackedBook m×n

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"StackedBook",{m,n}},"TetrahedronCount"],
0
},{n,10},{m,n}],1],{_,x_,x_}]

StackedBook n×n

A000004

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

StackedPrism m×n

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"StackedPrism",{m,n}},"TetrahedronCount"],
0
},{m,3,8},{n,5}],1],{_,x_,x_}]

StackedPrism n×n

A000004

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

Star

A000004

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

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

Sun

A000332

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

FindSequenceFunction[seq,n]

Table[Binomial[n,4],{n,3,20}]

Binomial[Range[20],4]

Sunlet

A000004

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

Tetrahedral

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

FindSequenceFunction[seq,n-5]//Head//Timing

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

TorusGrid n×n

A000004

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

Transposition

A000004

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

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

Trapezohedron

A000004

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

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

Triangular

A174002

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

FindSequenceFunction[seq,n-1]

seq=Table[(n-4) Binomial[n,4],{n,2,40}]

TriangularGrid

A000004

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

TriangularHoneycombAcuteKnight

A000004

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

TriangularHoneycombBishop

A277935

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

FindSequenceFunction[seq,n]//Factor

Table[2Binomial[n+1,5],{n,20}]

TriangularHoneycombObtuseKnight

A000004

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

TriangularHoneycombQueen

A253942

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

FindSequenceFunction[seq,n]//Factor

Table[3Binomial[n+1,5],{n,20}]

TriangularHoneycombRook

A000389

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

FindSequenceFunction[seq,n]

Table[Binomial[n+1,5],{n,20}]

TriangularSnake

A000004

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

Turan

GraphData["Turan","TetrahedronCount"]

Web

A000004

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

Wheel

A000007

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

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

WheelComplement

A005582

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

FindSequenceFunction[seq,n-3]

gf=(x^4)FindGeneratingFunction[seq,x]

Simplify[SeriesCoefficient[gf,{x,0,n}],n>5]

n*(n+1)*(n+2)*(n+7)/24/.n->n-8

Table[(1/24) (-8+n) (-7+n) (-6+n) (-1+n),{n,4,20}]

WhiteBishop n×n

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

(gf=(x^2)FindGeneratingFunction[seq,x])//Timing

Simplify[SeriesCoefficient[(((x^4) (1+2 x+16 (x^2)+6 (x^3)+7 (x^4)))/((((1-x))^6) (((1+x))^4))),{x,0,n}],n>0]

WhiteCamel n×n

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

Zebra n×n

A000004

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

Algorithms

Not the cliques we're looking for:

FindKClique[CompleteGraph[6],4,Infinity,All]

GraphData["K6","CliquePolynomial"][x]//Expand

<<IGraphM`

IGCliqueSizeCounts[CompleteGraph[6]]

IGCliqueSizeCounts[CompleteGraph[6],{4}]

IGCliques[CompleteGraph[6],{4}]

Length[%]

{
#,
With[
{
cnts=IGCliqueSizeCounts[IGAdjacencyGraph[GraphData[#,"AdjacencyMatrix"]],{4}]
},
If[cnts==={},0,cnts[[-1]]]]
}&/@GraphData[;;6]//Timing//AbsoluteTiming

ChapGPT