Delsarte Clique

Author

Eric W. Weisstein
December 1, 2023

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

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Source

https://arxiv.org/abs/2311.09001

GraphDataCountTable

GraphDataCountTable["DelsarteCliqueCount"]

Parametrized families

<<MathWorld`Graphs`

Alkane

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

FindSequenceFunction[seq,n]

AlternatingGroupGraph

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

Andrasfai

seq=Table[GraphData[{"Andrasfai",n},"DelsarteCliqueCount"],{n,10}]

Antelope n×n

seq=Table[GraphData[{"Antelope",{n,n}},"DelsarteCliqueCount"],{n,15}]

Antiprism

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

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

Apollonian

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

Barbell

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

FindSequenceFunction[seq,n]

BlackBishop n×n

seq=Table[GraphData[{"BlackBishop",{n,n}},"DelsarteCliqueCount"],{n,10}]

FullSimplify[FindSequenceFunction[Rest@seq,n-1],n∈Integers]

Bishop n×n

seq=Table[GraphData[{"Bishop",{n,n}},"DelsarteCliqueCount"],{n,10}]

Book

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

FindSequenceFunction[seq,n]

Bruhat

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

FindSequenceFunction[seq,n]

Camel n×n

seq=Table[GraphData[{"Camel",{n,n}},"DelsarteCliqueCount"],{n,10}]

Centipede

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

FindSequenceFunction[seq,n]

CocktailParty

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

DelsarteCliques[GraphData[{"CocktailParty",4}]]

FindSequenceFunction[DeleteMissing@Drop[seq,4],n-4]

Complete

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

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

GraphData["K1","Geometric"]

DelsarteCliques[CompleteGraph[1],Debug->True]

CompleteBipartite m×n

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

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"CompleteBipartite",{m,n}},"DelsarteCliqueCount"],
Piecewise[{{n^2,m==n}},0]
},{n,10},{m,n}],1],{_,x_,x_}]

CompleteBipartite n×n

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

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

FindSequenceFunction[seq,n]//FullSimplify

CompleteTripartite k×m×n

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

DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"DelsarteCliqueCount"],
Piecewise[{{n^3,k==m==n}},0]
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CompleteTripartite n×n×n

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

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

FindSequenceFunction[seq,n]//FullSimplify

CrossedPrism

seq=Table[GraphData[{"CrossedPrism",2n},"DelsarteCliqueCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]//Factor

Crown

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

CubeConnectedCycle

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

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

Cycle

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

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

Table[Piecewise[{{1,n==3},{n,Mod[n,2]==0}},0],{n,3,30}]

CycleComplement

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

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

DiagonalIntersection

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

Dipyramid

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

DoubleCone

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

Empty

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

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

GraphData["SingletonGraph","Geometric"]

FibonacciCube

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

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

Flower

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

FoldedCube

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

FindSequenceFunction[seq,n]

Gear

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

FindSequenceFunction[seq,n]

Giraffe n×n

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

Goldberg

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

Grid m×n

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

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"Grid",{m,n}},"DelsarteCliqueCount"],
Piecewise[{{4,m==n==2},{1,m n==2}},0]
},{n,10},{m,n}],1],{_,x_,x_}]

Grid n×n

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

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

Grid k×m×n

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

DeleteCases[Sort@Flatten[Table[{
{k,m,n},
GraphData[{"Grid",{k,m,n}},"DelsarteCliqueCount"],
Piecewise[{{12,k==m==n==2},{1,k m n==2},{4,Min[k,m,n]Max[k,m,n]==2}},0]
},{n,5},{m,n},{k,m}],2],{_,x_,x_}]

Tally[GraphData[{"Grid",{2,3,4}},"DistanceMatrix"]//Normal//Flatten]

Grid n×n×n

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

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

FindSequenceFunction[seq,n]//FullSimplify

HalvedCube

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

Hanoi

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

Helm

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

FindSequenceFunction[seq,n]

Hypercube

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

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

FindSequenceFunction[seq,n+1]

Keller

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

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

FindSequenceFunction[seq,n]

King n×n

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

FindSequenceFunction[seq,n]//Factor

Knight m×n

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

Knight n×n

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

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

Ladder

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

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

FindSequenceFunction[seq,n]//Factor

LadderRung

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

LucasCube

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

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

MengerSponge

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

MoebiusLadder

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

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

Mycielski

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

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

Odd

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

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

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

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

Pan

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

Path

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

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

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

PathComplement

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

Pell

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

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

Prism

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

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

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

Queen n×n

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

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

Rook n×n

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

FullSimplify[FindSequenceFunction[DeleteMissing@Rest@seq,n-1],n∈Integers]//Factor

RookComplement n×n

seq=Table[GraphData[{"RookComplement",{n,n}},"DelsarteCliqueCount"],{n,10}]

FindSequenceFunction[DeleteMissing@Drop[seq,2],n-2]//FunctionExpand

n!//FunctionExpand

SierpinskiCarpet

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

FindSequenceFunction[seq,n]

SierpinskiGasket

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

FindSequenceFunction[seq,n]

SierpinskiTetrahedron

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

FindSequenceFunction[seq,n]

Spoke n×n

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

FindSequenceFunction[seq,n]

StackedBook

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

StackedPrism

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

Star

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

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

Sun

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

FindSequenceFunction[seq,n-2]//Factor

Sunlet

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

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

TorusGrid n×n

seq=Table[GraphData[{"TorusGrid",{n,n}},"DelsarteCliqueCount"],{n,3,10}]

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

Transposition

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

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

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

Trapezohedron

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

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

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

Triangular

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

FindSequenceFunction[DeleteMissing@Drop[seq,3],n-4]

TriangularGrid

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

FindSequenceFunction[seq,n+1]//FullSimplify

TriangularHoneycombAcuteKnight

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

TriangularHoneycombBishop

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"DelsarteCliqueCount"],{n,9}]

FindSequenceFunction[seq,n]//Factor

TriangularHoneycombObtuseKnight

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

TriangularHoneycombQueen

seq=Table[GraphData[{"TriangularHoneycombQueen",n},"DelsarteCliqueCount"],{n,10}]

FindSequenceFunction[seq,n]

TriangularHoneycombRook

seq=Table[GraphData[{"TriangularHoneycombRook",n},"DelsarteCliqueCount"],{n,10}]

TriangularSnake

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

Web

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

Wheel

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

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

FindSequenceFunction[seq,n-3]//FullSimplify

WheelComplement

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

FindSequenceFunction[seq,n-3]//FullSimplify

WhiteBishop n×n

seq=Table[GraphData[{"WhiteBishop",{n,n}},"DelsarteCliqueCount"],{n,2,10}]

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

Factor[%]

Zebra n×n

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

Checks

<<MathWorld`Graphs`

DeleteCases[
Monitor[{gn=#,GraphData[#,"DelsarteCliqueCount"],Length@DelsarteCliques[GraphData[#]]}&/@GraphData["Geometric"],gn],
{_,x_,x_}]//Timing